Exploring Bohmian Mechanics Using Maple V { David Bohm's Interpretation of Qunatum Mechanics William S. Page Daneliuk & Page 8 Aug 1995 Abstract Bohmian mechanics (Berndl, et. al.[1]), also known as the pilot wave model (de Broglie[4], J.S. Bell[5]), the causal interpretation (Bohm[3]) and the ontological interpretation (Bohm and Hiley[2]), relates the motions (dx=dt) of classical particles to the quantum mechanical wave function ( ) through the non-linear guidance equation: dx = h Im
dt
m
A modi ed form of the classical Hamilton-Jacobi equation which includes a non-classical quantum potential term can be derived from Schrodinger's equation. This quantum HamiltonJacobi equation applies directly to the detailed motions of particles which in Bohmian mechanics have well-de ned deterministic trajectories. Further, the role of probablity in Bohmian mechanics is reduced to the statistical accuracy of measurements. And the process of measurement, itself is completely de ned within this formalism. The ablity of Maple to symbolically and numerically solve the resulting non-linear equations of motion together with Maple's extensive animation and 3-dimensional graphics features allows one to explore the complex trajectories of particles in simple quantum mechanical systems such as the "particle in a box", Gaussian wave packets, Airy wavefunctions and harmonic oscillator problems as well as some many-body problems. Some of these problems approach the limits of what is possible in the current generation of mathematical software. Bohmian mechanics together with tools such as Maple provide a concrete realization of quantum mechanical systems which avoids the dicult philosophical problems of the relationship between conventional quantum mechanics and reality. For example, in Bohm's interpretation of the ground state of the hydrogen atom the electron is at rest relative to the proton in contrast with the usual interpretation of the QM in which the ground state as some sort of "minimum" allowed kinetic energy. Bohm's interpretation is that the quantum potential implies an additional force that exactly balances the classical Coulomb potential in the stationary states corresponding to electron orbitals. Thus, the electron does not raditate energy and decay into the nucleus by virtue of quantum number constraints but rather by the fact that there is an equilibrium of forces at these locations. Release 3 Patchlevel 3
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Contents 1 Introduction 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
Maple preliminarys : : : : : : : : : : : : : : : : : : : : Schrodinger's Equation : : : : : : : : : : : : : : : : : : Polar Form of the Schrodinger Equation : : : : : : : : Probability conservation : : : : : : : : : : : : : : : : : The Quantum Hamilton-Jacobi equation : : : : : : : : The Equation of Motion and the Guidance Condition The Ground State of Hydrogen : : : : : : : : : : : : : Waves, Particles and Conservation Laws : : : : : : : :
2 Particle in a box 2.1 2.2 2.3 2.4 2.5 2.6 2.7
Maple Quantum Methods : : : Schrodinger's Equation : : : : : Non-stationary states : : : : : : The motion of the wave packet The Quantum Potential : : : : The Motion of the Particle : : The Trajectory of the Particle :
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3 The Gaussian wave packet and particle system 3.1 3.2 3.3 3.4 3.5 3.6
Evolution of the Gaussian wave packet : : : : : : : : : : : : : : : : : : : : : : : : Derivation of the wave packet as a superposition of stationary states : : : : : : : Deriving the Bohm Velocity Function for particles in the Gaussian Wave Packet : The Motion of a particle within the wave packet : : : : : : : : : : : : : : : : : : The Quantum potential : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Discussion of the wave packet : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4 The Airy Wavefunction 4.1 4.2 4.3 4.4
What Maple knows about the Airy functions and Bessel functions Schrodinger's equation in free space : : : : : : : : : : : : : : : : : Bohm's Interpretation of the Airywavefunction : : : : : : : : : : : Comments on the Airy Wavefunction : : : : : : : : : : : : : : : : :
5 More than one particle
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5.1 Two particle, one dimensional Schrodiner equation : : : : : : : : : : : : : : : : : : : 47 2
5.2 5.3 5.4 5.5
Quantum Hamilton-Jacobi Equation and the Quantum Potential Probability Conservation and Probability Currents : : : : : : : : The Two Particle Equation of Motion and Guidance Conditions : Many-body Equations : : : : : : : : : : : : : : : : : : : : : : : :
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List of Figures 1 2 3 4 5 6 7 8 9 10 11 12
A few eigen-functions for a particle in a box Prob. Dens. and Quantum Pot. : : : : : : : Quantum rattle velocity eld : : : : : : : : Trajectory near a pole : : : : : : : : : : : : Range of trajectories : : : : : : : : : : : : : Motion of the Gaussian Wave Packet : : : : Trajectories of an ensemble of particles : : : Quantum Potential of Gaussian Packet : : : The Airy Wave Function : : : : : : : : : : : Airy function and its derivative : : : : : : : Evolution of the Airy Wavefunction : : : : Trajectory quided by an Airy wavefunction
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1 Introduction The main goal of this paper is to demonstrate the use of Maple for exploring quantum mechanics by exploiting Maple's ability to do a wide range of symbolic and numeric calculations as well as its ability to present results in two- and three-dimensional graphics and animations. In particular, we emphasize the de Broglie/Bohm causal interpretation of quantum mechanics. The more well known Cophenhagen interpretation denies the possibility of a detailed (non-probablistic) discussion of the motion of particles. But the causual interpretation allows us to discuss both particles and wavefunctions. It is not our intention here to review the long history of this approach and its rivals. For this we refer the reader to the excellent discussions listed in the bibliography. We hope that even for readers who do not have a speci c interest in the physics of this paper, that this paper will demonstrate some techniques in Maple that will be useful in other applications. In developing the Maple worksheets for this project, we have found that it necessary to extend Maple's capabilities in several ways and sometimes to work around problems in the current release. We have also addressed the need to produce good quality printed results directly from the output of the Maple worksheet (via LaTEX). In the sections which follow, we investigate the application of Bohm's interpretation to simple single and multiple-body bound state problems like the harmonic oscillator and the "particle in a box". I have carried out these calculations largely based on the approach presented in the book "Quantum Methods" by James M. Feagin[6]. Consideration of these simple types of problems has proven to be a good way to gain some new intuitions about quantum mechanical systems from Bohm's perspective. And also a way of exploiting Maple to generate a lot of neat graphics and animations!
1.1 Maple preliminarys This paper was prepared using Maple V (Release 3, patchlevel 3). Each section roughly corresponds to a Maple worksheet. After executing each worksheet, the results where saved as a LaTEX le. These les were pre-processed by a simple perl script to provide equation numbering and to generate equation labels and then concatonated into a single LaTEX le for typesetting. The graphics generated by Maple in postscript form have been included as gures in the text, but obviously it is not possible to include the animations as such. For this reason the reader is encouraged to read this document together with the step by step execution of the Maple worksheets on a suitable computer system. We also hope that this "open" use of Maple will encourage readers to experiment with the calculations themselves in order to obtain a deeper appreciation of the subject. Maple is a large, complex and very exible tool for the manipulation and solution of mathematical equations. Similarily, LaTEX is also a complex and exible tool for the typesetting of mathematical equations. The marrage of these two tools necessarily involves some compremises. There are may di erent ways in which to use this combination. It is our belief, however, that the only way to learn to use Maple and LaTEX with style is to apply them as directly as possible to actual problems and to attempt, where possible, to emulate the style of others. In reading the following, please note that in Maple commands start with the prompt character >. Commands are what you type in order to get Maple to do its thing. In a command anything following # is a comment. In the worksheet, Maples results always follow a command of some kind and are indented or centered on the page. The symbol := is used when a name is being given to an expression or an equation. For printing, the name is converted to a LaTEX equation label which is used when referencing equations in the text. For the calculations that follow, please refer to Maple worksheet bohm1.ms. >
restart;
# Clear the work space
5
> > >
read `notilde.txt`;
# Avoids use of Maple's ~ character
`&@`:=(expr,eqn)->subs(eqn,expr):
# another notation for substitution
constants:=op({constants} minus {E}):# allow use of E as a variable
First we need to modify Maple's built-in de nitions for complex symbolic mathematics. > > > > > > > > > > > >
alias(I=I,i=sqrt(-1)):
# We like to use i for imaginary
`print/conjugate` := complex->complex^`*`: # How to print conjugate `conjugate/diff` := proc(x,v) diff(conjugate(x),v); end: `print/re`:=proc(x) `Re `(x); end:
# How to print re()
re:=proc(x) if type(x,'equation') then
# Works for equations
expand((lhs(x)+conjugate(lhs(x)))/2)= expand((rhs(x)+conjugate(rhs(x)))/2) else expand((x+conjugate(x))/2);
# Depends only on conjugate
fi; end: 're(x)'=re(x);
Re ( x ) = 21 x + 12 x > > > > > > > > >
`print/im`:=proc(x) `Im `(x); end:
# How to print im()
im:=proc(x) if type(x,'equation') then
# Works for equations
expand((lhs(x)-conjugate(lhs(x)))/2/i)= expand((rhs(x)-conjugate(rhs(x)))/2/i) else expand((x-conjugate(x))/2/i);
# Depends only on conjugate
fi; end: 'im(x)'=im(x);
Im ( x ) = 21 i x + 12 i x And also de ne a convenient form of the gradient operator. > > > > > > > > >
Delta:=proc(expr) global x;
#Usage: Delta[vars]
if not type(procname,'indexed') then diff(expr,x);
#assumes x if not specified
elif nops(procname)=1 then linalg[grad](expr,[op(procname)])[1]; else linalg[grad](expr,[op(procname)]); fi; end:
This is a more sophisticated way of printing partial derivatives using the gradient operator. > > >
(1)
`print/diff`:=proc(expr,v)
# When to use Delta
global x; if {args[2..nargs]}={x} then
6
(2)
> > > > > > > > > > > > >
if nargs=2 then if op(0,expr)='Delta' then 'Delta@@2'(op(1,expr)); elif op(0,op(0,expr))='`@@`' and op(1,op(0,expr))='Delta' then `print/diff`(op(expr),x$op(2,op(0,expr))+1); else 'Delta'(expr); fi; else 'Delta@@(nargs-1)'(expr); fi; else 'diff'(args); #'Delta[v](expr)'; fi; end:
Planck's constant h is declared as follows > > >
hbar:=`\\hbar`:
# Planck's constant /2/Pi
assume(hbar,real); constants:=op({constants} union {hbar}):
The Hamiltonian operator for a given potential V can be de ned as > >
H:=V -> unapply('-hbar^2/(2*m) * '(Delta@@2)'(psi)+V*psi',psi): 'H(V)(_(x))'=H(V(x))(_(x));
2 (2) H( V )( ( x ) ) = 12 h m( ( x ) ) + V( x ) ( x ) (3) The syntax of this de nition may seem a little obscure. 'unapply' is a Maple operator that takes an expression (in this case the Hamiltonian) and a variable name ( ) as arguements and returns an operator that substitutes for that variable where it occurs in the the expression. Here we want to distinguish between V as a parameter of the operator H, versus which is what H operates on.
