St. Joseph’s College of Arts & Science (Autonomous) St. Joseph’s College Road, Cuddalore – 607001 PCH703S - QUANTUM MECHANICS AND MOLECULAR STRUCTURE

Time : 3 hrs

Max Marks :75

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SECTION – A (20X1=20) Answer ALL Questions I. Choose the correct answer 1. The minimum uncertainty in the speed of an electron in a 1-D region of length 2a 0 is a) 100 kms -1 b) 500 kms -1 c) 1000 Kms -1 d) 5000 kms -1 R

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2. The temperature of the sun’s surface is 5900K. Assuming that sun to a black body, λ max of solar radiation is R

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a) 480 A

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b) 480nm

c) 148nm

d) 148 A

3. The fine-structure constant, α, plays a special role in the structure of matter; its approximate value is 1/137. What is the wavelength of an electron travelling at a speed αc, where c is the speed of light? (Note that the circumference of the first Bohr orbit in the hydrogen atom is 331 pm.) a) 3.36m b) 3.36 pm c) 3.36 nm d) 3.36Å 4. Which among the following is an acceptable function 2 i) ψ = x ii) ψ = x iii) ψ = Sin x iv) ψ = e − x a) (i) & (ii) b) (iii) c) (iv) d) (iii) & (iv) 5. A particle in a ring is in a state with Ψ = 1/ π cos 2φ . The average value for the angular momentum (ρφ) where the operator ˆ = h / 2π id / d φ is ρφ a) h/2π b) 0 c) –h/2πi d) 1

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Q7/15C/11-11 6. The wavefunction, Ψ(Φ), for the motion of a particle in a ring is of the form Ψ=Ne im/Φ. Determine the normalization constant, N. b) 2/√2Π b) 1/√2Π c) 5/√4Π d) 3/√2Π P

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7. Which among the following do have zero point energy? i) Particle in 1-D box ii) Circular harmonics iii) Harmonic oscillator iv) Hydrogen a)(ii) only b) (i),(ii) & (iv) c) (i),(iii)& (iv) d) (ii),(iii)& (iv) 8. The vibrational frequency of C-H bond is 2200cm -1 and that of C-D bond is _________cm -1 . a) 1555.8 b) 4400 c) 3110.8 d) 2200 P

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9. The term for any closed shell electronic configuration is a) 1 S 0 b) 3 S 1 c) 3 P 2 d) 3 P 1 P

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10. In general ∫ψ *ψ dτ , what are the dimensios of ψ a) unit less

b) m 3 P

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c) m -3/2 P

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d) m +1/2 P

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II. Fill up the blanks 11. The degeneracy of a particle in cubic box that has an energy 14/3 times that of the lowest level is _________. 12. The Hamiltonian operator for He atom is __________. 13. According to Born-Oppenheimer approximation, the complete form of electronic schrodinger equation is __________. 14. The expression for the slater type orbitals for 2s electron in nitrogen is ________. 15. The general formula to find out the number of radial node is _____.

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Q7/15C/11-11 III. Match the following I

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16. [L 2 ,L z ]

III 2

non-zero

LZ /2I

17. ∫ ψ 1( − e ) xψ 2 dT

Kinetic energy

( e -zr/a 0 )

18. ω2 I 2 /2I

-2(z/ a 0 ) 3/2

19. R 10

27.21eV

Allowed transition

0

Specified simultaneous

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+∞

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20. E=-1/2 a.u

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1Hartree

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SECTION – B (10X2=20) Answer any TEN Questions 21. Calculate the zero-point energy of a harmonic oscillator consisting of a particle of mass 5.16x10 -26 kg and force constant 285 Nm -1 . P

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22. Calculate the wavelength of a photon needed to excite a transition between neighbouring energy levels of a harmonic oscillator of effective mass equal to that of an oxygen atom (15.9949) and force constant 544 cm -1 . P

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23. Determine the normalization constant N, the wave function for the 2s orbital of a hydrogen atom is N(2-r/a 0 )e -r/2a 0 . R

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24. Calculate the density function for 2p Z orbital. R

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25. For particle in 1-D box, the reasonable guess is Ψ=Nx (L-x), Calculate the expected ground state average energy.

