2 Quantum Mechanics. 2.1 The Strange Lives of Electrons
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1 2 Quantum Mechanics A philosopher once said, It is necessary for the very existence of science that the same conditions always produce the same results. Well, they don t! Richard Feynman Today, we re going to talk about quantum mechanics the physics of the very small. The importance of quantum mechanics cannot be overstated. It explains the structure of atoms, describes how elementary particles interact (Lecture 5) and even may provide the ultimate origin of all structure in the universe (Lecture 8). And yet it s so mind-boggling strange and bizarre that it s often very difficult to come to terms with. To the point where it suggests that, ultimately, our brains may not be best equipped to understand what s going on! On the other hand, if we just shut up and calculate, quantum mechanics simply works. In fact, it is arguably the most successful scientific theory ever developed. 2.1 The Strange Lives of Electrons Electrons (and other elementary particles) behave like nothing you have ever seen before. This has recently been demonstrated in the following experiment: electrons are fired at a screen with two slits. In a world governed by classical (i.e. non-quantum) mechanics, there is a clear expectation for what should happen: electrons should behave like bullets. The bullets go through either slit A or slit B and we expect them just to pile up behind the two slits: In particular, we expect the pattern simply to be the sum of the patterns for each slit considered separately: if half the bullets were fired with only the left slit open and then half were fired with just the right slit open, the result would be the same. 13
2 14 2. Quantum Mechanics Let s compare this to the actual experiment. 1 The electrons, like bullets, strike the target one at a time. We slowly see a pattern building up yet, it is not the pattern that we would expect if electrons behaved like bullets: Instead the pattern we get for electrons looks more like the pattern we would get for waves: With waves, if the slits were opened one at a time, the pattern would resemble that for particles: two distinct peaks. But when the two slits are open at the same time, the waves pass through both slits at once and interfere with each other: where they are in phase they reinforce each other; where they are out of phase they cancel each other out. Electrons seem to do the same. However, if each electron passes individually through one slit, with what does it interfere? Although each electron arrives at the target at a single place and time, it seems that each has passed through both slits at once. Remember that in Lecture 1 we claimed that electrons sniff out all possible paths. In the double slit experiment we see this in action. 1
3 2.1 The Strange Lives of Electrons 15 This behaviour of electrons determines the structure of atoms: In classical physics, we have a simple (and wrong!) mental picture of the hydrogen atom: proton electron The electric force keeps an electron in orbit around a proton, just like the gravitational force keeps the Earth in orbit around the Sun. According to this view of the atom, the electron follows a single classical path with no restriction on the size or eccentricity of the orbit (which only depend on the initial conditions). In quantum mechanics, the answer is very different. The electron sniffs out many possible orbits and its position smears out into quantum orbits : Moreover, there are only a finite number of allowed quantum orbits, with specific energies. The orbits and their energies are quantized. In this way, quantum mechanics explains the periodic table of elements and therefore underlies all of chemistry. In this lecture, I will give you a glimpse of the quantum world.
4 16 2. Quantum Mechanics 2.2 Principles of Quantum Mechanics Quantum mechanics doesn t use any fancy mathematics. Everything that we will need was described in your course on Vectors and Matrices. In this section, we will use some of that knowledge to introduce the basic postulates of quantum mechanics. We don t have time to waste, so let s jump right in: Superposition of States The state of a system consists of all the information that we have to specify about a system so that we know its future for all times. For example, in classical mechanics the state of a system is given by the positions and momenta of all particles. Now, in quantum mechanics, the state of a system is given by a vector Ψ. Often this is a vector in an infinite dimensional vector space known as a Hilbert space. Usually Ψ is a complex vector and sometimes Ψ is called the wavefunction. Since states are vectors, we can add and subtract them (something which makes no sense in classical mechanics). In other words, a quantum mechanical system can be in a superposition of states. You have probably heard of the famous example of a cat that is both dead and alive, Ψ = dead + alive. The possibility of a superposition of states is the origin of all the weirdness of quantum mechanics Measurements So, states are vectors. What about matrices? In quantum mechanics, observables are given by (Hermitian) matrices X. We have a different matrix X for each kind of measurement we can make on a system: energy, position, momentum, spin, etc. Moreover, the possible outcomes of a measurement are the eigenvalues x i of the matrix X. Since the measurement matrices are Hermitian, the eigenvalues are always real. There are certain states for which you always get a definite answer for the observable X: The measurement of X leads to definite values if the states are eigenvectors Ψ i of X, i.e. X Ψ i = x i Ψ i. (2.2.1) In other words, if the system is prepared in an eigenvector of X, you will always measure the corresponding eigenvalue x i, with certainty! This is important, so let me say it again: Although we will see that quantum mechanics is all about probabilities and uncertainties, if the system is prepared in an eigenvector of a particular observable and you measure that observable, you will always just get the eigenvalue.
