Notes on quantum mechanics (under construction)

 (updated 10/11/04)

NOTE: These are supplementary notes, you will still need to read your text book.
Caution, I've used symbol font in a few places- if your browser does not do symbol font these letters will show up as roman letters. I've tried to avoid this as much as I could.

References:

The 'weirdness' of quantum mechanics.

Most of the difficulty in understanding quantum mechanics comes because of its probabilistic nature. For example, the electron wave function squared gives us the probability of finding the electron at some location.  Yet in some cases we measure the particle like properties and actually find it in a specific location (it arrives as a lump in a collision). How do we reconcile the idea that the location is described by a probability yet electrons can arrive in lumps as particles? The same problem is true of photons and any other sub atomic particle (and even larger objects such as buckyballs). The same is also true about other properties of a subatomic particle such as polarization, spin, etc. (they have only a probabilistic distribution of values until a measurement is made of them, at which time we now know the specific value). As Griffiths and Greene point out, when faced with dual wave/particle descriptions of an electron we are tempted to take one of four positions:
  1. The realist position: The electron really is at a specific location all the time, but quantum mechanics can't tell us precisely where. Therefore quantum mechanics is incomplete, there must be hidden variables which someday might be found and which will tell us where the electron is all the time. This was Einstein's view and there is at least one current group of physicists who are pursuing this idea, lead by David Bohm (thus far inconclusively).
  2. The orthodox position: Until we actually make a measurement that determines the particle like behavior the electron isn't anywhere in particular. The observation of an electron in a particular location causes its wave function to collapse; after measuring there is no longer any probability but before measuring the electron doesn't exist as a particle. This is called the Copenhagen interpretation and Bohr was its main proponent.
  3. The agnostic position: Since we can't know the position of the particle like behavior until we make the measurement we can just say that it doesn't matter if the particle exists before we measure. It doesn't make sense to ask questions which have no answers.
  4. The many worlds position: The wave function represents a collection of possibilities. Each of these possibilities comes true in a different universe. When we make a measurement of a probability and find that the electron has a definite location (arrives in a lump) we are merely determining which of the many possible universes we and the electron are actually in. This position is defended by Hugh Everett and others.
The Bell inequalities (mentioned below) discovered by John Bell in 1964 suggested experiments which would be able to decided which of these positions is correct. Those experiment have been done and they strongly suggest that position 2. is correct (they in fact rule out the possibility of 1.).

The EPR paradox (AKA entanglement).

 IN 1935 Einstein, Podolsky and Rosen presented a paradox which was meant to show that quantum mechanics had to be incomplete (essentially the realist position). They suggested a hypothetical experiment which would show that quantum mechanics would have to be nonlocal if the orthodox position were true. Nonlocality means that the wave function must react instantly over large distances. This doesn't conflict with special relativity because no information can be transmitted faster than the speed of light but it does suggest that some quantum states are very strange because they can be spread out over extremely large regions of space (be nonlocal) and yet remain in communication so that the wave function can instantly collapse when a measurement is made of the particle like property. When the experiments were finally done they did not show quantum mechanics to be incomplete but rather showed that quantum mechanics IS nonlocal.

Here is a version of the experiment. A neutral pion with zero spin decays into a positron and an electron each traveling in opposite directions. To maintain conservation of angular momentum we must have one decay product (positron or electron) with spin and the other with the opposite spin (so that the total is zero which is what we started with). But after the decay we do not know which particle (positron or electron) has the spin up and which has the spin down (quantum mechanics doesn't tell us which one gets a spin in any particular direction). So we must describe the two decay particles with a wave function with a 50% probability of the electron with a spin and the positron with opposite spin and 50% chance of the other way round. However if we measure one of the spins and find out what it is we then know exactly what the other particle spin is (it has to be opposite of the what we measured and in fact when we measure the other one it is always the opposite of the first). Suppose the two particles are produced and move very far apart. Now when we measure one of them we still know what the other one's spin is, even if it is very far away. How did the second particle know what the first one's spin was and how did it get that information instantly (faster than the speed of light)?

A realist would say that the electrons had a particular spin when they left but the theory didn't tell us, therefore quantum mechanics is incomplete. The orthodox position says the wave function is nonlocal (stretches from one particle to the other, even if they are very far apart) and the measurement of one particle causes the wave function to collapse so that the other particle instantly 'knows' what spin to have. A test of Bell's inequalities show that the outcome of a large number of measurements is slightly different if there are hidden variables than if there is not. Experiments have shown that the realist position is wrong, there cannot be hidden variables.

Bell's inequalities: a simple example borrowed from Chapter 4 in Greene.