1.2 Schrodinger's Equation The time-dependent Schrodinger wave equation for one body in one dimension can be written >
SCHROD:=i*hbar*diff(psi(x,t),t)-H(V(x))(psi(x,t))=0;
@
1 h ( ( x; t ) ) i h @t ( x; t ) + 2 V( x ) ( x; t ) = 0 m 2
(2)
(4)
where the potential V (x) and mass m are real valued > >
`conjugate/V` := proc(x) V(x); end: # tell conjugate about V assume(m>0);
# Mass
1.3 Polar Form of the Schrodinger Equation The following discussion is based on Bohm and Hiley[2], "The Undivided Universe", Chapter 3. "Causal interpretation of the one body system". Now we get to some actual calculations. Bohm and Hiley motivate the causal interpretation by demonstrating that the Schrodinger equation can be manipulated into a form which admits an interpretation as a classical Hamilton-Jacobi equation for the dynamics of a particle subject to 7
the usual classical potential eld plus a new quantum potential eld. We begin by de ning the wavefunction in polar form and separating the real and complex parts of the wave equation. We shall nd that the imaginary component results in the well recognized conservation of probability equation while the real component will comprise the new quantum Hamilton-Jacobi equation. >
POLAR:=psi(x,t)=R(x,t)*exp(i*S(x,t)/hbar);
i S( x;t )
( x; t ) = R( x; t ) e h (5) where R(x,t) and S(x,t) are real-valued functions. S(x,t) is called the "phase" of the wavefunction. > >
`conjugate/R` := proc(x,t) R(x,t); end:
# Tell conjugate that
`conjugate/S` := proc(x,t) S(x,t); end:
# these are real functions
In polar form, Schrodinger's equation (4) becomes >
SCHROD2:=expand(SCHROD&@POLAR);
@
i S( x;t )
@
i S( x;t )
R( x; t ) @t S( x; t ) e h i h @t R( x; t ) e i S( x;t ) 2 (2) h 1 h ( R( x; t ) ) e +2 m i S( x;t ) h i h ( R( x; t ) ) ( S( x; t ) ) e + m i S( x;t ) (2) h + 21 i h R( x; t ) (mS( x; t ) ) e i S( x;t ) i S( x;t ) 1 R( x; t ) ( S( x; t ) )2 e h h = 0 V( x ) R( x; t ) e (6) 2 m Considering just the real part of equation (173) and dividing through by the polar form of (x; t) we get >
h
SCH2RE:=re(-combine(SCHROD2)/psi(x,t)&@POLAR);
1 h ( R( x; t ) ) 1 ( S( x; t ) ) + V( x ) = 0 @t S( x; t ) 2 R( x; t ) m + 2 m
@
2
(2)
2
(7)
Similary, the imaginary part is >
SCH2IM:=im(-SCHROD2/psi(x,t)&@POLAR);
h ( R( x; t ) ) ( S( x; t ) ) 1 h ( S( x; t ) )
h
@ @t R( x; t )
(2)
R( x; t ) =0 This can be manipulated into the form >
R( x; t ) m
2
m
(8)
SCH2IM2:='diff(R(x,t)^2,t)+Delta(R(x,t)^2*Delta(S(x,t))/m)=0';
@
R( x; t ) ( S( x; t ) ) =0 @t R( x; t ) + m 2
2
(9)
as can be shown by rst performing the derivatives in this new form >
SCH2IM2A:=SCH2IM2;
@ R( x; t ) + 2 R( x; t ) ( S( x; t ) ) ( R( x; t ) ) 2 R( x; t ) @t m 2 (2) + R( x; t ) m ( S( x; t ) ) = 0 8
(10)
and on multiplying equation (8) by 2R(x; t)=h and observing that the two equations are now equal: >
is(expand(lhs(SCH2IM2A))=expand(-2/hbar*R(x,t)^2*lhs(SCH2IM)));
true
(11)
1.4 Probability conservation First, lets consider the meaning of the imaginary part of the Schrodinger equation. The probability density (the square of the absolute value of a given wavefunction ) is given by >
rho:=psi -> conjugate(psi)*psi: 'rho(psi)'=rho(psi);
( ) = (12) In the usual interpretation, the probability density for the location of a particle is given by >
'abs(psi(x,t))^2'=simplify(rho(psi(x,t)&@POLAR));
j ( x; t )j2 = R( x; t )2
(13) and the probability current (as in the usual treatment of quantum mechanics) can be de ned as > > >
j:=unapply('hbar/2/i/m*('conjugate'(psi)*'diff'(psi,x)'diff'('conjugate'(psi),x)*psi)',psi): 'j(psi(x,t))'=j(psi(x,t));
j( ( x; t ) ) = 21 i h ( ( x; t ) ( ( x; t )m) ( ( x; t ) ) ( x; t )) Conservation of probability requires that >
CONSPROB:='diff(R(x,t)^2,t)+Delta(j(psi(x,t)))=0';
@
In polar form this becomes, >
(14)
@t R( x; t ) + ( j( ( x; t ) ) ) = 0 2
(15)
eval(CONSPROB,1)&@('j(psi(x,t))'=simplify(j(psi(x,t))&@POLAR));
@
R( x; t ) ( S( x; t ) ) =0 @t R( x; t ) + m 2
2
(16)
So we see that the imaginary part of the polar Schrodinger equation, that is equation (8), is just the well known continuity equation that expresses the conservation of probability. What role does probabilty play in this new viewpoint? Bohm and Hiley indicate that in their theory the function P = j j2 is not taken apriori to be the probability distribution of the particle but rather the ultimate distribution of the particle given an uncertain initial distribution. We can think of this as the steady state probability density of a chaotic process. That is, given a deterministic iteration procedure that has however some chaotic behavior and an exactly known initial condition, the value of the process after n iterations is known exactly. For a perfectly known initial condition we can state the value at iteration n as a unit delta function at the known value. Now, if instead the initial value is known to only within some probabilty distribution, then for n suciently large (and under certain conditions placed on the iteratation process, and for an initial distribution which is suciently wide) the nal result is known to within a probabilty distribution which is invariant with respect to the starting distribution and the iterate number n). More over, this distribution will be exactly that predicted by Born's interpretation of j j2. 9
Bohm and Hiley also note that even in the case of an exactly known particle state, this conservation of probability equation plays a role. It can be thought to describe the necessary behavior of an ensemble of particles prepared so that the statistics match j j2.
1.5 The Quantum Hamilton-Jacobi equation Now, consider the real part of the polar form of the Schrodinger. Note that this component strongly resembles that of the classical Hamilton-Jacobi equation for a particle in a potential eld V(x) where the momentum of the particle is taken to be >
momentum:=p='Delta'(S(x,t));
p = ( S( x; t ) ) (17) Given this representation for the momentum we have the classical Hamilton-Jacobi equation given by >
HJ:='diff(S(x,t),t)+(Delta(S(x,t)))^2/(2*m)+V(x)';
1 ( S( x; t ) ) + V( x ) @t S( x; t ) + 2 m
@
2
(18)
By noticing the similarity between the real parto of the Schrodinger equation in polar form (7) and the classical Hamilton-Jacobi equation (18), we may de ne the modi ed quantum Hamilton-Jacobi equation by: >
QHJ:='diff(S(x,t),t)+(Delta(S(x,t)))^2/(2*m)+V(x)+Q(x,t)=0';
@
1 ( S( x; t ) ) + V( x ) + Q( x; t ) = 0 @t S( x; t ) + 2 m 2
(19)
so that if we de ne the quantum potential as: >
QP:= 'Q(x,t) = -hbar^2/(2*m)*(Delta@@2)(R(x,t))/R(x,t)';
2 (2) ( R( x; t ) ) (20) Q( x; t ) = 12 hbar mR( x; t) then we have that equation (7) is the same as equation (19). Thus, the real component of the Schrodinger equation is equivalent to a classical description of the system but for the presence of an additional potential eld, the quantum potential (20). Note that in the WKB limit where the wave packet width is much greater than the wave length, Q is very small so the Quantum Hamilton Jacobi equation approaches the Hamilton Jacobi equation. Thus we see that the new discription enjoys a smooth and naturally obtained transistion from the quantum to the classical domain.
1.6 The Equation of Motion and the Guidance Condition From the Quantum Hamilton-Jacobi equation, the equation of motion of a particle with velocity v(t) (i.e. Newton's F=ma) is seen to be given by >
newton:='m*diff(v(t),t)=-Delta(V(x))-Delta(Q(x,t))';
@
m @t v( t ) = ( V( x ) ) ( Q( x; t ) )
10
(21)
Also, from the de nition of momentum in terms of the phase of the wavefunction (equation 17) we also have the following relationship, known as the quidance condition: >
guide1:=v(t)=solve(momentum&@(p=m*v(t)),v(t));
v( t ) = ( S(mx; t ) ) and from its relationship to the probability current >
guide2:=v(t)='j(psi(x,t))'/rho(psi(x,t));
v( t ) = ( j(x; t ()x; t()x;) t ) >
(23)
is(guide1&@simplify(guide1&@POLAR));
true Another useful expression for the guidance condition is >
(22)
(24)
guide3:=v(t)='hbar/m*im(Delta(psi(x,t))/psi(x,t))';
v( t ) =
( ( x; t ) ) hbar Im m
( x; t )
(25)
as can be seen by substituting the polar expression for , evaluating the partial derivative and simplying >
is(guide1&@simplify(eval(guide3,1)&@POLAR));
true
(26)
1.7 The Ground State of Hydrogen We will now examine the implications of this interpretation in a stationary state. Recall (from most elementary treatments of QM) that a special situation arises when the potential function V(x) is not a function of time. In this case, the Schrodinger equation can be partially solved by separation of variables. The resulting pair of equations yield the fact that psi can be factored >
psi(x,t) = phi(x)*eta(t);
( x; t ) = ( x ) ( t ) (27) into a time independent function and a time harmonic, spatially constant function of the form >
eta(t)=C*exp((-i/hbar)*E[n]*t);
i En t ( t ) = C e( h ) then the time-independant Schrodinger equation for
>
(28)
H(V(x))(phi[n](x))=E[n]*phi[n](x);
1 h 2 ( 2 ) ( n ( x )) + V( x ) ( x ) = E ( x ) n n n 2 m
11
(29)
is an eigenvalue-eigenfunction equation. n (x) is an eigenfunction of the Hamiltonian operator with real eigen-values En corresponding to the allowed values of the total energy. We call the states that arise from such a situation, stationary states because the probability density function j j2 is xed with respect to time as is the expected energy value for that state. Now, lets examine the case where our solution represents the ground state of a hydrogen atom, which is such a stationary state because of the time independence of the electrostatic potential eld. Once again, if we consult an elementary treatment of QM we can nd the ground state (1s state) equation for the hydrogen atom and nd that it is given by the psi function: > > >
psi1s :=
(1/(Pi^(1/2)))*(1/a0)^(3/2)*exp(-r/a0)*exp((-i/hbar)*E[0]*t) =phi[0](x)*exp((-i/hbar)*E[0]*t);
1=
3 2
a0
r e( a0 ) e( p
i E0 t )
h
= 0 ( x ) e(
i E0 t )
h
(30)
where a0 is the Bohr radius, and r is the radial distance from the nucleus. Thus we see that in the ground state, psi factors into a real, time independent function phi0 and a harmonic time function. Thus in terms of our polar representation of we have > > >
RH0:=R(x,t)=phi[0](x); SH0:=S(x,t)=-E[0]*t; POLAR&@RH0&@SH0;
R( x; t ) = 0( x )
(31)
S( x; t ) = E0 t
(32)
( x; t ) = 0 ( x ) e(
i E0 t )
h
(33)
Note that S is independent of x, so that > >
'Delta(S(x,t))'=Delta(subs(SH0,S(x,t))); eval(QHJ&@RH0&@SH0);
( S( x; t ) ) = 0
(34)
E0 + V( x ) + Q( x; t ) = 0
(35)
E0 = V( x ) + Q( x; t )
(36)
that is, >
E[0]=solve(",E[0]);
12
and the momentum is >
momentum&@SH0;";
p = ( E0 t)
(37)
p=0
(38)
which means the particle is at rest! This is very much at odds with the usual interpretation of the QM ground state. Understanding this is an important key to understanding Bohm's interpretation. Bohm's interpretation is that the quantum potential eld generates a new force in this system that balances that of the classical potential V in this 'orbital' position. Thus, the electron is xed in position not by virtue of quantum number constraints but by the fact that there is indeed an equilibrium of forces at this location. In the next following sections we will compute the equation of motion of some simple systems and extend this interpretation to the many-body Schrodinger equation. In the orksheets which follow, we make use of the de nitions listed below that have been de ned in this worksheet. > >
save `&@`,`print/conjugate`,`print/diff`,re,im,`print/re`,`print/im`,
hbar,SCHROD,POLAR,Delta,H,rho,j,QP,QHJ,guide1,guide2,guide3,`save1.m`;
1.8 Waves, Particles and Conservation Laws Before we go too far with the theory in this abstract form, now is a good time to stop and consider what Bohm's interpretation provides. It is now possible to treat both particle motions and wavefunctions according to one consistent formalism! We no longer have to be concerned with a probabilistic interpretation. When we talk about a "particle" we really do mean a particle located exactly at x(t). Later, when we want to consider whether tragectories exist for two (or more) particles for which the particles are suciently close for a suciently long time for fusion to take place, this ability of talk about the exact location of particles will be very convenient. It means that at the scale of electronic interactions, we can treat the (putative) nuclear interactions as an interaction of particles with much the same intuitions as in high energy physics. But we must be aware that the motions of these particles are now a ected by an additional force due to the quantum potential. We will discover that the quantum potential has some very unusual properties (compared to the potentials usually considered in classical mechanics) and it is these properties that give rise to quantum mechanical phenomena in spite of the fact that the motions of the particles is still described by essentially classical equations of motion. Bohm&Hiley point out that because appears in both the numerator and denominator of Q, the e ect of the quantum potential on the motion of particles is independent of the strength of the quantum eld. Instead it depends primarily on its form 2 R, i.e. the "curvature" of the wavefunction. This is in contrast to classical elds (waves) which always produce e ects which are more or less proportional to the strength of the eld (wave). We think of this as a transfer of energy and momentum from the eld to the particle. Obviously we need to think of the quantum potential in a di erent way. In their book, Bohm&Hiley[2] suggest that the appropriate way to think of the quantum potential is in terms of what they call "active information". The analogy is with a ship running on autopilot, the autopilot being quided by radio waves. We are asked to think of the quantum potential as "informing" the particle about its environment. The particle, in turn, is thought to move (at least in part) under its own energy, in the same way that a ship on autopilot moves "on its own energy" in spite of being quided by the radio beacon. This is certainly a radical departure from the implicit assumptions of classical mechanics but Bohm&Hiley argue that this notion is fairly common at the 13
level of ordinary experience and therefore easily put to work as an "intuition", that is, a way of thinking about reality. It is important to realize that in saying that the particle moves "on its own energy", we are not saying that its motions are arbitrary. No. We have already stated the exact equations of motion for the particle. It does, however mean that in classical terms, the total energy and momentum of the system (wavefuntion and particle) will not necessarily be conserved in the detailed motions of the particle. The particle will be accelerated and decelerated by the e ects of the time varying quantum potential. In spite of this, Bohm&Hiley show that we do still have energy and momentum conservation on the average for a statistical ensemble of possible particle trajectories as required by the Schrodiner's equation and the expected value of the Hamiltonian operator (total energy). The extent of the non-conservation is just that allowed by the Heisenberg uncertainty principle. More over, we will see that the expected values of particle position and momentum also obey Erenfest's theorem which states that the expected values will obey classical laws of motion.
