In this video, we will discuss indistinguishability. In classical physics, one can in principle track individual particles accurately even if they look alike. If you have access to the most accurate equipment that you need, in principle, you can track down individual particles without confusion. In quantum mechanics, however, particles are fundamentally indistinguishable. Tracking particles would mean that you have to make position measurements. We all know that the position measurements unavoidably disturb the system. This is fundamentally related to the uncertainty principle. Particles are not distinguishable. You can't tell one from the other. Suppose a system that's looking like this. You have one electron coming from this direction, top right, and another electron coming from bottom-left. Then they somehow interact with each other in this gray region. Monitoring the system, you observed that one electron came out here, and one electron came out there. Let's call this path a. Let's call this path b. What we're saying here is that you cannot tell whether electron 1 collided and scattered off of this electron and came out in path b, and electron 2 came out in path a and electron 2 in path b. Or electron 1 came out in path b and electron 2 came out in path a. These two situations cannot be distinguished. To be more specific, let's try to write down the wave function for this two particle system. The two particle system wave function can be expressed as a product of the 2,1 particle wave functions. Explicitly, we can write it like this. The total wave function of the two particle system is a superposition of these two product states of a single particle system. In one case, you have electron 1, r_1, this subscript 1 represents electron 1, electron one in path a, and electron 2 in path b. This product state represents this situation on the left here. The other case is electron 2 in path a, and electron 1 in path b. This product state represents the situation shown here in the right. The most general state that we can come up with is the superposition of these two states with some complex coefficients, c_1 and c_2. That would be our total wave function for the two electron system. Now, the fact that the two electrons are truly indistinguishable means that there is no difference, even when we swap the two electrons in our wave functions, the wave functions shouldn't change. But we cannot measure wave function directly. We can only measure the probability. What really needs to remain invariant is not the wave function itself, but the absolute value squared of the wave function. That's the equation shown here. The total wave function, r_1 and r_2 shown here in this order, absolute value squared, should be equal to the total wave function where these two variable, two position variable for electron 1 and electron 2 are swapped like this. They should remain the same. That's what we mean by indistinguishability. What this means is that these two functions, Psi of r_2, r_1 and Psi of r_1, r_2, these two wavefunctions are related to each other through a constant Gamma here, which is a complex number in general with a unit magnitude, as long as this Gamma is a phase factor in a complex number with a unit magnitude, they will satisfy this equation that we require. Now, we can do the swapping operation twice. Swap 1 and 2, and swap 1 and 2, again. Obviously, it will come back to the same situation. When you have Psi function with r_1 and r_2 in this order, so r_2 in the first and r_1 in the second, and that wave function is related to this r_1, r_2 order, the swapped state through a phase factor Gamma, and if you do it one more time, you will have another Gamma factor. This function should be related to the original function by Gamma square, but we know that they should be the same. Which means that these Gamma square should be one, and therefore Gamma should be either plus one or minus one. The operator that rearranges particle as we just did, is called the permutation operator. The simple operation of swapping two electrons that we just discussed, we define the permutation operator for that swapping two electrons, P_12, as this. If you apply P_12 permutation operator for two electron to a state two-electron wave function as shown here, it simply switches the order of these two variables for electron 1 and 2. If you do it twice, then it comes back to itself, the same system, so the permutation operator squared should be equal to the identity operator. If you express what we just discussed in the previous slide in terms of the permutation operator, what we have said in the previous slide essentially, is that our two-electron system is an eigenstate of the permutation operator, because if you apply permutation operation to this state, it becomes the same state with a number Gamma, and this number Gamma will be the eigenvalue of the permutation operator. The fact that permutation squared is identity, means that there are only two possible values for the eigenvalue, Gamma plus one or minus one. Now, we can substitute the explicit form of the two-particle wave function to be more specific in our discussion. Go back to the second slide, and take the explicit form of the two-particle wave function, expressed in terms of the product of the one particle wave functions. Then the most general expression for the two-particle wave function was the superposition of these two products states. Remember, electron 1 in path A times electron 2 in path B, that's one situation. Then electron 2 in path A, electron 1 in path B, that product is other situation, and we form the superposition of the two, c_1 and c_2 being the complex coefficients. Because Gamma can be either plus one or minus one, this state is related to the swapped state through either plus one or minus one here, multiplied to these coefficients. Rearranging them gives you this, and then now the requirement is, this equation should be valid for any function. Any function at all positions, and that requirement simply translates that these coefficients to be zero, so c_1 should either be equal to c_2, that will be the top minus sign there, or c_1 is equal to minus c_2, that will be this plus sign here in this equation. Explicitly, what we can do is we can write down two total wave function for the two particle system. One corresponding to Gamma equals plus one, the other corresponding to Gamma equals minus one. For the case of Gamma equals plus one, c_1 is equal to c_2. There's some constant to be determined by the normalization condition, but the main point is that these two products states have the same phase, they're added together. For Gamma equals minus one, these two products states, they have 180 degree phase difference or one is subtracted from the other. Those are the two possible total wave function for the two particle system. In principle, to generalize what we have just discussed, in general, a system of identical particles, system of electrons, system of protons, a system of identical particles is an eigenstate of a permutation operator. The particles corresponding to Gamma equals plus one. The eigenvalue of the permutation operator can be either plus one or minus one. The states or the particle system of particles with eigenvalue plus one for the permutation operator is called bosons. Okay? Photons and all other particles with integer spins are bosons. Bosons are symmetric under permutation up to particle, as we have seen in the equation in the previous slide, the two products states are added together in our example, they are symmetric. As you exchange those two electrons, the state remains the same. The system of particles corresponding to Gamma equals negative one are called fermions. All particles, including electron and protons with half integer spins are fermions. The fermion wave functions are anti-symmetry under exchange of two particles. We will later generalize to this concept to more than two particles, a system of more than two particles. For now, we want to look into some more details of the system of fermions. Fermion, as we just discussed, the wave function of fermions is anti-symmetric under permutation or exchange of two particles. Let's look at our example once again. These total wave function of the two particles that we constructed, corresponding to Gamma equals negative one, the fermion system look like this. If the spatial state a, so if previously we had a and b here, a and b there. If they were both a, in other words, the spatial state these two electrons occupy are identical. If that's the case, then these two terms are identical. They subtract each other to subtract from each other to make the total wave function identically zero What does that mean? That means there are no particles. The probability is zero everywhere, those states are not allowed, those states are not possible. No two fermions can occupy the same single-particle state. This statement is the famous Pauli's Exclusion Principle. The Pauli's Exclusion Principle only applies to fermions. Bosons can occupy the same single-particle states. Many bosons can occupy the same single-particle states.
