Quantum Mechanics: Note 10

The Holy Grail: Quantum Gravity The microscopic world obeys the laws of quantum mechanics. The macroscopic world obeys the laws of Einstein’s gravity. These laws seem to be incompatible. Finding a way to unify them into a deeper theory of “quantum gravity” is the greatest challenge facing theoretical physicists today. In this activity, you will review what we have learned over this course by conducting a simple thought experiment that uses ideas from both quantum theory and Einstein’s theory of gravity. By combining ideas from both theories, we will discover a profoundly important information limit, which shows how bizarre a “quantum gravity” understanding of our universe might turn out to be. Thinking Like a Physicist: Quantum Gravity You will focus on the question posed in class: “How much information can we cram into a ball of radius R, using photons to encode the bits of information?” The microscopic world obeys the laws of quantum mechanics. The macroscopic world obeys the of Einstein’s gravity. These two laws seem to be incompatible. Finding a way to unify them into laws deeper theory of “quantum gravity” is the greatest challenge facing theoretical physicists today. a Part 1 – Quantum Fact: More Information Means More Energy In this group activity you will review what we learned today by conducting a simple thought 1. From the photoelectric effect, we learned that a light wave of frequency f is experiment that uses ideas from both quantum theory and Einstein’s theory of gravity. By composed of a shower of quantum particles called photons, where each photon combining ideas from both theories we will discover a profoundly important information limit, has energy E = hf. which shows how bizarre a “quantum gravity” understanding of our universe might turn out to be. a. Using the universal wave equation, v = fλ, express the energy of a photon You will focus on the question posed in class: “How much information can we cram into a ball of (E = hf) in terms of their wavelength radius R, using photons to encode the bits of information?” Part 1—Quantum Fact: More Information Means More Energy 1. In the photoelectric effect lab we learned that a light wave of frequency f is composed of a b. What happens to the energy of the photons as the wavelength is shower of quantum particles called photons, where each photon has energy E = hf. increased? If we want to minimize the amount of energy each bit of a. Using the universal wave equation, f = c/21, express the energy of the photons in terms information carries with it into the ball, should we use short or long of their wavelength, )i. wavelength photons? b. What happens to the energy of the photons as the wavelength is increased? If we want to minimize the amount of energy each bit of information carries with it into the ball, should we use short or long wavelength photons? When electrons are confined to small spaces (such as a ball), they will behave 2. 2. In the quantum dots lab we learned that electrons confined to a nano-bead behave like standing waves,like standing waves, with only certain wavelengths being allowed: with only certain wavelengths being allowed.

n a.

According to quantum mechanics, photons confined to our ball will behave the same way as electrons confined to a nano-bead. Thus estimate the maximum wavelength a photon can have and still “fit” into our ball of radius R.

b. Put this result together with your answer to Question 1(a) to estimate the minimum amount of energy each bit of information will necessarily carry with it into the ball.

Quantum Mechanics: Note 10



3.

a. According to quantum mechanics, photons confined to this space will behave the exact same way as electrons will. Estimate the maximum wavelength a photon can have and still “fit” into our ball of radius R.

b. Put this result together with your answer to Question 1(a) to estimate the minimum amount of energy each bit of information will necessarily carry with it into the ball.

Final result; if 2b. describes the energy of one bit of information, say we cram N bits of information into the ball, what does the quantum nature of the universe tell us is the minimum amount of energy that will be added to the ball?

Part 2 – Gravity Fact: Enough Energy Will Create a Black Hole 1. Using Einstein’s idea of the equivalence between energy and mass, E = mc2, and your final result from Part 1, estimate the minimum amount of mass the ball will have after we have crammed it full of N bits of information. (Yes, a ball of light has inertial mass!) 2. In the special relativity part of the course, we discussed Einstein’s discovery that gravity is not a force, but a warping of spacetime, and that this warping is caused by mass. If we start with an imaginary ball of radius R in empty space, and keep filling it with more and more mass, the warping of spacetime will eventually become severe enough to create a black hole. a. First, we need to create an expression for the mass of a black hole. Essentially, a black hole has warped spacetime so severely that photons are not even able to escape. Remembering the equation derived for escape velocity, we can determine the mass of a black hole (if the speed to escape is equal to c). 2Gm vesc = r

Quantum Mechanics: Note 10



b. Equate this critical mass (black hole mass) with your mass from Question 1 to estimate the maximum number of bits of information that can be crammed inside a ball of radius R before a black hole forms. This is the INFORMATION LIMIT. Adding more information will make the black hole bigger, so the information no longer fits inside a ball of radius R.

c. Up to small numerical factors like π, Stephen Hawking and Jacob Berkenstein showed that the amount of information that could be crammed into a ball of radius R can be described by the following equation: 1A N= 2 4 lp where A = 4πR2 is the area of the ball, and l p = hG / 2π c 3 ≈ 10 −35 m is



called the Planck length. It is believed that ideas from this still elusive theory of quantum gravity will be crucial to understand how the universe works on the ultra-microscopic scale of the Planck length.

d. Estimate the number of bits of information it is theoretically possible to cram inside a ball with a surface area of 1m2. With present technology, estimate how many terabyte drives could be fit inside the same ball. How many bits of information does this represent? By how many orders of magnitude (powers of 10) must our technology improve to reach the theoretical information limit?

The information limit depends on the surface area of the ball, as opposed to its volume. This bizarre fact has led to an important new idea called the Holographic Principle.