2 Particle in a box Lets start with a simple example. In the following, we are going to apply the treatment in Quantum Methods [6] of a particle in a box to the equations of motion of the particle according to Bohm's interpretation. Think of a particle constrained to move in only one dimension (for simplicity) and walled in by two impenetrable (in nite potential) barriers.
2.1 Maple Quantum Methods Please refer to Maple worksheet bohm2a.ms in what follows. First, we need to establish some basic de nitions. These are similar to those we used above. > > > >
restart;
alias(I=I,i=sqrt(-1)):
# Use i for complex numbers
read `notilde.txt`;
# Don't use ~
read `save1.m`;
# Recall some definitions
We will need these variables. > > > >
assume(x,real);
# location
assume(t,real);
# time
assume(m>0);
# mass
assume(L>0);
# length of the box
And these constants. > >
assume(hbar,real);
# Planck's constant/2/Pi
constants:=op({constants} minus {E} union {\hbar}):
For numeric calculations we will ususually assume units of measure in which >
units:={hbar=1,L=1,m=1};
f h = 1; L = 1; m = 1 g
(39) Now we can make a series of de nitions based on standard quantum mechanics. In position space the position operator is just a multiplication >
X:= psi -> x*psi;
and the momentum operator is
!x 14
(40)
>
P := psi -> -i*hbar*Delta(psi): P=P(_(x,t));
P = i h ( ( x; t ) )
The kinetic energy operator is >
K:=psi -> 1/(2*m)*(p@@2)(psi): K=K(_(x,t)); (2) K = 21 p ( m( x; t ) )
>
(42)
'conjugate'(bra);
bra We can de ne Dirac's < brajcjket > operator notation as follows: > > > > >
(41)
(43)
bracket:=(bra,c,ket)->int(conjugate(bra)*c(ket),boundry): `print/bracket`:=proc (bra,c,ket)
# How to print bracket
Int('conjugate'(bra)*'c'(ket),boundry); end: 'bracket'(bra,c,ket);
Z
conjugate( bra ) c( ket ) dboundry
(44)
And we will need the plotting functions > >
with(plots): read `post.txt`;
# provides postscript graphics
Maple can also produce "animations" by displaying a series of graphs (frames) in the same window. On my 486 25 MHz PC, Maple can display a maximum of about 10 frames per sec. More than fast enough to get a sense of "movement". I have included some animation commands in this worksheet but of course you will only be able to see these animations if you are reading this arcticle as a worksheet in Maple on an appropriate machine.
2.2 Schrodinger's Equation For a particle of mass m in a one dimensional box of length L with the barriers at x=0 and x=L, >
boundry:=x=0..L;
x = 0::L (45) we know from elementary quantum mechanics that the eigenfunctions of the time independent Schrodinger wave equation are given by: >
phibox:=phi[n](x)=sqrt(2)/sqrt(L)*sin(n*Pi*x/L);
>
H(0)(phi[n](x))=E[n]*phi[n](x);eval("&@phibox);
2 sin n L x p n( x ) = (46) L Inside the box the potential is zero and so if Schrodinger's equation is satis ed we must have p
1 h 2 ( 2 ) ( n ( x )) = E ( x ) n n 2 m 15
(47)
p
p
nx 2 2 nx 2 1 h 2 sin L n = En 2 sin p L 2 m L 5=2 L
(48)
We can divide both sides of this equation by box to obtain >
Schbox:=expand("/phi[n](x)&@phibox);
Therefore the eigen-energies are >
1 h 2 n2 2 = E n 2 L2 m
(49)
eigenbox:=E[n]=eval(solve(Schbox,E[n]));
2 2 2 (50) En = 12 h Ln2 m At the boundries V(0) and V(L) are in nite and the wavefunction must vanish. For all integers n we have
>
sin(n*Pi)=0; assign(");
sin( n ) = 0
(51)
so that > >
phi[n](0)=eval(phi[n](x)&@phibox&@(x=0)); phi[n](L)=eval(phi[n](x)&@phibox&@(x=L));
n( 0 ) = 0
(52)
n( L ) = 0 (53) Now we are ready to take a look at the rst few eigen-functions (stationary states) of this system. The eigen-functions are displayed below, shifted by their eigen-energies and scaled by an arbitrary factor of 4. > > > >
postscript(plot(
{((E[n]+4*phi[n](x)) &@ eigenbox &@ phibox &@ units)$n=1..3}, x=0..1), `fig2a1`,`A few eigen-functions for a particle in a box`,`3.5`,`3.5`);
2.3 Non-stationary states Now lets take a look at a non-stationary state. The following wavefunction is an equal weighting superpostion of the rst two eigenfunctions which Qunatum Methods[6] calls the "Quantum Rattle". First lets de ne the time harmonic as discussed in Part 1. >
harmonic:=eta[n](t)=1/sqrt(2)*exp(-i*E[n]*t/hbar);
p i En t n( t ) = 21 2 e( h ) Then the wavefunction of the desired non-stationary state is
16
(54)
50
40
30
20
10
0
0.2
0.4
0.6
1
0.8
x
Figure 1: A few eigen-functions for a particle in a box >
psi(x,t)='Sum(eta[n](t)*phi[n](x),n=1..2)';
( x; t ) = >
X 2
n=1
n( t ) n ( x )
(55)
psibox:=value("&@harmonic&@eigenbox&@phibox);
( x; t ) =
e
= iLh mt
2 1 2 2
p
x sin L e +
L
ih t L m
2 2 2
x sin 2
p
L
L
(56)
The probability density function for quantum rattle is >
>
rho[box]=simplify(rho(psi(x,t)&@psibox));
2 2 2 box = 2 sin Lx sin 2 Lx cos 23 hL2 mt + 2 cos Lx cos 2 Lx L
(57)
rattle:={eigenbox&@(n=1),eigenbox&@(n=2)};
2 2 2 2 E2 = 2 hL2 m ; E1 = 21 hL2 m
The frequency of oscillation of this system is given by
17
(58)
>
freqbox:=frequency=((E[2]-E[1])/hbar)&@rattle;
2 frequency = 23 Lh2m and in units where we take h , L and m to be 1, the period is
>
(59)
peribox:=period=(2*Pi/frequency)&@freqbox; 2 period = 34 L hm
(60)
2.4 The motion of the wave packet If you are using Maple to read this you will be able to view a live animation of the evolution of the probability density function. For those reading this without Maple, the please refer to gure 2 below. > > >
eval('animate'(
rho(psi(x,t)),x=0..1,t=0..period,color=blue ) &@ psibox&@peribox&@units);
Now lets consider how the expected values of the momentum and position change over time. The expected value of momentum is >
ExpPpsibox:='ExpP(t)'=simplify(bracket(psi(x,t),P,psi(x,t))&@psibox);
3 h 2 t h sin 2 L2 m ExpP( t ) = 38 L and the expected value of location is >
(61)
ExpXpsibox:='ExpX(t)'=expand(bracket(psi(x,t),X,psi(x,t))&@psibox);
3 h t 2
L cos 2 L2 m ExpX( t ) = 12 L 16 (62) 9 2 [Note that ( > x ) is just a cryptic way in which we can tell Maple to treat x as an operator. is a placeholder for the expression on which the operator operates and in this representation the location x operates by a multiplication.] In conventional quantum mechanics Ehrenfest's theorem states that these expected values are related by a quantum analogue of a the classical equation >
Ehrenfest:=m*diff(ExpX(t),t)=ExpP(t);
@ ExpX( t ) = ExpP( t ) m @t where momentum and position are replaced by their expected values. Lets test this. >
(63)
is(Ehrenfest&@ExpXpsibox&@ExpPpsibox);
true (64) But this is not the equation of motion that is used in Bohm's interpretation! Bohm's interpretation deals with the exact location and exact momentum of an individual particle under the in uence of both the classical and the quantum potentials. 18
2.5 The Quantum Potential Recalling the polar form of the wavefunction (5) that we discussed previously we can nd R as the square root of the absolute value of box squared. >
Rbox:=R(x,t)=sqrt(expand(simplify(rho(psi(x,t)&@psibox))));
x
sin L R( x; t ) = 4
2
3 h t
2 2 cos 2 L2 m cos Lx cos Lx 1 + L +3 L L
4 cos Lx !1=2 4 L Then the quantum potential (20) is >
(65)
Qbox:=simplify(QP&@Rbox);
3 2 4 Q( x; t ) = 21 64 cos Lx + 104 %1 cos Lx + 36 %12 cos Lx 2 + 24 cos Lx + 14 %1 cos Lx + 4 %12 3 2 h 2 3 2 2 4 16 cos Lx + 32 %1 cos Lx + 16 %12 cos Lx + 8 cos Lx + 8 %1 cos Lx + 1 L2 m 3 h 2 t %1 := cos 2 L2 m (66) [Note that Maple uses the placeholder names %1, %2, etc. in order to split a long expression into a manageable form. Think of the %1, %2 etc. as just temporary names for a given subexpression.] Obviously, the quantum potential is rather complex (and very non-linear) even in this very simple case. But thanks to tools like Maple, we can still get a good idea of what it looks like. In the following animation we show both the probability density function and the quantum potential. Note how the quantum potential has a large (negative) value when the amplitude of the wavefunction is small but varying signi cantly over a short distance. In the animation, the quantum potential appears to "sweep" back and forth between the points where the probability density is greatest, but out of phase with the probability density. If we imagined a large number of possible instances of the particle in the box (a large number of trials with the random initial locations), it appears as if the quantum pontential is "dragging" them rst one way and then the other, making the probability of nding a particle at a given location vary in exactly the way the probability density is varying! > > > >
postscript(animate( {100*rho(psi(x,t)),Q(x,t)}&@psibox&@Qbox&@units, x=0..1,t=0.1..1.1*period&@peribox&@units), `fig2a2`,`Prob. Dens. and Quantum Pot.`);
2.6 The Motion of the Particle So nally, lets calculate the motion of the particle according to Bohm's interpretation. 19
250 160
140
200
120
150
100
80 100 60
40 50 20
00
0.2
0.4
0.6
0.8
00
1
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
300
250 250
200 200
150 150
100
100
50
50
00
0.2
0.4
0.6
0.8
00
1
Figure 2: Prob. Dens. and Quantum Pot.
20
The form of the classic force in the case of a particle in a box presents some diculties when it comes to solving the classical Hamilton-Jacobi equation of motion because of its non-analytic form at the boundries. However, we can still nd the motion of the particle by using the guidance conditon23. The expression for the probability current is >
'j[box]'=simplify(evalc(j(psi(x,t)&@psibox)));
2 2 jbox = cos 12 hL2 mt cos Lx sin 2 hL2 mt sin 2 Lx 2 2 + 2 cos 2 hL2 mt cos 2 Lx sin 21 hL2 mt sin Lx
2 2 sin 21 hL2 mt cos Lx cos 2 hL2 mt sin 2 Lx 2 2 2 sin 2 hL2 mt cos 2 Lx cos 21 hL2 mt sin Lx h ( L2 m) Therefore the velocity of the particle varies according to >
(67)
vbox:=simplify(combine(eval(guide2&@psibox),trig));
2 mL v( t ) = 21 h sin 3 Lx 3 sin Lx sin 23 hL2 mt x x 3 h 2 t 2 2 cos 2 Lx 2 sin L sin 2 L cos 2 L2 m + 2 cos Lx
(68)
Because of the way we constructed the wavefunction, (x; t) is real valued at t=0, therefore the momentum and the velocity are necessarily initially >
simplify(vbox&@(t=0));
v( 0 ) = 0 The following animation shows how the velocity varies with location over time. >
(69)
eval('animate'(v(t),x=0..1,t=0.1..1.1*period)&@vbox&@peribox&@units);
Or we can display the velocity as a function of time and position (Figure 3) > > > > >
postscript(eval('plot3d'(v(t),x=0.1..0.9*L,t=0.01..0.99*period, axes=boxed,style=PATCHCONTOUR,orientation=[30,50], numpoints=3000,view=-20..20) &@vbox&@peribox&@units),`fig2a3`,`Quantum rattle velocity field`, `3.5`,`3.5`);
We can note that there are two locations (at di erent times) where the velocity is very large and suddenly switches to the opposite direction. This is related to the sigularities in the velocity function where the denominator goes to zero. Maple can actually compute exact values for the location of these poles, once we have a good guess as to where they are. >
{solve(denom(v(t)&@vbox)&@(t=period/2)&@peribox&@units=0,x)};
1
3 ; 0; 1 21
(70)
20
10
0
-10
-20
0.1 0.2 0.1
0.3 0.4 0.2 t
0.6 0.3
0.5 x
0.7 0.4
0.8 0.9
Figure 3: Quantum rattle velocity eld >
{solve(denom(v(t)&@vbox)&@(t=period)&@peribox&@units=0,x)};
(71) 0; 1; 32 We can double check that the velocity does indeed change direction rapidly at these points by a simple calculation. > > >
'v(`~period`/2)[`x=1/3 +.001`]'= [evalf(v(t)&@vbox&@{x=1/3+.001,t=period/2-0.001}&@peribox&@units), evalf(v(t)&@vbox&@{x=1/3+.001,t=period/2+0.001}&@peribox&@units)];
1
= [ 325:8794693; 325:8794873 ] v 2 ~period x =1 =3 +:001 > > >
(72)
'v(`~period`)[`x=2/3 + .001`]'= [evalf(v(t)&@vbox&@{x=2/3+.001,t=period-0.001}&@peribox&@units), evalf(v(t)&@vbox&@{x=2/3+.001,t=period+0.001}&@peribox&@units)];
v( ~period )x =2 =3 + :001 = [ 321:7401470; 321:7401389 ] (73) Apparently the particle undergoes an in nite acceleration at these points due to the quantum potential. Actually it can be shown that such motions never occur because the probability ux into such regions is 0.