5 2.2 Principles of Quantum Mechanics Structure of Atoms A special matrix X = H (called the Hamiltonian) represents a measurement of energy. The possible outcomes of the corresponding measurement are the energy eigenvalues E i, and eq. (2.2.1) becomes the famous Schrödinger equation, H Ψ i = E i Ψ i. (2.2.2) For a given system, you need to figure out the matrix H, then solve its eigenvalues. This is usually hard! Sometimes, like for the hydrogen atom it can be done exactly. Most of the time (e.g. for all other atoms) it is done in some approximation scheme. Solving the eigenvalue equation (2.2.2) explains the discrete energy levels of atoms Probabilities We have seen that if the system is in an eigenstate Ψ i of X, then we always measure the corresponding eigenvalue x i. However, if the state is not an eigenvector of the observable X, then the outcomes of measurements of X will be probabilistic. The measurement could give any one of the eigenvalues x i. Each with some probability. We can expand any arbitrary state Ψ in a basis of eigenvectors of X, Ψ = i c i Ψ i, (2.2.3) where c i are complex constants, that we are free to normalize as i c i 2 1. The probability of the measurement giving the eigenvalue x i then is Prob(x i )= c i 2, (2.2.4) i.e. the probability is simply the square of the expansion coefficient c i. If the eigenvalue x i has been measured, then the state of the system after the measurement is the corresponding eigenvector Ψ i. If we now were to repeat the measurement we would get the same value x i with certainty. This is called the collapse of the state-vector (or collapse of the wavefunction). If we next performed a measurement of a different observable corresponding to a matrix Y then the outcome will again be probabilistic unless Ψ i is also an eigenvector of Y Heisenberg Uncertainty This now has an important implication. Most matrices have different eigenvectors. (This is the case if the matrices don t commute, i.e. XY = YX.) So if you re in a state that is an eigenvector of one matrix, it is unlikely to be an eigenvector of a different matrix. If one type of measurement is certain, another type is uncertain. This is Heisenberg s uncertainty relation. For example, if we know the position of a particle, its momentum becomes uncertain. And vice versa. Similar uncertainty relations exist for other observables (such as energy and time).
6 18 2. Quantum Mechanics 2.3 Quantum Mechanics in Your Face Let s look at one of the most bewildering aspects of quantum mechanics.2 It is the fact that strange correlations can exist between causally disconnected experiments. This is called entanglement. The punchline will be that the universe is a much stranger place than you might have imagined GHZ Experiment Consider three scientists: They are sent to three different labs. Every minute, they receive a package sent from a mysterious central station: 1 C 3 2 Each scientist has a machine that performs measurements of the packages. The machine has two settings, X or Y, and each measurement can give two outcomes, +1 and X Y 2 This section is based on a famous lecture by Sidney Coleman ( which itself was based on Greenberger, Horne, and Zeilinger (GHZ) (1990) Bell s theorem without inequalities, Am. J. Phys. 58 (12): 1131, and Mermin (1990) Quantum mysteries revisited Am. J. Phys. 58 (8):
7 2.3 Quantum Mechanics in Your Face 19 The scientists are told what they have to do: 3 1. Choose the setting X or Y on the machine. 2. Insert the package into the machine. 3. Make a measurement. 4. Record whether the result is +1 or Go back to Step 1. Each measurement is recorded until each scientist has a list that looks like this: X X Y X Y Y X Y X X X Y After they each have made a bazillion measurements, the scientists get together and start looking for correlations in their measurements. (Since their packages came from the same source, it isn t unreasonable to expect some correlations.) They notice the following: Whenever one of them measured X, and the other two measured Y, theresultsalways multiply to +1, i.e. 4 X 1 Y 2 Y 3 = Y 1 X 2 Y 3 = Y 1 Y 2 X 3 =+1. (2.3.5) Maybe this occurred because all three got the result +1; or perhaps one got +1 and the other two got 1. Since the central station doesn t know ahead of time which setting (X or Y) each scientist will choose, it has to prepare each package with definite states for both property X and property Y. The observed correlation in eq. (2.3.5) is consistent with only 8 different shipments from the central station: = X 1 Y 1 X 2 Y 2 X 3 Y 3 = (2.3.6) Now, notice that eq. (2.3.5) gives a prediction... if all three scientists measure X, theresults multiplied together must give +1. You can see this simply by multiplying the entries of the first columns in the matrices in (2.3.6). Alternative, we can prove the result using nothing but simple arithmetic: (X 1 Y 2 Y 3 )(Y 1 X 2 Y 3 )(Y 1 Y 2 X 3 )=X 1 X 2 X 3 (Y 1 Y 2 Y 3 ) 2 = X 1 X 2 X 3 =+1. (2.3.7) In the first equality we used that the product is associative, while the second equality follows from (±1) 2 = +1. The final equality is a consequence of (2.3.5). 