Suppose two people (A and B) each get a series of numbered boxes. Each box has a door on it and when the door opens a light flashes either blue or red, randomly. A and B notice that whenever A gets a red flash, B also gets a red flash. This is true of all the boxes they open. The orthodox quantum view is that the light really is random, a random decision as to which light will flash is made just before the door on the box is opened, but because of quantum mechanics (quantum weirdness), A's box always correlates to the color B gets. Einstein called this 'Spooky action at a distance' because it seems like the measurement of A affects the measurement of B, even though they may be far apart. A realist view is that the boxes are actually programed to give the same colors in some random order with both boxes having the same program. The program that causes the correlation is the 'hidden variable'.  This is exactly the same as the pion decay mentioned above.

Now consider a slight modification. Each box has three doors instead of one but only one door can be opened at a time (once one is opened the others remain forever locked). Call the top door 1, the side door 2 and the front door 3. A and B notice that if A opens door 1 and sees blue, B will also see blue if she opens door 1. This is true of all the boxes they open, if they open the same door they get the same color. Again, the realist view is that there is some kind of program inside each pair of boxes which tells it what color to flash, depending on which door is opened. Such a program might be 'blue, blue, red' meaning that if door 1 is opened you get blue, door 2 gives blue and door 3 gives red. The next pair of boxes might have a different program from blue, blue, red but each has the same program.

Now suppose A and B decide to open the doors randomly (they don't both agree to open door 1 but rather randomly choose which door to open). Notice that there are 9 possible outcomes; A opens door 1 and B opens door 1, A opens 1 and B opens 2, etc.:  (A, B) = (1,1); (1, 2); (1, 3); (2, 1); (2, 2); (2, 3); (3, 1); (3, 2); or (3, 3).  Now look at the outcomes if there is a program (hidden variable) 'blue, blue, red'. In this case there are 5 different ways that A and B could get the same color: (1,1); (2, 2); (3, 3); (1, 2) or (2, 1) and only four ways to get different colors. The last two cases of getting the same color occur because there are only two colors but three doors, so there has to be two ways to get the same color without opening the same door. Said another way; we know we always get the same color if we open the same door and there are two more cases where we will get the same color IF (!!!!) there is a program. Notice that the same argument would hold for any other program because there are only two colors but three doors (try 'red, blue, red' for example). So IF (!!!!) there are hidden variables (a program) we expect to get the same color on average 5 out of 9 times if A and B randomly open a door on each new pair of boxes. So IF (!!!!) there is a program telling the lights how to flash we MUST get a matching color more than 50% of the time on average. If we get matching colors less than 50% of the time there can be no program, no hidden variable. If there really is a quantum correlation between the boxes (we always get the same color if A and B open the same door) then we expect to get matches 3 out of 9 times but less than 5 out of 9 times. So 33% = or < number of correlations = or < 55%. This is one version of Bell's inequality.

A and B correspond to the positron and the electron in the above pion decay. The three doors correspond to the three coordinate axises (x, y, z). Opening a door corresponds to measuring the spin in the direction of one of the axises (one you measure the component along a certain axis you have disturbed the particle so you cannot go back and measure the y component). Red corresponds to a clockwise spin, blue to counterclockwise (in the box example the colors correlate, if A measures blue, so does B; here the spins anti-correlate, if the positron is clockwise on the x axis the electron is counterclockwise on the x axis). Bell's inequalities say that if we randomly look at the spin along different axis we MUST get anti-correlation 55% of the time (or more) on average IF (!!!!) there is a hidden variable (a program) that the electron and positron carry with them, telling them which way to spin). So quantum mechanics says 33% = or < number of anti-correlations of spin = or < 55% whereas a hidden variable theory MUST have the number of anti-correlations of spin = or > 55%.

The experiments show a correlation LESS than 55% of the time! There can be NO hidden variable!

Schrodinger's cat.

Once Bell's inequalities were tested and hidden variables ruled out we have to take the orthodox position seriously. Wave functions evolve as described by Schrodinger's equation but only give probabilities we cannot say the electrons have any position or any particular spin. When we make a measurement however, the probabilities go away and we see the particle like behavior of the electron. In some sense, then, when we make a measurement (find the electron as a lump) and the wave function collapses we somehow cause the physical state we measure. Beforehand it was just a wave function, afterwards we have a specific value for the property we measured.

Example: We put a cat in a box with a Geiger counter and a small amount of radioactive material and a vial of cyanide. Radioactive decay is random (a quantum mechanical process), put enough in that within an hour there is a 50% chance one atom has decayed. If one does decay it is detected by the Geiger counter which (through some mechanism) breaks the vial of cyanide, killing the cat. Now how do we describe the state of the whole system? Since quantum mechanical states are involved we have to have a wave function which shows a 50% probability of having the cat alive and a 50% chance of the cat being dead after one hour. And quantum mechanics says we cannot say the cat has any particular state before we make a measurement. So if we now peek into the box and find out if the cat is alive or dead, that measurement is what collapses the wave function and CAUSES the cat to be either alive or dead. The cat is alive or dead or neither, depending on our measurement of its state.

Pretty bizarre, huh?