22
2.7 The Trajectory of the Particle We can express the quidance condition as a di erential equation and solve for the particle position as follows. >
M1:=dsolve(combine(vbox&@(v(t)=diff(x(t),t)),trig),x(t)):M1;
3 h t 2
x( t )
x( t )
3
x( t )
3
cos L L m sin L 4 cos 2 L2 m L m sin L 3 + x( t ) L m = C1 (74) Maple returns an implicit solution relating t and x(t) by an arbitrary constant C1. In units where h, L, and m are 1, and given a location x(t) at time t, we can solve for the value of C1. For example, for a particle located (approximately) at one of the poles we have >
2
2
C1pole:=_C1=fsolve(subs(x(t)=1/3,t=period/2,peribox,units,M1),_C1);
C1 = :02321082978 (75) Then we can use this value to compute the trajectory of the particle. In the following graph (Figure 4) time is advancing through two complete cycles from bottom to top. The horizontal axis gives the position of the particle. > >
postscript(eval('implicitplot'(M1&@C1pole,x=0..1,t=0..2*period) &@(x(t)=x)&@peribox&@units),`fig2a4`,`Trajectory near a pole`);
0.8
0.6
t 0.4
0.2
0
0.1
0.12
0.14
0.16
0.18 x
0.2
0.22
0.24
0.26
Figure 4: Trajectory near a pole Lets try to display a set of trajectories over a range of intial locations (see Figure 5) > > > >
postscript(eval( 'implicitplot'({(M1&@(_C1=j/7))$j=1..6},x=0..1,t=0..2*period) &@(x(t)=x)&@peribox&@units),`fig2a5`,`Range of trajectories`); save P,X,K,bracket,`print/bracket`,postscript,`save2a.m`;
23
0.8
0.6
t 0.4
0.2
0
0.2
0.3
0.4
0.5 x
0.6
0.7
0.8
Figure 5: Range of trajectories
3 The Gaussian wave packet and particle system Bohm and Hiley discuss the behavior of a particle inside of a Gaussian wavepacket, a discussion that could not take place within the context of the ordinary interpretation of quantum mechanics. We will develop here the equations that appear in the text as well as others that describe this situation and explore the behavior of the particle in the new interpretation. Furthermore we will o er a further extension of the interpretation that we believe more completely describes the relationship between the wave, the particle and the observer. First, we need to establish some basic de nitions. These are similar to those we used above. Please refer to the Maple worksheet bohm2b.ms for the calculations that follow. We greatfully acknowledge the contributions of David Cyganski[7] to this section of the paper. > > > > > >
restart;
read(`notilde.txt`); alias(I=I,i=sqrt(-1)):
# Use i for complex numbers
read `save1.m`; read `save2a.m`; with(plots):
The following variables must be assumed to be real (and positve where appropriate) > > > > > > > >
assume(hbar,real);
# Planck's constant /2 /Pi
constants:=op({constants} minus {E} union {hbar}); assume(t,real);
# time
assume(x,real);
# position
Dx:=`\\Dx`: assume(Dx>0);
# deviation of position
Dp:=`\\Dp`: assume(Dp>0);
# deviation of momentum
assume(m>0);
# mass
assume(k,real);
# wave number
24
; 1; Catalan ; FAIL; false; ; true; h The boundry conditions that we will use below are > >
inf:=`\\inf`:
(76)
# We use our own definiton of infinity to avoid a bug
boundry:=x=-inf..inf;
x = 1::1
(77)
3.1 Evolution of the Gaussian wave packet Bohm and Hiley introduce the Gaussian wave packet as an example of a Quantum system in a nonstationary state, that is, having a quantum potential that changes with time. Such a packet is a linear combination of stationary state functions, such as an integral combination of complex exponential wave functions. Such packets arise in the usual discussions of QM as soon as we wish to describe the location of a particle to within a given probability distribution. We nd immediately that nonclassical behavior is asserted in so much as our uncertainty with respect to the particle's position generates a reciprocal uncertainty with respect to its momentum. Thus we cannot specify a location distribution without regard to a momentum distribution that ful lls the Heisenberg uncertainty principle. Furthermore, usual treatments show that if we solve for that distribution that minimizes the uncertainty product x p, the product of the location and momentum standard deviations, then the solution is a Gaussian wave packet with initial probability distribution given by: >
GaussProb:='rho(psi(x,0))'=1/2*2^(1/2)*exp(-1/2*x^2/Dx^2)/Pi^(1/2)/Dx;
p
x2
1=2 x 2 (78) ( ( x; 0 ) ) = 21 2 ep x We will brie y sketch this derivation here owing to the complexity of some of the symbolic manipulations that would otherwise be required. Following this derivation, we will generate another less intutive but complete derivation using Maple's capabilities fully. This second course is that taken in fact by the Bohm and Hiley book[2]. Given a wave function that describes a particle with zero mean location < jX j >= 0 and zero mean momentum < jP j >= 0, we can express the mean square location uncertainty by
>
MSL:=Dx^2='bracket'(psi(x,t),X@@2,psi(x,t));simplify(")&@(inf=infinity);
x 2 =
Z1
1
conjugate( ( x; t ) ) X ( 2 ) ( ( x; t ) ) dx
Z1
x 2 = ( x; t ) x2 ( x; t ) dx 1 And likewise the momentum uncertainty is given by >
(79) (80)
Dp^2='bracket'(psi(x,t),P@@2,psi(x,t));``=simplify(rhs("))&@(inf=infinity);
p = 2
=
Z1
1
conjugate( ( x; t ) ) P ( 2 ) ( ( x; t ) ) dx
h
2
Z1 1
( x; t ) ( 2 ) ( ( x; t ) ) dx
(81) (82)
This last expression can be integrated by parts in closed form by using the boundary conditions 25
that must be obeyed by all wave functions >
psi(infinity,t)=0; psi(-infinity,t)=0;
resulting in the new form >
( 1; t ) = 0
(83)
( 1; t ) = 0
(84)
MSM:=Delta_p^2=-hbar^2*'int'(conjugate(Delta(psi(x,t)))*Delta(psi(x,t)),boundry);
Delta p = h 2
2
Z1
1
( ( x; t ) ) ( ( x; t ) ) dx
(85)
We will now express the product of the position and momemtum variances in terms of two functions f and g given by: >
alias(f='diff(psi(x,t),x)',g='X(psi(x,t))'):
where we have taken f and g to be >
'f'=f; 'g'=g;
f = ( ( x; t ) )
(86)
g = x ( x; t ) The squared uncertainty product is now of the form: >
(87)
'Dx^2*Dp^2'=(Dx^2*Dp^2)&@MSL&@MSM&@(inf=infinity);
Dx Dp = 2
2
Z1
1
( x; t ) x2 ( x; t ) dx p 2
(88)
But the uncertainty product is of the form of one side of the well known Schwarz inequality, > > >
schwarz:=
'(1/2*(Int(f*conjugate(g),x)+Int(g*conjugate(f),x)))^2 <= Int(f*conjugate(f),x)*Int(g*conjugate(g),x)';
1 Z
f g dx + 1
Z
g f dx
2
2 2 which is known to take on its minimum value when > >
Z
Z
f f dx g g dx
(89)
'f(x,t)'=a(t)*'g(x,t)';
f=a(t)*g;
f( x; t ) = a( t ) g( x; t )
(90)
( ( x; t ) ) = a( t ) x ( x; t ) (91) The equation above constitutes a di erential equation for the (x; t) that yields the minimum uncertainty product. The only solution to this di erential equation has the form where N(t) and a(t) are arbitrary time functions: >
Psi_1:=psi(x,t)=N(t)*exp(-a(t)*x^2/2);
( x; t ) = N( t ) e( 1=2 a( t ) x2 ) (92) We note for general interest that it so happens that with f and g as de ned above, the left hand 26
side of the Schwarz inequality in this case can be evaluated in closed form[11] yielding for any psi function: >
lhs(schwarz)=h^2/4;
1 Z
2 Z dx + 1 x ( x; t ) ( ( x; t ) ) dx = 1 h2 ( ( x; t ) ) x ( x; t ) (93) 2 2 4 Thus we have here the value of the exact lower bound on the uncertainty product even without nding the wave functions associated with that minimum uncertainty. Thus from the above we already know that: > >
'Dx^2*Dp^2' <= h^2/4; 'Dx*Dp' <= h/2;
Dx 2 Dp 2 14 h2
(94)
Dx Dp 21 h (95) We will continue our derivation with the purpose of nding the psi function that results in the minimum value given by this bound. The wave function, , must ful ll the Schrodinger equation which dictates the allowed form of its time evolution. Thus substituting our current form of into the Schroedinger equation we can obtain N(t) to within a constant factor. Now recall that Schrodinger's wave equation (4) for a free body in one dimension can be written >
SCHROD&@(V(x)=0);
@ ( x; t ) + 1 h 2 ( 2 ) ( ( x; t ) ) = 0 i h @t 2 m >
Sch1:=SCHROD&@(V(x)=0)&@Psi_1;
@ N( t ) e( i h @t >
=
t x
1 2 a( ) 2 )
1 h ( N( t ) e + 2
(2)
2
(
m
=
(96) t x
1 2 a( ) 2 )
) =0
(97)
Sch1s:=normal(2*m*Sch1/(hbar*exp(-1/2 *a(t)*x^2)));
@ N( t ) i m N( t ) @ a( t ) x2 h N( t ) a( t ) 2 i m @t @t 2 2 + h N( t ) a( t ) x = 0 (98) As this must be true for all x, we can separate this into two independent equations that must be satis ed (the coecient of unity and the coecient of x2) >
Sch1u:=Sch1s&@(x=0);
@
nhbar N( t ) a( t ) = 0
2 i m @t N( t ) >
>
(99)
Sch1x2:=normal((Sch1s-Sch1u)/(N(t)*x^2));
@ a( t ) + nhbar a( t )2 = 0 i m @t
(100)
dsol_a:=map(<1/_>,dsolve(Sch1x2,a(t)));
a( t ) = i h t +mC1 m 27
(101)
>
Sch1ua:=Sch1u&@dsol_a;
@
h N( t ) m i h t + C1 m = 0
2 i m @t N( t ) >
dsol_N:=dsolve(Sch1ua,N(t));
N( t ) =
( h t + C1 m ) 2 2
2
C2s
=
2 1 4
Hence the general solution for psi is given by: >
(102)
p i 2h t2+ C1 m2 2 h t + C1 m
Psi_gen:=Psi_1&@dsol_N&@dsol_a;
x2 = i h tm + C1 m
C2 e
1 2
s
(103)
(104) i h t + C1 m ( h t + C1 m2 )1=4 p 2 2 h t + C1 2 m2 We can x the value of the arbitrary constants by choosing C1 and C2 so that obeys the requirement that the magnitude square integral is unity (since this quantity is interpreted in QM as the total probability of nding the particle anywhere and hence must be unity) and so that we begin at time zero with a speci c distribution with respect to location, lets say that with a statistical variance of x2 . We will rst set C2 = (m=)1=4 and just establish the value of C1 from the initial variance criterion. With the above substitution, at time zero we have: ( x; t ) =
2 2
>
2
Psi_gen0:=Psi_gen&@(t=0);
x2
=2 C1 C2 e 1r (105) ( x; 0 ) = C1 m 2 2 1 = 4 p 2 2 ( C1 m ) C1 m This is obviously of the form of a Gaussian distribution with variance given by C1, hence we ful ll the second condition by making
>
Psi_varfixed:=Psi_gen&@(_C1=Dx^2);
x2 = i h tm +x 2 m
1 2
C2 e
( x; t ) =
s
(106) 2 i h t + x m p22 ( h t + x m ) h t + x 4 m2 We can obtain the proper value for the constant C2 indirectly by merely recalling that the form of a Gaussian distribution is given by: 2 2
>
=
4
2 1 4
Psi_Gaussian:= 1/(sqrt(2*Pi)*sigma) * exp(-1/2*x^2/sigma^2);
p
x2
1 2 ep 1=2 2 2 Hence since the exponential that appears in our function is > >
(107)
assume(c, real); psi_exp:=psi(x,t)=c*numer(psi(x,t)&@Psi_varfixed)/_C2;
( x; t ) = c e
x2 = i h tm +x 2 m
1 2
28
(108)
>
psi_mag:=simplify(int(rho(psi(x,t)&@psi_exp),x=-infinity..infinity))=1;
p
p
c2 h 2 t2 + x 4 m2 = 1 m x >
c_roots:=solve(psi_mag,c);
rp p %1
(109)
rp p %1
m nDx m x p pm x ; p p %1 %1 2 2 4 2 %1 := h t + x m
m x
>
>
(110)
c_pos:=c=radsimp(c_roots[1]);
pm px c= 2 2 ( h t + x 4 m2 )1=4 1=4
psi_fi:=psi_exp&@c_pos;
(111)
x2 2 pm px e 1=2 i h tm+x m ( x; t ) = ( nhbar 2 t2 + x 4 m2 )1=4 1=4
(112)
The above is a unique description, to within a unit magnitude complex multiplier which is not a function of x, of the function that satis es the minimum uncertainty constraint for a particle that has variance x2 at time 0. The important notion to be obtained from the above is that QM does not allow a description of a particle as a localized distribution without a reciprocal distribution of momentum. This is clearly evident in that the above minimization of the uncertainty product can yield no smaller value for this product than h=2. This behavior is counter to all classical intuition.
