In this video, we will learn exchange energy. Consider two electrons with identical spin states. The Hamiltonian is given by this. This term here represents the kinetic energy of electron number 1 and this term is the kinetic energy for electron number 2, and this is the coulomb potential energy between the two, the repulsion between those two electrons. The total wave function for the two electron system can be expressed as a linear combination of the product states, the product of the two single electron wave functions as before and explicitly we can write down the total wave function of the two electron system as an anti-symmetric combination of the product states of the single electron state. This first term here represents electron 1 in quantum state a, single electron state a, and electron 2 in single electron quantum state b. Then you switch electron 1 and 2 and then make an anti-symmetric combination corresponding to Gamma equals negative 1, the anti-symmetric eigenspace of the permutation operator. The normalization constant is given by 1 over the square root of 2 so that the total wave function is properly normalized. To simplify our notation, I'm going to drop all these Phi wave function notation and simply write a ket. Number 1 and 2 represents electron 1 and electron 2, and a and b represents the single electron quantum state. This equation is simply written as the two products states represent by two kets. Now, we want to estimate the energy of the two-electron system and to do that, we evaluate the energy expectation value and by definition that is just the Hamiltonian operator sandwiched by the total wave function Psi. Substituting the expression for the total wave function Psi that we just obtained in the previous slide, we can obtain this equation, equation number 1. It has four terms. The first term in equation 1, by substituting the equation for Hamiltonian, have these three terms and these three terms corresponding to the kinetic energy for electron 1, kinetic energy for electron 2, and the potential energy. We can simply write it out like this, kinetic energy for electron 1, kinetic energy for electron 2, and potential energy between the two. Of course, we can write out the kinetic energy term explicitly, so the first term in the Hamiltonian, then this is a derivative, this a differential operator acting on r_1 only, so this ket point and bra involving two can be separated out and they're are assumed to be normalized, so this product simply becomes one, you're left with this and you can explicitly write it out in terms of single-particle wave function and integral as shown here. Similarly, you can do the same for the second term of the Hamiltonian. You get a very similar expression, except that the Psi a, the wave function for quantum state a, is now replaced by the wave function for quantum state b. Finally, the potential energy term is given by this and once again, expressing these kets and bras as the wave functions explicitly, we obtain this integral expression for the potential energy. This is the evaluation of the first term in equation 1. Evaluating the 2nd term in equation 1 gives you exactly the same result. The first two terms in equation 1 gives you two identical results, but equation 1 has this factor of 1/2 out in front, and so these first two terms simply add up to give you these three terms: kinetic energy for electron occupying quantum state a, kinetic energy for electron occupying quantum state b, and the potential energy coulomb interaction between the two electrons. This is what we expected from the system. There are two electrons, kinetic energy for each electron, and the coulomb interaction between the two, that's all there is. Classically, that should be all there is. However, because we have started out with the anti-symmetric combination for the total wavefunction of the two-electron system, there are additional terms. There are then 4th term in equation 1, and they produced additional terms and we call those additional energy exchange energy. It has to do with the permutation or exchange of electrons, and therefore the name exchange energy. The last term in equation 1, if you recall, was this, and that simply is the complex conjugate of this term, because Hamiltonian is a Hermitian. What that means is that the 4th term is just a simple complex conjugate of the third term in equation 1. So the remaining two terms of equation 1 simply gives you the real part of this term here, the third term in the original equation, and you can write it out explicitly in terms of the single electron wavefunctions and integration as shown here. This is the explicit expression for the exchange energy. Finally, the total energy of the system is the kinetic energy of the two electrons, coulomb potential plus exchange energy. If you look at the explicit expression, the integral expression of exchange energy in the previous slide, you will notice that the exchange energy is appreciable only when there is an appreciable overlap between the two single-electron wavefunction; Phi_A and Phi_B, when they overlap appreciably only when that integral will have an appreciable value. This exchange energy, we have to emphasize, is a distinctly quantum mechanical phenomenon, there is no analogy in classical physics. Exchange energy is critical in quantum mechanics dealing with a system of identical particles. A simplistic example of a system of identical particle is a helium atom, which has two electrons. The excited state of a two-electron system of helium atom can be expressed as a linear combination of the product of single-electron wavefunction, hydrogen wavefunction basically. As before, one electron is in the ground state, Phi_0, another electron is in an excited state. Some nth state represented by Phi_n here. Of course, we need to make an anti-symmetric combination of these two products states of the single electron states, and the normalization constant is once again 1 over root 2. Now, when we construct this anti-symmetric combinations, we ignored spin, which means that we assumed implicitly that the electrons have the same spin state. That's why their spatial state has to be anti-symmetric. But if we include spin states, we have to consider two different cases. One is the case when the spin state is anti-symmetric under permutation operation or under exchange operation. In that case, your spin wave function is anti-symmetric, which means that your spatial part of the wavefunction has to be symmetric so that the total wavefunction comes out anti-symmetric under exchange. If the spin state is symmetric as shown here, then, of course, the spatial part of the wave function must be anti-symmetric to make the total wave function anti-symmetric once again. In the previous slide, we considered this case, assuming a symmetric spin state. But in general, we have to consider these two cases. Now, ignoring spin-spin interaction, the Hamiltonian operator acts only on the spatial part. Hamiltonian has the kinetic energy term, which is a differential operator acting on the spatial part of the wavefunction and the Coulomb interaction also acting on the spatial part of the wavefunction only has no spin component. Energy of these two electron system depends only on the spatial part of the wavefunction. They are different depending on their spin state, which means that the energy of this two electron system, helium atom, depends on the spin state, whether their spin state is symmetric or anti-symmetric under exchange. The two cases will produce different exchange energies with opposite sign in fact. That results in energy level splitting. Even though this is despite the fact that these two electrons occupy the same single electron state, one electron in the ground state and another electron in the nth excited state. But depending on their spin state, exchange energy becomes different and therefore, the total system exhibits different total energy. The anti-symmetric spin state is called the spin singlet. The symmetric spin state is called the spin triplet. Depending on the spin state, the energy levels of helium are different and the helium atom in spin singlet state, and therefore symmetric spatial part of the wave function is called a parahelium and the helium atom in spin triplet state is called the orthohelium. These guys exhibit different energy level spectrum. So far, we only discussed the case of two fermions electrons, but you can imagine that you can perform the same procedure for the case of bosons. Bosons also produce exchange energy. However, usually the exchange energy for bosons is negligibly small and they rarely become important.