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Q7/15C/11-11 26. A hydrogen atom in exposed to an electric field of strength F applied in the αdirection. Calculate the first order and the second order effects for the ground state of the atom. 27. An electron moving in a simple harmonic potential V=1/2Kx 2 is P

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subjected to a perturbation H = Ex where E is the strength of electric field which is applied in x-direction. Determine the effect of first order perturbation on the energy. 28. Which of the following four functions are symmetric? (i) ϕ(1)α(1)ϕ(2)α(2) (ii) ϕ(1)α(1)ϕ(2)β(2) (iii) ϕ(1) β (1)ϕ(2)β(2) (iv) ϕ(1) α (2)ϕ(2)β(1) 29. Verify by the use of determinantal wave function that for Li atom a configuration like 1s 3 is impossible. P

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30. Calculate the first ionisation potentials for Be on the basis of slater’s rule. 31. Determine the term symbols for L=2, S=1/2 32. Find the expression for the expectation value of the Hamiltonian for H 2 in terms of atomic integrals. R

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SECTION - C (5X7=35) Answer any FIVE Questions 33. a) The size of the nucleus is 10 -12 cm. Treating it as 1-D box show why electron does not exist in the nucleus. (3) b) Show that the probability of finding the particle in 1D box in the region L/4 and 3L/4 is ½ if n is even. (2) c) For an electron in 3d rectangular box of dimension L x =1x10 -15 m, L y =1.5x10 -15 m and L z =2x10 -15 m, calculate the zero point energy. (2) P

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Q7/15C/11-11 34. a) Write the Eigen value of the harmonic oscillator for the second excited state (1) b) The Eigen function corresponding to this Eigen value may be 2 written as Ψ 2 = N ( x2 + bx + c)e − β x /2 Where Nis normalization factor and β=2Π(mk) 1/2 /h Knowing that this function must be even (symmetric) and that it is orthogonal to the ground state eigen function, determine the value of b and c. (1) c) Determine the normalization constant. (2) P

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d) Show that the resulting function is an eigenfunction of H and calculate the eigen value. (3) 35. a) Show that Y 1,1 (θ,ϕ) and Y 2,0 (θ,ϕ) are orthogonal. b) Find out the radial function for 3s, 3p orbitals. c) Normalize the following Ψ = re -r/2 cosθ R

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(3) (3) (1)

36. Take a trial function for Helium atom as, Ψ = ϕ(1) ϕ(2) with ϕ(1) = (z’ 3 /Π) 1/2 e -z’r 1 ϕ(2) = (z’ 3 /Π) 1/2 e -z’r 2 a) Find the best value of Z’ (2) b) Obtain the ground state energy for the corresponding to the given functionΨ. (3) c) Calculate the first and second ionization potentials. (2) P

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37. a) Calculate the most probable distance r m electron in the 2p state of H-atom (2) b) Calculate the average value of x for the 1s electron in H-atom. (2) c) Make a reasonable guess at the ground state wave function for H-atom. Use variation method to calculate the energy and normalized wave function. Show that the function is an Eigen function of Hamiltonian operator. (3) R

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Q7/15C/11-11 38. a) Construct an antisymmetric wave function for Be atom i) As a linear combination of products of spin orbitals. (1) ii) As a slater determinant (1) b) Calculate the effective nuclear charge for the 2s electron in Li atom, given that the first ionization potential is 5.4eV. (2) c) For a one electron homonuclear diatomic molecule the values of some relevant integrals are given below ^

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∫ϕA H ϕA dT=-2a.u., ∫ϕB H ϕB dT=-2a.u., R

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∫ϕA H ϕB dT=-1a.u., ∫ϕA ϕB dT= 0.25a.u., Where ϕA and ϕB are the normalised set of basis function for an LCAO wave function. Find out the upper bound for the electron (3) energy. R

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39. a) How many microstates exist for d 3 configuration? (1) b) Determine all possible term symbols for this configuration. (4) c) Show that there can be no term symbol like 3 D 0 . (2) P

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