3 The scientists are not told what s in the packages: They could be blood samples, with the machine testing for high/low glucose when the switch is on X, and high/low cholesterol when the switch is on Y. They could be elementary particles. Or, the whole thing could just be a hoax with the machine flashing up +1/ 1 at random. 4 Here, the notation X 1Y 2Y 3 means that scientist 1 measured X and scientists 2 and 3 measured Y.
8 20 2. Quantum Mechanics Completely Nuts, But True! The GHZ experiment has been done 5. The things measured were the spins of elementary particles. Here is the astonishing truth: The observed correlations are X 1 Y 2 Y 3 = Y 1 X 2 Y 3 = Y 1 Y 2 X 3 = +1 (2.3.8) and X 1 X 2 X 3 = 1. (2.3.9) The prediction (2.3.7), based on a basic (classical) intuition for how the universe works, is wrong! Desperate Times, Desperate Measures? What could possibly explain this strange result? An implicit assumption is that the measurements are performed independently, so that experiment 2 has no way of knowing whether the switch on experiment 1 is set to X or Y. We would think that this is guaranteed to be true, since the scientists are at space-like separated points: t x C To communicate between the scientists and conspire to give the strange correlations would require information to be transmitted faster than light! Desperate times, call for desperate measures. But are we desperate enough to give up relativity? Quantum Reality We assumed that the packages leaving the central station had definite assignments for the quantities X and Y we listed all possibilities in eq. (2.3.6). But in the quantum world, we cannot give definite assignments to all possible measurements, but have to allow for the possibility of a superposition of states and experimental outcomes being probabilistic. It turns out that this special feature of quantum mechanics resolves the puzzle. More concretely, the three scientists were, in fact, measuring the intrinsic spins of particles. In this case, the measurement matrices are X = and Y = 0 i i 0. (2.3.10) 5 Pan et al. (2000), Experimental test of quantum non-locality in three-photon GHZ entanglement Nature 403 (6769)
9 2.3 Quantum Mechanics in Your Face 21 You can check that these matrices both have eigenvalues +1 and 1, corresponding to the measurements. (But, X and Y do not have the same eigenvectors.) Now, define two special state vectors corresponding to a particle spinning up or down (relative to say the z-axis), 1 0 and 0 1. (2.3.11) These states are not eigenstates of X and Y. It is easy to see that acting with the matrix X on the up-state turns it into a down-state and vice versa X = and X =. (2.3.12) Similarly, acting with the matrix Y also exchanges up- and down-states (up to factors of i and i) Y = i and Y = i. (2.3.13) Here is the state that the central station was actually sending out: Ψ =. (2.3.14) This corresponds to the superposition of two states: a state with all spins up, and a state with all spins down. I am using a notation where the arrows are ordered: the first arrow corresponds to the spin of the first particle, the second arrow corresponds to the spin of the second particle, etc. The measurement matrix X 1 therefore acts on the first arrow of each state, X 2 one the second arrow, etc. The state in (2.3.14) is an eigenvector of X 1 Y 2 Y 3 and Y 1 X 2 Y 3 and Y 1 Y 2 X 3. And, importantly, it is also an eigenvector of X 1 X 2 X 3. Let us check that this gives rise to the observed correlations: For instance, X 1 Y 2 Y 3 Ψ = X 1 Y 2 Y 3 = (1)(i)(i) (1)( i)( i) = + =+1 Ψ. (2.3.15)