3.2 Derivation of the wave packet as a superposition of stationary states We will obtain the same expression now by following a development that is chosen by many texts, including Bohm and Hiley's, so as to tie these developments together. This development emphasizes the nature of the wave packet as a superposition of stationary states, while somewhat obscuring the starting point, used above, in which the emphasis is on the inevitability of the uncertainty of momentum given a speci ed location distribution. On the other hand, this development generates the complete, time varying, expression for (x; t) in a simpler fashion. As is discussed in most elementary QM texts, the eigenfunctions of a free system (classical potential = 0) with a precisely known momemtum, k, and mass m are plane waves >
exp(i*k*x)*exp(-i*'omega'*t);
with frequency given by >
e( i k x ) e(
i! t)
(113)
planefreq:=omega= hbar*k^2/(2*m);
(114) ! = 21 hmk Now these plane waves ll a somewhat strange place in any discussion of a QM system: They 2
29
describe a physically unrealizable solution of the Schroedinger wave equation. That these are physically unrealizable should be evident after the previous discusion in so much as they disobey the uncertainty principle by demonstrating an absolutely known value of momentum. More directly, the unrealizability is further evidenced by not having a magnitude squared integral of unity, a requirement of the probability distribution interpretation of the function. However, since each such wave function does satisfy the Schroedinger di erential equation, any linear combination of these functions also solves this equation. Hence, if we can assemble a linear combination of these wave functions that satis es the nite squared integral constraint, then this new solution will be physically realizable state of the free space system. By weighting the functions corresponding to a range of momentum values by a Gaussian weighting function of width determined by a value x2 (the width of the momentum distribution being chosen inversely proportional to x2 ) and further choosing the mean of the momemtum distribution to be zero, we can write a solution of Schrodinger's equation as the following linear combination of eigenfunctions > > >
psiform:=psi(x,t)=c*int( exp(-k^2*Dx^2/2)*exp(i*k*x)*exp(-i*omega*t)&@planefreq, k=-infinity..infinity);
( x; t ) = c
Z1 1
e(
= k
1 2 2
= i hmk t
x 2 ) e( i k x ) e
2 1 2
dk
(115)
where the normalization constant c is chosen to obtain a unit probabilty (the integral of the magnitude squared psi function) for any time, t. >
psiform1:=combine(psiform,exp);
( x; t ) = c
Z1 1
e
= k
1 2 2
x 2 +i k x
= i hmk t
2 1 2
dk
(116)
Unfortunately, Maple doesn't by iteself see the means to analytically integrate this function. Nor have we suciently constrained the domain of some of the variables to allow general closed form solution. We will assist Maple in the evaluation by completing the square in the exponent and changing variables so that our integral becomes expressed as the well known error function (or probability integral). We rst load the Maple Student package that provides tools for the manipulation of the integral. >
with(student, completesquare, changevar):
We complete the square on k to reveal the error function form of the integral. >
psiform2:=completesquare(psiform1,k);
( x; t ) = c
Z1 1
0 @
e
=
h x 2 m )
( i t+
1 2
ixm2 k i h t+ x m m
2
m x2 1=2 i t+ 2 m
h x
1 A
dk
(117)
Now lets de ne the constant alpha so as to simplify the form of the equation >
ALPHA:=Dx^2*m+i*hbar*t=alpha;
i h t + x 2 m =
30
(118)
>
psiform4:=psiform2&@ALPHA;
Z1
=
1 2
(k i x m )2 m
= m x2
1 2
( x; t ) = c e dk (119) 1 Now despite having this familiar form, Maple will still not integrate this function as stated because the convergence of this integral depends upon the argument (angle with respect to the real axis in the complex plane of the complex constant alpha). Speci cally, for a purely imaginary alpha, this integral does not exist. We, however can still proceed as we have additional information regarding the nature of alpha, that is, that alpha always has a non-zero real component (as seen above in its de nition.) The next step in our symbolic integration process involves a change of variables resulting in the formation of new complex limits of integration at in nity. We will show this by executing the change of variables ourselves. We need to begin by de ning the integral with nite limits of which the above is the limiting form. > > >
psifinite:=psi(x,t)=c*Int(
exp(-1/2*alpha/m*(k-i*x/alpha*m)^2-1/2/alpha*m*x^2), k = -R .. R);
ZR
=
1 2
(k i x m )2 m
= m x2
1 2
( x; t ) = c e dk (120) R now apply the change of variables that manipulates this into the form of the well known erf function or so-called error integral: >
changevar(sqrt(alpha)*(k-i*x*m/alpha)=u,psifinite,u);
0 BB @
Z p R ipx m e ( x; t ) = c
=
1 2
ipx m u i x m p m
2
=
1 2
1 C mx C A 2
p
xm p R ip
du
(121)
As this is a de nite integral over a nite contour of integration in the complex plane of an entire function (that is, it is analytic everywhere in the complex plane) the result is independent of the path connecting the endpoints, Maple can supply a closed form solution (in terms of the special function erf) >
psidefinite:=value(psifinite);
0 p e B 1 ( x; t ) = c B @ 2
p e + 21
= m x2
1 2
= m x2
1 2
p
0 1 p B1 2 ( r R + i x m )C CA 2 erf B @2 m m r
m 0 1 p B 1 p2 ( R + i x m ) C r CA 1 2 erf B @2 m m CC r A m
31
(122)
>
psidefsimp:=combine(radsimp(psidefinite));
m x2 p p ( x; t ) = 21 e 1=2 2 m ! !! p p p p (123) 2 m ( R + i x m ) 2 m ( R + i x m ) 1 1 erf 2 erf c m 2 m Since the limit of erf(z) as jz j approaches in nity is well de ned for all complex valued arguments z such that jarg(z)j <= =2 (see Special Functions by N.N Lebedev, section 2.2) and approaches the value 1 for Re(z) > 0 and -1 for Re(z) < 0, we have that
> > > >
psiforms:=subs({ erf(1/2*2^(1/2)*((m*alpha)^(1/2)*(alpha*R+i*x*m)/(m*alpha)))=1, erf(1/2*2^(1/2)*((m*alpha)^(1/2)*(-alpha*R+i*x*m)/(m*alpha)))=-1}, psidefsimp);
( x; t ) = e
= m x2
1 2
p p 2 m c
(124) We can now nd the value of the constant c. Since the integral of the square magnitude of the Psi function must always be unity, and since c can always be taken to be real : >
psiform5:=Int(rho(psi(x,t)&@psiforms/c),x=-infinity..infinity)=1/c^2;
Z1
0 @e
= m x2
1 2
1
p
m A e
= m x2
1 2
p
m
dx = c12
(125) To carry out this evaluation we must substitute back the expression that alpha represents 1
>
2
psiform5value:=value(psiform5&@(rhs(ALPHA)=lhs(ALPHA)));
q
q
m ( i h t + x 2 m ) m ( i h t + x 2 m ) p s 2 = c12 2 2 ( i h t + x 2 m ) ( i h t + x 2 m ) 2 2m x 4 2 h t + x m >
csolve:=radsimp(evalc(combine(psiform5value,radical,symbolic)));
2 x = c12 Selecting the positive root we obtain a solution for c =
3 2
>
(126)
csol:=c=radsimp(1/sqrt(lhs(csolve)));
(127)
p p
c = 21 2 3=x (128) 4 Thus, substituting alpha and the value of c we just found into the unnormalized expression for , we obtain the explicit expression for , >
gwpexp:=radsimp(psiforms&@csol&@(rhs(ALPHA)=lhs(ALPHA)));
( x; t ) = e or the simplier expression
x2 2 = i h t+mx m
1 2
p
pm px
i h t + x 2 m 1=4 32
(129)
> > >
gwp:=psi(x,t)=radsimp( numer(psi(x,t)&@gwpexp)*sqrt(m))/ denom(psi(x,t)&@gwpexp)/sqrt(m);
x2 2 = i h t+mx m
1 2
( x; t ) = e
p
pm px
(130) i h t + x 2 m 1=4 This expression matches that which we gave in the introduction as they di er only by the allowed unit magnitude complex factor as shown below: > >
cfactor:=(psi(x,t)&@gwp)/(psi(x,t)&@psi_fi); ``=simplify(abs(cfactor)); 2 2 ( h p t + x 4 m2 )1=4 i h t + x 2 m
(131)
=1
(132)
3.3 Deriving the Bohm Velocity Function for particles in the Gaussian Wave Packet Now we are in the position to express the psi function in polar form5, which is an essential rst step towards deriving the expression for the velocity function that guides the progress of any particle within the Gaussian wave packet according the Bohm interpretation of QM. Obviously we have that the magnitude function, R, for the psi function is given by: >
Rgwp:=R(x,t)=evalc(abs(psi(x,t)&@gwp));
R( x; t ) = e
=
1 2
Dx x
m2 x2 n 2 4 m2 2 t2 +
h
pm pnDx
(133) ( h 2 t2 + x 4 m2 )1=4 1=4 Now, note that the function has the argument (phase angle of the polar expression) given by (which is di erent than that given by Bohm and Hiley as equation 3.21 in which they have ignored the phase contribution of the rst factor of the expression for in equation 3.20) > > >
phigwp:=phi=evalc(simplify(expand( ln(combine(eval(psi(x,t)&@gwp/R(x,t)&@Rgwp),exp)) )));
=i
!
!
1 1 1 h t h t 2 2 4 2 2 2 arctan nDx 2 m h t 2 arctan nDx 2 m x m + 2 h t m x ( h 2 t2 + x 4 m2 ) (134) But the phase of the polar form of the function is de ned by Bohm as having a form in which the
33
function S plays a signi cant role in determining the velocity function that indicates how individual particles are guided by the quantum potential: >
'phi=i*S(x,t)/hbar';
Thus we can solve for S as >
x; t ) = i S(hbar
(135)
Sgwp:=S(x,t)=simplify(solve(",S(x,t))&@phigwp);
S( x; t ) = 21 arctan h2t h 2 t2 + arctan h2t x 4 m2 h t m x2 h x m x m ( h 2 t2 + x 4 2 m ) (136) But from S we can obtain the velocity function for a particle at postion x since according to Bohm this is given by: >
vgwp:=v(t)=(1/m)*diff(S(x,t)&@Sgwp,x);
h 2 t x (137) h 2 t2 + x 4 m2 (This equation that we have obtained is in agreement with Bohm and Hiley[2] because the di erence in the phase terms, for the two derivations noted earlier, was constant with respect to x and hence constributes nothing to the above derivative with respect to x.) We will now graphically examine the time evolution of the function and the related motion of a particle guided by this function as de ned by the above velocity guidance relationship. We can visualize the behavior of the magnitude squared (probability) wave function related to the function we found by setting v( t ) =
>
units:={Dx=1,m=1,hbar=1};
f h = 1; m = 1; nDx = 1 g
and plotting its magnitude squared function, R(x; t) , versus time:
(138)
2
> >
postscript(animate((R(x,t)^2)&@Rgwp&@units,x=-20..20,t=0..12), `fig2b1`,`Motion of the Gaussian Wave Packet`);
We see in these plots (Figure 6) a non-classical behavior of a particle in the standard interpretation of QM: the spreading of its probability distribution with respect to time. In the following we will see the ontological explanation for this behavior in terms of the guidance relationship that Bohm's work contributes.
3.4 The Motion of a particle within the wave packet The motion of a particle within the wave packet in the Bohm interpretation can be obtained by integrating the partial di erential equation we obtain from the relationship between the velocity function and the position function to obtain a position function versus time. >
M1:=dsolve(vgwp&@(v(t)=diff(x(t),t)),x(t)): M1;
p
x( t ) = h 2 t2 + x 4 m2 C1 Now solve for the constant so we can express xpos in terms of intial position x(0) = x0. 34
(139)
0.3
0.5 0.25
0.4 0.2
0.3 0.15
0.2
0.1
0.1
-20
-10
00
0.05
10
20
-20
-10
00
10
20
10
20
0.16 0.1 0.14
0.12
0.08
0.1 0.06 0.08
0.06
0.04
0.04 0.02 0.02
-20
-10
00
10
20
-20
-10
Figure 6: Motion of the Gaussian Wave Packet
35
00
>
xpos:=subs(_C1=subs({t=0,x(t)=x0},solve(M1,_C1)),M1);
p
2 2 4 m2 x0 x( t ) = h tp + x x 4 m2
And check the solution >
(140)
is(diff(x(t)&@xpos,t)=rhs(vgwp&@(x=x(t))&@xpos));
true (141) Now we can plot the postion of an ensemble of particles distributed between -5 and 5 with respect to time. > >
postscript(plot({(x(t)&@xpos&@units)$x0=-5..5},t=0..5), `fig2b2`,`Trajectories of an ensemble of particles`);
20
10
00
1
2
3
4
5
t
-10
-20
Figure 7: Trajectories of an ensemble of particles As can be seen in the plot (Figure 7), each particle travels along a non-linear trajectory which is determined by the wave function for the packet to which it belongs. The "force" involved in this guidance comes from the quantum potential that we will nd in the next section.
3.5 The Quantum potential As discussed above the quantum potential (equation 20), according to Bohm, is given by >
Qgwp:=Q(x,t)=simplify((-(hbar^2/(2*m))*(diff(R(x,t),x,x)/R(x,t)))&@Rgwp);
2 2 2 2 4 2 2 2 2 Q( x; t ) = 21 m x h ( h 2t 2+ x 4m 2 2m x x ) ( h t + x m ) This can also be obtained by solving the quantum Hamilton-Jacobi equation.
>
(142)
is(Q(x,t)&@Qgwp=simplify(solve(QHJ&@Sgwp&@(V(x)=0),Q(x,t))));
true (143) The following graphic (Figure 8) shows how the quantum potential causes the particle in the wave 36
packet to be swept outwards from either side of the center owing to a negative potential eld which changes with time. As time progesses, the quantum potential decreases in size until the particle acceleration approaches zero and each particle travels at a xed velocity. > > > >
postscript(plot3d(Q(x,t)&@Qgwp&@units, x=-20..20,t=0..2.0,orientation=[-70,75], axes=BOXED,style=PATCHCONTOUR), `fig2b3`,`Quantum Potential of Gaussian Packet`,`3.5`,`3.5`);
0
-50
-100
-150
-200 -20
0 -10
0.5 1
0
t 1.5
x
10 2 20
Figure 8: Quantum Potential of Gaussian Packet The force on the particles due to this quantum potential varies with both location and time >
F(x,t)=-diff(Q(x,t)&@Qgwp,x);
F( x; t ) =
m3 x 4 h 2 x ( h 2 t2 + x 4 m2 )2
(144)
3.6 Discussion of the wave packet Bohm's ontological interpretation of the wave packet introduces the notion that it is not the particle itself that is spreading out with time like the wave packet's psi function. The classic interpretation which considers the notion of the particle itself, when in this state, to be a nonsequitor, must introduce the collapse of the wave packet to the particle upon the act of observation of the particle. In Bohm's interpretation, the particle is always a true particle located somewhere inside the guiding wave packet. On observation, we nd the particle at a location which is the same as that is possessed just prior to the observation, (and we also disturb the wave packet, changing its progress from this point onwards). According to Bohm and Hiley, the distribution of momentum (that is, the fact that we are not allowed to have a particle with zero momentum and instead associate a range of 37
momenta with the packet) associated with the wave packet does not derive from some breakdown in the meaning of a particle as dictated by the uncertainty principle, but rather from the application of forces derived from the quantum potential acting on the very real particle in accord with its positions within the packet which de nes this distribution of quantum potential and not a distribution of the particle itself. The resulting distribution of momentum from this force is in compliance with the uncertainty principle. When we rst considered Bohm and Hiley's interpretation of the wave packet, we were struck by a new problem that it introduced (at least for us). In this new interpretation, the particle can be found within the wave packet at a speci c location. The trajectory that the particle takes is governed by its particular location within the packet. This begs the question of the causal relationship between the packet and the particle. For if the particle is the cause of the wave packet, why is the wave packet's spatial relationship with respect to the particle not xed. That is, we naively expected that the wave packet would always be centered on the particle, but instead the wave packet and the particle must now be considered separate entities. So, if the particle does not cause the wave packet, but is merely a passenger on this vehicle, then what does determine the location and behavior of the packet? Consider the following resolution of the above question. Suppose a low emission source of electrons is placed behind an adjustable aperture. The aperture is controlled so that it is closed in the in nite past and future, but proceeds to slowly open to some maximum size at time zero and then executes the reverse of this motion. Suppose further that the area of the opening presented to the source is a Gaussian function of time. Now, the passage of an electron through this aperture, has a probability density that is distributed in a Gaussian fashion with respect to time, with the peak of the probability occuring at time zero. The wave packet that describes this situation is exactly that which we derived above, since it describes just such a distribution of probabilty for nding an electron on the other side of the aperture versus position beyond the aperture. Thus we see clearly in this case that the wave packet is a function of the state of the experimental apparatus and not a function of the particular electron which is launched through the system or its location within the wave packet. That is, the wave packet is describing the quantum elds produced by the opening and closing of the aperture and the simultaneous presence of a source capable of launching electrons that sits behind it. These packets evolve regardless of the presence or position of an actual electron in the system. Thus, by this interpretation, when we presuppose a wave packet or other wave function for a particle, we are presupposing the form of the experimental appartus that delivers the particle to us. The wave function is a form of sucient statistic (to draw from the language of statistical communications theory) that communicates the minimum information between pieces of appartus needed to generate component solutions that would have arisen from a joint solution of the complete description of the entire experiment including the original source of the particle. That is, suppose the electrons that pass through our aperture were used next for an experiment that that involves interference phenomena. We could model the entire experimental appartus, solving for the psi function that describes the source, the aperture and the interfence experimental appartus. Or, we could instead obtain the wave packet description of the electron leaving our emitter/aperture appartus; then, solve for the interfence experiment result using this as a sucient description of the e ect of the emitter/aperture system on the conditioning of the quantum potential in our experiment. The latter course of action is actually an approximation since, as Bohm and Hiley point out, every experiment can only be completely analyzed as a whole owing to the the non-local character of quantum behavior.
38
4 The Airy Wavefunction The Airy wavefunction is a non-dispersive wave packet-like solution to the potential free Schrodinger's equation that implies a constant non-zero force arising from the quantum potential. According to Bohm's interpretation, a particle guided by such a wavefunction is accellerated in a non energy conserving manner. Please refer to Maple worksheet bohm2c.ms in what follows. > > > > > > > > >
restart;
alias(I='I',i=sqrt(-1)): read `notilde.txt`; with(plots): read `post.txt`; read `save1a.m`; read `save2a.m`; assume(hbar,real); constants:=op({constants} minus {E} union {hbar}):
4.1 What Maple knows about the Airy functions and Bessel functions Maple knows the Airy function as Ai. There is another Airy function, Bi, which we do not consider. > >
postscript(plot(Ai(x),x=-50..10,numpoints=1000), `fig2c1`,`The Airy Wave Function`);
0.4
0.2
-50
-40
-30
-20
-10
00
10
x
-0.2
-0.4
Figure 9: The Airy Wave Function The Airy function can also be de ned in terms of Bessel functions as follows[8] > > >
aip:=x->1/Pi*sqrt(x/3)*BesselK(1/3,2/3*x^(3/2)): ain:=x->sqrt(-x)/2*(BesselJ(1/3,2/3*(-x)^(3/2))-1/sqrt(3)*BesselY(1/3,2/3*(-x)^(3/2))): ai:=x->piecewise(x<0,ain(x),x>0,aip(x),0.3550280539): 'ai(x)'=ai(x);
39
ai( x ) = piecewise x < 0;
1 p x BesselJ 1 ; 2 ( x )3=2 1 p3 BesselY 1 ; 2 ( x )3=2 ; 0 < x; 2 3 3 3 31 32 p p ! 3=2 1 3 xBesselK 3 ; 3 x ; :3550280539 3 But apparently Maple uses a slightly di erent method for numerical evaluation. >
evalf(Int(abs(ai(x)-Ai(x)),x=-10..10));
:9943982614 10 Maple evaluates the derivative of the Airy function as >
dAi:='Diff(Ai(x),x)'=diff(Ai(x),x);
(146)
11
p 2 2 3=2 x @ Ai( x ) = 1 3 BesselK 3 ; 3 x @x 3 which is the same as the derivative of the de nition of ai(x) for positive x. >
(145)
(147)
daip:='Diff(aip(x),x)'=simplify(diff(aip(x),x));
22 = @ aip( x ) = 1 3 x BesselK 3 ; 3 x @x 3 p
3 2
(148) Note: BesselK(n,x)=BesselK(-n,x) for fractional n, but Maple doesn't do this simpli cation automatically. Oddly, Maple does not numerically evaluate the BesselK function with an imaginary argument. For example >
lhs(dAi)[x=-1]=evalf(rhs(dAi)&@(x=-1));
@
@x Ai( x )
x=
1
= :1837762985 BesselK( :6666666667; :6666666667 i )
(149)
But since BesselK can be de ned in terms of BesselJ and BesselY, we have >
evalf(eval(subs(BesselK=((n,x)->Pi/2*I^(n+1)*(BesselJ(n,I*x)+I*BesselY(n,I*x))),")));
@
1:666666667 ( @x Ai( x ) x= 1 = :2886751347I BesselJ( :6666666667; :6666666667 i I ) + I BesselY( :6666666667; :6666666667 i I )) (150) In any case, this is not the same as the derivative of the expression of the Airy function as de ned for negative arguments.