10 22 2. Quantum Mechanics Similarly, we can show that Y 1 X 2 Y 3 Ψ = Y 1 Y 2 X 3 Ψ =+1 Ψ. (2.3.16) So, whenever exactly one scientist measures X, the results indeed multiply to give +1. However, when all three scientists measure X, we get X 1 X 2 X 3 Ψ = X 1 X 2 X 3 = (1)(1)(1) (1)(1)(1) = = 1 Ψ. (2.3.17) The classical expectation, eq. (2.3.7), is wrong. But, it makes total sense in the quantum world. Notice that it was important that the spin states of the three particles weren t independent, but that they were entangled in the state (2.3.14) a superposition of all spins up and all spins down. No matter how far the scientists are separated in space, this entanglement of spin states is reflected in their measurements. In the next section, we will discuss another dramatic consequence of entanglement. 2.4 Beam Me Up, Scotty Just when you thought it couldn t get any weirder let me tell you that entanglement implies that teleportation à la Star Trek is possible in the quantum world Heisenberg and No-Cloning At first, it seems that teleportation is impossible precisely because of quantum mechanics. To see this, consider what Scotty has to do to beam you from A to B:
11 2.4 Beam Me Up, Scotty At A, he determines the states of all atoms in your body. 2. He sends that information to B. 3. At B, this information is used to reassemble you. But, in quantum mechanics Step 1 is impossible! The Heisenberg uncertainty relation tells us that we can never have complete information about the state of a system. Moreover, each measurement can change the state. The Star Trek producers knew about this, so they invented the Heisenberg compensator. When Michael Okuba, the technical advisor of the Star Trek series, was once asked by Time magazine, And how does the Heisenberg compensator work? He replied, It works very well, thank you Entanglement Quantum mechanics itself comes to the rescue. Entanglement makes teleportation possible. As we have seen, entangled particles are intimately connected with each other. When one is measured, the state of the others is instantly influenced, no matter how far they are separated. Let me explain this a bit more. Consider a particle without spin decaying into two particles with spin. The two particles are then separated by a large distance (as large as you wish). When you measure the spin of each particle there is a certain probability that the result will be spin-up, and a certain probability that it will be spin-down. However, once the spin of one of the particles is measured, the spin of the second particle is determined instantly. For instance, if the first particle is spin-up, the second has to be spin-down, so that the total spin of the system remains zero. As an even simpler, if not real, example, think about two entangled dice. When we throw the dice, they each produce random numbers. However, once the first die is observed to show, the second die also has to show. This is the rule of entanglement for quantum dice.
12 24 2. Quantum Mechanics Quantum Teleportation The basic idea of quantum teleportation (or entanglement-assisted teleportation) is quite simple. I will sketch an experiment that has recently been carried out on the Canarian Islands: teleported X Tenerife BOB B EPR A ALICE La Palma X original A central station (EPR) produces a pair of entangled photons. Photon A is sent to Alice on La Palma; photon B is sent to Bob on Tenerife. The entanglement means that any measurement that Alice performs on A instantly determines the state of B. However, instead of making a measurement, Alice entangles A with another photon X. SinceA and B are entangled, it can be arranged that the entanglement between X and A instantly teleports the state of X to Bob. Let s see how this would work for our entangled dice: Alice and Bob first produce an entangled pair of dice. Then, Alice takes her die to La Palma and Bob takes his to Tenerife. Before making a measurement, none of the faces show any numbers. But if the dice were to be observed, they would show the same numbers on their top faces. Chris initially has a die that shows on its top face. He passes it over to Alice, asking her to teleport its state over to Bob. Alice entangles her original die with the one she just got from Chris. (Importantly, Alice doesn t actually observe Chris die.) Through this procedure the faces of Bob s die obtain well-defined numbers. The top face of Bob s die will be. Bob could now meet with Chris and show him his die. This would prove to Chris that the teleportation succeeded. More details on quantum teleportation can be found in Anton Zeilinger s book, Dance of the Photons.
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