>
dain:=Diff('ain'(x),x)=simplify(diff(ain(-x),x));
@ ain( x ) = 1 x 3 BesselJ 2 ; 2 x3=2 p3 BesselY 2 ; 2 x3=2 (151) @x 6 3 3 3 3 Again, Maple does not use the Bessel function re ection formulas for fractional order automatically,
40
but
> > > >
>
lhs(eval(dain,1))=simplify(eval(subs( {BesselJ=((n,x)->cos(-n*Pi)*BesselJ(-n,x)-sin(-n*Pi)*BesselY(-n,x)), BesselY=((n,x)->sin(-n*Pi)*BesselJ(-n,x)+cos(-n*Pi)*BesselY(-n,x))}, rhs(dain))));
@ ain( x ) = 1 x 3 BesselJ 2 ; 2 x3=2 + p3 BesselY 2 ; 2 x3=2 @x 6 3 3 3 3
(152)
lhs(eval(dain,1))[x=1]=evalf(rhs(")&@(x=1));
@
(153) @x ain( x ) x=1 = :0101605672 It is apparently that the derivative of the Airy function as given by Maple is right only of x>0. However, with a little help (with the value at x=0), Maple can di erentiate the alternate piecewise de nition of the Airy function. >
dai:=Diff('ai'(x),x)=subs(undefined=-0.25881940,simplify(diff(ai(x),x)));
@ ai( x ) = piecewise 1; x < 0; @x 1 x 3 BesselJ 2 ; 2 x p x + p3 BesselY 2 ; 2 x p x ; x = 0; 6 3 3 3 3 p ! 3 x BesselK 32 ; 32 x3=2 1 ; 0 = x; :25881940; 0 (154) :25881940; 0 < x; 3 > >
postscript(plot({ai(x),rhs(dai)},x=-10..10), `fig2c2`,`Airy function and its derivative`);
4.2 Schrodinger's equation in free space Recall Schrodinger's equation for a single particle in one dimension in empty space > >
assume(m>0);assume(x,real);assume(t,real); schrod:=i*hbar*diff(psi(x,t),t)=-hbar^2/2/m*diff(psi(x,t),x,x);
@ 2 (2) i h @t ( x; t ) = 21 h (m ( x; t ) ) Berry and Balazs[9] give the following solution > > >
(155)
assume(B,real); airywave:=psi(x,t)= Ai(B/hbar^(2/3)*(x-B^3*t^2/4/m^2))*exp((i*B^3*t/2/m/hbar)*(x-(B^3*t^2/6/m^2)));
0 1B t BB x 4 m ( x; t ) = Ai B @ =
3 2 2
h 2 3
1 CC e = A
1 2
i B3 t x 1=6 Bm3 2t2 m h
(156)
which they state is easily veri ed by direct substitution and use of the Airy function's di erential 41
1
0.5
-5
-10
00
5 x
10
-0.5
-1
Figure 10: Airy function and its derivative equation >
airyde:=diff(A(x),x,x)=A(x)*x;
( 2 ) ( A( x ) ) = A( x ) x >
(157)
dsolve(airyde,A(x));
1 2 1 2 p = A( x ) = C1 x BesselI 3 ; 3 x + C2 x BesselK 3 ; 3 x = p
3 2
3 2
(158)
We can verify that our de nition of ai of the Airy function is indeed a solution of this di erential equation. > >
is(airyde&@(A(x)=aip(x))); is(airyde&@(A(x)=ain(x)));
true
(159)
true (160) and show that the airy wavefunction is a solution of Schrodinger's equation. We substitute A for Ai to avoid Maple's evaluation of the derivative of the Airy function, which isn't necessary in the proof. >
sol:=schrod&@airywave&@(Ai=A);
0 B@ i h B @
@t
0 1B t B x 4 m B B A@ =
3 2 2
h 2 3
1 CC e = A 1
42
i B3 t x 1=6 Bm3 2t2 2 m h
1 CC = A
0 0 1B t B x 4 m B B B B h @A @ =
3 2
2
1 2 >
2
(2)
h 2 3
1
i B3 t x 1=6 Bm3 2t2 2 m h
1 CC A (161)
m
sol&@((D@@2)(A)=(x->A(x)*x));
0 1B t B x 4 m B B D( A ) @ =
3 2
0 i h @ 1
1 CC e = A
2
1 CC B t %1 A 4
h 2 3
2
h 2=3 m2 1 B3 t2 1 1 0 1 B3 t2 1 0 3 B B x 4 m2 CC BB 1 i B x 6 m2 1 i B 6 t2 C CA %1A = 1 + AB @ A 3 @ 2=3 2 m h 6 m h 2 h
0 1B t B x 4 m B B ( x ! A( x ) x ) @ = 0
3 2 2
h
h 2 @
+
2
h 4=3 0 1 B3 t2 1 B B x 4 m2 CC B4 t %1 i D( A ) B @ A h 2=3
%1 := e >
2 3
1 CC B %1 A
h 5=3 m =
1 2
i B3 t x 1=6 Bm3 2t2 m h
0 1B t BB x 4 m AB @ =
3 2 2
1 4
h
2 3
m2 h 2
1 CC B t %1 A 1 A m 6 2
(162)
simplify(");
1 i(2 m t D( A )( %1 ) %2 B 4 nhbar 2 i B 3 %2 A( %1 ) h 2=3 x m2 4 1 2=3 2 6 3 2=3 2 3 + i B %2 A( %1 ) h t ) ( m h ) = 2 m x A(%1 ) %2 B + 41 t2 A( %1 ) %2 B 6 12 i h 1=3 B 4 %2 D( A )( %1 ) t m m3 2 + B 4 t2 %1 := 41 4 B x m m2 h 2=3 %2 := e
>
=
1 12
i B3 t (
6 x m2 +B 3 t2 ) m3
h
(163)
is(");
true To see how the airy wavefunction evolves, we can do an animation. 43
(164)
> > > >
postscript(animate( (abs(psi(x,t))^2)&@airywave&@{B=1,m=1,hbar=1}, x=-5..30,t=0..10,numpoints=200,frames=16), `fig2c3`,`Evolution of the Airy Wavefunction`);
4.3 Bohm's Interpretation of the Airywavefunction Since the Airy function is real >
`conjugate/Ai` := proc(x) Ai(conjugate(x)) end:
we can write >
where >
POLAR;
( x; t ) = R( x; t ) e
i S( x;t )
h
1 B ( 4 x m + B t ) R( x; t ) = Ai 4
3 2
h 2=3 m2
1 B 3 t ( 6 x m2 + B 3 t2 ) S( x; t ) = 12 m3
(167)
trajectory:=expand(dsolve(m*diff(x(t),t)=diff(S(x,t)&@Sairy,x),x(t))); 3 2 x( t ) = 14 Bm2t + C1
> >
(166)
Sairy:=S(x,t)=simplify(hbar/i*ln(psi(x,t)&@airywave/R(x,t)&@Rairy));
Equation of motion >
(165)
Rairy:=R(x,t)=sqrt(simplify(rho(psi(x,t)&@airywave)),symbolic); 2
>
(168)
postscript(plot(x(t)&@trajectory&@{B=1,m=1,_C1=0},t=-10..10), `fig2c4`,`Trajectory quided by an Airy wavefunction`);
We can use the quantum Hamilton Jacobi equation (19) to solve for the quantum potential Q(x,t) >
Qairy:=Q(x,t)=solve(eval(QHJ&@(V(x)=0)&@Sairy),Q(x,t));
3 6 2 Q( x; t ) = 21 Bmx + 18 Bm3t and the constant "quantum" force
>
(169)
diff(Qairy,x); 3 ( Q( x; t ) ) = 12 Bm
44
(170)
-5
-5
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
00
5
10
15
20
25
-5
30
00
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
00
5
10
15
20
25
-5
30
00
5
10
15
20
25
30
5
10
15
20
25
30
Figure 11: Evolution of the Airy Wavefunction
45
25
20
15
10
5
-10
-5
00
5 t
10
Figure 12: Trajectory quided by an Airy wavefunction
4.4 Comments on the Airy Wavefunction Both Berry and Balazs[9] and P. Holland[10] point out that the apparent contradiction with Ehrenfest's theorem which guarantees that the expected value of location and momentum must satisfy classical conservative laws of motion (63) is avoided because the Airy wavefunction is not square integrable - a property which it shares with the well known plane wave solutions to the Schrodinger equation in free space. None the less, such solutions are meaningful and useful in situations involving large numbers of particles, in particular in solid state physics and in scattering theory. The question which arises is: Is it conceivable that we might be able to "engineer" situations in which the Airy wavefunction applies and therefore that would accellerate particles in a non-energy conserving manner?
5 More than one particle Now we are going to start again, this time considering Schrodinger's wave equation for more than one particle. It will turn out that we can derive the quantum Hamiltonian-Jacobi equations of motion and the quantum potential in very much the same way as in the one particle case. But before we get into the mathematics, lets review what we have learned so far. David Albert, in his introductory book on the quantum theory of measurement[12] introduces Bohm's theory in this way: There's an entirely di erent way of trying to understand all this stu [measurement] (a way of being absolutely deviant about it, a way of being polymorphously heretical against the standard way of thinking, a way of tearing quantum mechanics all the way down and replacing it with something else) which was rst hinted at a long time ago by Louis de Broglie (1930), and which was rst developed into a genuine mathematical 46
theory back in the fties by David Bohm (1952), and which has recently been put into a particularly clear and simple and powerful form by John Bell (1982), and that's what this chapter [Chapter 7] is going to be about. Bohm's theory has more or less (but not exactly) the same empirical content as quantum mechanics does, and it has much the same mathematical formalism as quantum mechanics does too, but the metaphysics is di erent. The metaphysics of this theory is exactly the same as the metaphysics of classical mechanics. Here's what I mean: This theory presumes (to begin with) that *every material particle in the world invariably has a perfectly determinant position*. And what this theory is about is the evolution of those postions in time. ... And it turns out that the account which Bohm's theory gives of these motions is *completely deterministic*. And so, on Bohm's theory, the world can only appear to us to evolve probabilistically (and of course it does appear that way to us) in the event that we are somehow ignorant of its exact state. And so the very idea of probability will have to enter into this theory as some kind of epistemic idea, just as it enters into classical statistical mechanics. What the physical word consists of besides particles and besides force elds, on this theory, is (oddly) *wave functions*. That's what the theory requires in order to produce its account of the particle motions. The quantum mechanical wave functions are conceived of in this theory as geniunely physical *things*, as some what like force elds (but not quite), and anyway as something quite distinct from the particles; and the laws of the evolutions of these wave functions are stipulated to be precisely the linear quantum mechanical equations of motion (always, period; wave functions never collapse in this theory); and the *job* of these wave functions in this theory is to sort of *push the particles around* (as force elds do), to guide them along their proper courses; and there are additional laws in the theory (new ones, un-quantum mechanical ones) which stipulate precisely how they do that [the guidance conditions]. Albert continues with his discussion of measurement, the details of which need not concern us at this time. But he concludes with the following statement that is both amusing and enlightening: ... And so if this theory is right (and this is one of the things about it that's cheap and unbeautiful, and that I like), then the fundamental laws of the world are cooked up in such a way as to systematically mislead us about ourselves. Here's what's so cool about this theory: This is the kind of theory whereby you can tell an absolutely low-brow story about the world, the kind of story (that is) that's about the motions of material bodies, the kind of story that contains nothing cryptic and nothing metaphysically novel and nothing ambiguous and nothing inexplicit and nothing evasive and nothing unintelligible and nothing inexact and nothing subtle and in which no questions ever fail to have answers and which no two physical properties of anything are ever "incompatible" with one another and in which the whole universe always evolves *deterministically* and which recounts the unfolding of a perverse and gigantic conspiracy to make the world appear to be quantum mechanical.
47
First, lets recall the de nitions we used for complex symbolic mathematics previously. Please refer to Maple worksheet bohm3a.ms in wohat follows. > > > > > > > > >
restart;
read `save1.m`; alias(I=I,i=sqrt(-1)): read `notilde.txt`; assume(h,real); unprotect(conjugate); conjugate:=expr->eval(subs(i=-i,expr)): re:=expr->expand((expr+conjugate(expr))/2): im:=expr->expand((expr-conjugate(expr))/(2*i)):
We would like to extend the one particle case considered in parts 1 and 2 to the case of systems containing more than one particle. Lets start with the two particle case.
5.1 Two particle, one dimensional Schrodiner equation De ning the one dimensional gradient operators > >
del[1]:= psi -> diff(psi,x1); del[2]:= psi -> diff(psi,x2);
! @@x1
(171)
! @@x2
(172) for particles 1 and 2 respectively, Schrodinger's wave equation for two particles in one dimension can be written > > >
SCHROD2:=i*h*diff(psi(x1,x2,t),t)
+(h^2/(2*m))*(del[1]@@2+del[2]@@2)(psi(x1,x2,t)) -V(x1,x2)*psi(x1,x2,t)=0;
@ ( x1 ; x2 ; t ) i h @t h2 del 1 ( 2 ) ( ( x1 ; x2 ; t ) ) + del 2 ( 2 ) ( ( x1 ; x2 ; t ) ) 1 +2 m V( x1 ; x2 ) ( x1 ; x2 ; t ) = 0 (173) As in the one particle case, this two particle Schrodinger equation can be manipulated into a form which admits an interpretation as a classical Hamilton-Jacobi equation for the dynamics of a particle subject to the usual potential eld plus a new quantum potential eld. Again we de ne the wavefunction psi in polar form and separate the real and complex parts of the wave equation. We shall nd that the imaginary component results in the well recognized conservation of probability equation while the real component will comprise the new quantum HamiltonJacobi equation. >
POLAR2:=psi(x1,x2,t)=R(x1,x2,t)*exp(i*S(x1,x2,t)/h);
i S( x1 ;x2 ;t )
h (174) ( x1 ; x2 ; t ) = R( x1 ; x2 ; t ) e where R and S are real-valued functions. S is called the "phase" of the wavefunction. In polar form,
48
Schrodinger's equation (4.1) becomes >
expand(SCHROD2&@POLAR2);
@
i S( x1 ;x2 ;t ) h
@
i h @t R( x1 ; x2 ; t ) e R( x1 ; x2 ; t ) @t S( x1 ; x2 ; t ) e i S( x1h;x2 ;t ) @2 2 R( x1 ; x2 ; t ) e h @ x1 2 + 21 m i S( x1h;x2 ;t ) @ i h @ x1 R( x1 ; x2 ; t ) @@x1 S( x1 ; x2 ; t ) e + @2 m i S( x1h;x2 ;t ) i h R( x1 ; x2 ; t ) S( x1 ; x2 ; t ) e 2 @ x1 + 21 m i S( x1 ;x2 ;t ) 2 @ h 1 R( x1 ; x2 ; t ) @ x1 S( x1 ; x2 ; t ) e 2 m 2 @ R( x1 ; x2 ; t ) e i S( x1h;x2 ;t ) 2 h @ x2 2 + 21 m i S( x1h;x2 ;t ) @ i h @ x2 R( x1 ; x2 ; t ) @@x2 S( x1 ; x2 ; t ) e + @2 m i S( x1h;x2 ;t ) i h R( x1 ; x2 ; t ) S( x1 ; x2 ; t ) e 2 @ x2 + 21 m i S( x1 ;x2 ;t ) 2 @ h 1 R( x1 ; x2 ; t ) @ x2 S( x1 ; x2 ; t ) e 2 m i S( x1 ;x2 ;t ) h V( x1 ; x2 ) R( x1 ; x2 ; t ) e =0
i S( x1 ;x2 ;t ) h
(175)
5.2 Quantum Hamilton-Jacobi Equation and the Quantum Potential Considering just the real part of the Schrodinger equation (4.1) and dividing through by psi we get >
SCHROD2a:=re((-SCHROD2/psi(x1,x2,t))&@POLAR2);
@
1 S( x1 ; x2 ; t ) +
S( x1 ; x2 ; t ) 2 1 @@x2 S( x1 ; x2 ; t ) 2 +2 @t 2 m m@2 2 @ 2 2 h h R( x1 ; x2 ; t ) R( x1 ; x2 ; t ) 2 2 @ x1 @ x2 1 =0 + V( x1 ; x2 ) 12 m R( x1 ; x2 ; t ) 2 m R( x1 ; x2 ; t )
> >
@ @ x1
(176)
SCHROD42:=diff(S(x1,x2,t),t)+(del[1](S(x1,x2,t)))^2/2/m
@
+(del[2](S(x1,x2,t)))^2/2/m+V(x1,x2)+Q(x1,x2,t)=0;
1 S( x1 ; x2 ; t ) +
S( x1 ; x2 ; t ) 2 1 +2 @t 2 m + V( x1 ; x2 ) + Q( x1 ; x2 ; t ) = 0 where we de ne the quantum potential as: @ @ x1
49
@ @ x2
S( x1 ; x2 ; t ) m
2
(177)
> >
QP2:= Q(x1,x2,t) = -h^2/2/m*(del[1]@@2+del[2]@@2)(R(x1,x2,t)) /R(x1,x2,t);
h2 del 1( 2 ) ( R( x1 ; x2 ; t ) ) + del 2 ( 2 ) ( R( x1 ; x2 ; t ) ) 1 Q( x1 ; x2 ; t ) = 2 m R( x1 ; x2 ; t ) >
>
subs(QP2,SCHROD42);
@
1 S( x1 ; x2 ; t ) +2 @t
S( x1 ; x2 ; t ) 2 1 @@x2 S( x1 ; x2 ; t ) 2 +2 + V( x1 ; x2 ) m m (2) (2) 2 1 h del 1 ( R( x1 ; x2 ; t ) ) + del 2 ( R( x1 ; x2 ; t ) ) = 0 2 m R( x1 ; x2 ; t ) @ @ x1
(178)
(179)
is(lhs(expand(subs(QP2,SCHROD42)))=lhs(SCHROD2a));
true (180) Note that this component strongly resembles that of the classical Hamilton-Jacobi equation for a particle in a potential eld V where the momentum of each particle is taken to be > >
MOMENT1:=p[1]=del[1](S(x1,x2,t)); MOMENT2:=p[2]=del[2](S(x1,x2,t));
p1 = del 1 ( S( x1 ; x2 ; t ) )
(181)
p2 = del 2 ( S( x1 ; x2 ; t ) ) (182) Thus, the real component of the Schrodinger equation is equivalent to a classical description of the system but for the presence of an additional potential eld, the quantum potential. Bohm&Hiley state: "The above is evidently an extension of the guidance relationship for the two-body system. It implies that the particles are guided in a correlated way. ... As can be seen from equation (178), this potential contains R both in the denominator and the numerator, so that it does not necessarily fall o with distance. We have already seen that for the one-body system this means that the [motion of the] particle can depend strongly on distant features of the environment. In the two-body system we can have similar dependence on the environment, but in addition, the two particles can also be strongly coupled at long distances. Their interaction can therefore be described as non-local." They also note that: "While non-locality is described above as an important new feature of the quantum theory, there is yet another new feature that implies an even more radical departure from the classical onotology, to which little attention has generally been paid thus far. This is that the quantum potential Q depends on the 'quantum state' of the whole system in a way that cannot be de ned simply as a pre-assigned interaction between all the particles."
5.3 Probability Conservation and Probability Currents Similary, the imaginary part is >
SCHROD2b:=im((-SCHROD2/psi(x1,x2,t))&@POLAR2);
h
h
@ @t R( x1 ; x2 ; t )
R( x1 ; x2 ; t )
@ @ x1
R( x1 ; x2 ; t ) @@x1 S( x1 ; x2 ; t ) R( x1 ; x2 ; t ) m 50
@2 1 h @ x1 2 S( x1 ; x2 ; t ) h @@x2 R( x1 ; x2 ; t ) @@x2 S( x1 ; x2 ; t ) 2 m R( x1 ; x2 ; t ) m 2 @ 1 h @ x2 2 S( x1 ; x2 ; t ) = 0 2 m This can be manipulated into the form > > >
PROBCONS2:='diff(R(x1,x2,t)^2,t)+ del[1](R(x1,x2,t)^2*del[1](S(x1,x2,t))/m)+ del[2](R(x1,x2,t)^2*del[2](S(x1,x2,t))/m)=0';
R( x1 ; x2 ; t ) del ( S( x1 ; x2 ; t ) ) @t R( x1 ; x2 ; t ) + del R( x1 ; x2 ; t ) del ( S( x1 ; x2 ; t )m)
@
2
+ del 2
2
1
1
2
2 =0 m as can be shown by rst performing the derivatives in this new form
>
(183)
PROBCONS2;
(184)
@
2 R( x1 ; x2 ; t ) @t R( x1 ; x2 ; t ) R( x1 ; x2 ; t ) @@x1 S( x1 ; x2 ; t ) @@x1 R( x1 ; x2 ; t ) +2 m @2 2 R( x1 ; x2 ; t ) @ x1 2 S( x1 ; x2 ; t ) + m @ S( x1 ; x2 ; t ) @ R( x1 ; x2 ; t ) R( x1 ; x2 ; t ) @ x2 @ x2 +2 m R( x1 ; x2 ; t )2
@ x2 S( x1 ; x2 ; t ) + =0 (185) m and on multiplying equation (183) by -2*R/h and observing that the two equations are now equal:
>
@2
2
is(expand(lhs(PROBCONS2))=expand(-2/h*R(x1,x2,t)^2*lhs(SCHROD2b)));
true (186) Lets consider the meaning of this equation. In the usual interpretation, the probability density for the particle is given by >
abs('psi(x1,x2,t)')^2=simplify(rho(psi(x1,x2,t)&@POLAR2));
j ( x1 ; x2 ; t )j2 = R( x1 ; x2 ; t )2
(187) and the probability current (as in the usual treatment of quantum mechanics) is de ned as the sum of two components > > >
j[1]:=(x1,x2,t) -> simplify(h/2/i/m* (conjugate(psi(x1,x2,t)&@POLAR2)*diff(psi(x1,x2,t)&@POLAR2,x1) -psi(x1,x2,t)&@POLAR2*diff(conjugate(psi(x1,x2,t)&@POLAR2),x1)));
1 ( x1 ; x2 ; t ) ! simplify 2 i h(conjugate( ( x1 ; x2 ; t ) &@ POLAR2 )
di ( ( x1 ; x2 ; t ) &@ POLAR2 ; x1 ) ( ( x1 ; x2 ; t ) &@ POLAR2 ) di ( conjugate( ( x1 ; x2 ; t ) &@ POLAR2 ); x1 )) m 51
(188)
> > >
>
j[2]:=(x1,x2,t) -> simplify(h/2/i/m* (conjugate(psi(x1,x2,t)&@POLAR2)*diff(psi(x1,x2,t)&@POLAR2,x2) -psi(x1,x2,t)&@POLAR2*diff(conjugate(psi(x1,x2,t))&@POLAR2,x2)));
( x1 ; x2 ; t ) ! simplify 21 i h(conjugate( ( x1 ; x2 ; t ) &@ POLAR2 ) di ( ( x1 ; x2 ; t ) &@ POLAR2 ; x2 ) ( ( x1 ; x2 ; t ) &@ POLAR2 ) di ( conjugate( ( x1 ; x2 ; t ) ) &@ POLAR2 ; x2 )) m j[1](x1,x2,t)+j[2](x1,x2,t);
R( x1 ; x2 ; t )2 @@x1 S( x1 ; x2 ; t ) 1 + 2i m i S( x1 ;x2 ;t ) h @ @x2 R( x1 ; x2 ; t ) R( x1 ; x2 ; t ) e2 h h @@x2 R( x1 ; x2 ; t ) R( x1 ; x2 ; t ) i S( x1 ;x2;t ) + i R( x1 ; x2 ; t )2 @@x2 S( x1 ; x2 ; t ) e 2 h i R( x1 ; x2 ; t )2 @@x2 S( x1 ; x2 ; t ) m so that equation (4.4) can be written > > > >
(189)
(190)
'diff(R(x1,x2,t)^2,t)+ del[1](j[1](x1,x2,t))+del[2](j[2](x1,x2,t))=0'; 'diff(R(x,t)^2,t)'+ 'del[1]'(j[1](x1,x2,t))+'del[2]'(j[2](x1,x2,t))=0;
@
2 @t R( x1 ; x2 ; t ) + del 1 (j1( x1 ; x2 ; t )) + del 2 (j2 ( x1 ; x2 ; t )) = 0
@
(191)
!
R( x1 ; x2 ; t )2 @@x1 S( x1 ; x2 ; t ) 2 R( x; t ) + del 1 @t m ! R( x1 ; x2 ; t )2 @@x2 S( x1 ; x2 ; t ) =0 (192) + del 2 m Therefore the imaginary part of the polar Schrodinger equation, that is equation (4.4), is just the well known continuity equation that expresses the conservation of probability. Bohm&Hiley note that: "As in the one-body case, we may take P = R2 as the probability density, but this is now in the con guration space of all the particles."
5.4 The Two Particle Equation of Motion and Guidance Conditions From the Quantum Hamilton-Jacobi equation, the equation of motion of a paticles with velocity v[1](t) and v[2](t) is seen to be given by > >
m*diff(v[1](t),t)=-del[1](V(x1,x2))-del[1](Q(x1,x2,t)); m*diff(v[2](t),t)=-del[2](V(x1,x2))-del[2](Q(x1,x2,t));
52
@
@ @ m @t v ( t ) = @ x1 V( x1 ; x2 ) @ x1 Q( x1 ; x2 ; t ) 1
(193)
@ @ v2 ( t ) = @@x2 V( x1 ; x2 ) Q( x1 ; x2 ; t ) (194) m @t @ x2 Also, from the de nition of momentum in terms of the phase of the wavefunction (181,182) and its relationship to the probability current we also have the following relationship, known as the quidance conditions: > > > >
v[1](t)='j[1](x1,x2,t)'/abs('psi(x1,x2,t)')^2;
simplify(j[1](x1,x2,t)/(psi(x1,x2,t)*conjugate(psi(x1,x2,t)))); v[2](t)='j[2](x1,x2,t)'/abs('psi(x1,x2,t)')^2; simplify(j[2](x1,x2,t)/(psi(x1,x2,t)*conjugate(psi(x1,x2,t))));
v1 ( t ) = j1( x1 ; x2 ; t )2 j ( x1 ; x2 ; t )j @ @ x1
S( x1 ; x2 ; t ) m
v2 ( t ) = j2( x1 ; x2 ; t )2 j ( x1 ; x2 ; t )j
(195) (196) (197)
S( x1 ; x2 ; t ) (198) m In reference to an example of the motion of an electron in a hydrogen atom, Bohm&Hiley state: @ @ x2
The relationship between the parts of a system described above implies a new quality of wholeness of the entire system going beyond anything that can be speci ed solely in terms of the actual spatial relationships of all the particles. This is indeed the feature which makes the quantum theory go beyond mechanism of any kind. For it is the essence of mechanism to say that basic reality consists of the parts of a system which are in a preassigned interaction. The concept of the whole, then, has only a secondary signi cance, in the sense that it is only a way of looking at certain overall aspects of what is in reality the behaviour of the parts. In our interpretation of the quantum theory, we see that the interaction of parts is determined by something that cannot be described solely in terms of these parts and their preassigned relationships. Rather it depends on the many-body wave function (which in the usual interpretation, is said to determine the quantum state of the system). This many-body wave function evolves according to Schrodinger's equation. Something with this kind of dynamical signi cance that refers directly to the whole system is thus playing a key role in the theory. We emphasise that *this is the most fundamentally new aspect* of the quantum theory.
5.5 Many-body Equations It is clear from the above that the N-particle case will be a natural extension the two particle equations: >
del[n]=Diff(psi,x.n);
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> > > > >
i*h*Diff(psi,t)=(-h^2/2/m*sum(del[n]@@2,n=1..N)(psi))+V*psi; j[n]=h/2/i/m*('conjugate'(psi)*Diff(psi,x.n)psi*Diff('conjugate'(psi),x.n)); m*diff(v[n](t),t)=-del[n](V)-del[n](Q); v[n](t)='j[n]'/abs('psi')^2;
del n = @@xn
@
i h @t
= 12
ih jn = 21
@
h2
N X n=1 @ @ xn
(199)
!
del n ( 2 ) ( )
m
m
@ @ xn
+V
(200) (201)
m @t vn ( t ) = del n ( V ) del n ( Q )
(202)
vn ( t ) = jn2
(203)
j j
References [1] K. Berndl, et. al., A Survey of Bohmian Mechanics, Il Nuovo Cimento (1995). [2] D. Bohm and B.J. Hiley, The Undivided Universe: An Ontological Interpretation of Quantum Theory, Routledge, London (1993). [3] D. Bohm, A Suggested Interpretation of the Quantum Theory in Terms of "hidden" Variables, Physical Review, vol. 85 (1952), Part 1: page 166 and Part 2: page 180. [4] L. de Broglie Tentative d'Interpretation Causale et Non-lineaire de la Mecanique Ondulatorire, Gathier-Villars, Paris (1956). English translation: Nonlinear Wave Mechanics, Elsevier, Amsterdam (1960). [5] J.S. Bell, Speakable and unspeakable in quantum mechanics. Cambridge University Press (1987), page 159. [6] James M. Feagin, Quantum Methods with Mathematica, Springer-Verlag (1994). [7] David Cyganski, Worchester Polytechnique Institute, Ma. Personal communications and various postings to sci.physics.fusion newsgroup (1994). [8] W. Press, et.al., Numerical Recipes in C (The Art of Scienti c Computing), 2nd edition, Cambridge University Press (1992), page 250. [9] M.V. Berry and N. L. Balazs, Nonspreading Wave Packets, Am. J. Physics, vol. 47(3), page 264, Mar. 1979. 54
[10] P. Holland, The Quantum Theory of Motion (An account of the de Broglie-Bohm causal interpretation of quantum mechanics), paperback ed., Cambridge University Press (1995), page 167. [11] Amnon Yariv, Quantum Electronics, page 13. [12] David Z. Albert, Quantum Mechanics and Experience, Harvard University Press (1992)
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