 Let's summarize the properties of the bows on destruction and creation operators, a hat minus and a hat plus. These satisfy the commutation relation, a hat minus a hat plus minus a hat plus a hat minus equals one. The creation operator applied to the n-photon state produces square root n plus one times the n plus one photon state. The destruction operator applied to the n-photon state produces square root n times the n minus one photon state. The creation operator times the destruction operator forms the number operator, n hat. The number operator applied to the n-photon state produces n times the same state, and the allowable values of n are zero, one, two, three, and so on. The fermion destruction and creation operators satisfy an anti-commutator relation, b hat minus b hat plus plus b hat plus b hat minus equals one. We haven't yet determined the specific form of the operators, but we still need the creation operator times the destruction operator to form the number operator, n hat, which applied to the n-electron state produces n times that state. Finally, the exclusion principle limits the valid values of n to zero and one. From the anti-commutator relation, we have b hat minus b hat plus equals one minus b hat plus b hat minus, which is one minus the number operator. The number operator squared is b hat plus b hat minus b hat plus b hat minus. If we group the second and third operators, substitute from the relation above, and multiply out the expression, we get b hat plus b hat minus minus b hat plus b hat plus b hat minus b hat minus. The second term vanishes since b hat minus b hat minus is zero. And the first term is the number operator. So the number operator squared equals the number operator. The number operator applied to the n-electron state produces n times this state, and the number operator squared applied to this state produces n squared times this state. For these to be equal, requires n squared equals n. And this quadratic equation has two solutions, n equals zero or one, which is another statement of the exclusion principle. There can only be zero or one fermion in a given quantum state. Now let's consider the state psi produced by applying the creation operator to the n-electron state. Applying the number operator to that state, we have b hat plus b hat minus b hat plus applied to the n-electron state. Grouping the second and third operators, b hat minus b hat plus, substituting for this from the anti-commutator relation, one minus the number operator, and applying the number operator, we get b hat plus times one minus n times the n-electron state. Applying the number one minus n to the left, we end up with one minus n times the state psi. And this equals the number operator applied to that state. This tells us that the quantum state resulting from b hat plus applied to the n-electron state contains one minus n electrons. So the effect of the fermion creation operator applied to the n-electron state is to create the one minus n-electron state with some as yet unknown coefficient. Now let's consider the state psi produced by applying the destruction operator to the n-electron state. Applying b hat minus b hat plus to psi and substituting from the anti-commutator relation gives us one minus the number operator applied to psi. On the left, we can group the second and third operators which form the number operator and write this as b hat minus times the number operator applied to the n-electron state. The number operator produces a factor of n, which we move to the left, leaving n times the state psi. Therefore, one minus the number operator applied to the state psi equals n times the state psi, which can be solved for the number operator applied to the state psi equals one minus n times state psi. So the effect of the fermion destruction operator applied to the n-electron state is also to create the one minus n-electron state with some as yet unknown coefficient. To see why the one minus n state can be the result of both the creation and destruction operators, let's assume for now the unknown coefficients are one. When the creation operator applied to the n equals zero state produces the n equals one minus zero equals one state. The destruction operator applied to the n equals one state produces the n equals one minus one equals zero state. Both results are what we expect these operators to do. Clearly though, the coefficients cannot be one because then the operators would be identical and the creation operator applied to the n equals one state would produce the n equals zero state and not zero as it must to be compatible with the exclusion principle. Likewise, the destruction operator applied to the n equals zero state would produce the n equals one state and not zero as it must. We can use the Boson operators as a guide to what the fermion coefficients might be. Following the pattern, we guessed that the creation operator coefficient is square root one minus n and the destruction operator coefficient is square root n. Since n is either zero or one, these coefficients are either zero or one. We can readily verify that with these coefficients, the fermion operators satisfy all of the conditions we require, including the combination B hat plus B hat minus forming the number operator. Now we can complete our table of the creation and destruction operators for both Bosons and fermions. The mathematics of fermion operators was mostly worked out by Pesquale Jordan and collaborators in the mid-1920s. However, Jordan was perplexed by one remaining inconsistency of the theory and he sought the assistance of Eugene Wigner to find a solution. Jordan and Wigner's 1928 paper on the poly exclusion principle presented the final theory of fermion creation and destruction operators. Here's the problem. Suppose our atom has one spin orbital psi one containing n1 electrons and a second spin orbital psi two containing n2 electrons. The state of the entire atom is then specified by the list of occupation numbers n1, n2. And each orbital has its own set of creation and destruction operators. Suppose we start with no electrons and apply the second creation operator to create an electron in the second orbital, followed by the first creation operator to create an electron in the first orbital. We end up with the state 1, 1. If we had instead started with the first creation operator, followed by the second, we would have ended up with the same state. But if fermion operators satisfy anticommutated relations, the two creation operators should anticommute, which requires that changing their order of application introduces a minus sign. So the two results we obtained above should have been negatives of each other. Jordan and Wigner's solution was to introduce factors theta k equal to one or minus one to the kth creation and destruction operators. For our two state atom, we take theta one to be one and theta two to be minus one to the power n1. This makes the second orbital creation and destruction operators dependent on the number of electrons in the first orbital. Starting with the state 00, applying the second creation operator followed by the first produces the state 11 as before. But applying the operators in the opposite order does not. The first creation operator produces the state 10, but when it comes to applying the second creation operator, we now have n1 equal to one. So we have to add a factor of minus one to the first power, and we end up with minus one times the state 11. For a system with any number of orbitals, we take theta k equal to minus one to the power n1 plus n2 plus so on up to nk minus one. There is an interesting analogy with the quantum mechanics of a multi-electron atom. As we showed in the chemistry video of the quantum mechanics series, the exclusion principle requires that a multi-electron wave function is anti-symmetric with respect to the exchange of the coordinates of any two electrons. If in the wave function psi, we exchange the coordinates xj and xk, then the resulting wave function obtains a minus sign. The negative sign introduced when the order of fermion operators is swapped is an analogous relation in quantum field theory. We are now in a position to summarize the Hamiltonians and quantum states of the two types of quantum fields we have dealt with so far. If we have an atom with orbitals psi 1, psi 2, and so on, the state of the atom is specified by the list of electron occupation numbers, Na1, Na2, etc. Since electrons are fermions, these numbers can only be zero or one. The atomic Hamiltonian describing the atom's energy is the sum over all orbitals of the orbital energy ek times the orbital number operator b hat k plus b hat k minus. If we also have an electromagnetic field with various radiation modes, its state is specified by the list of photon occupation numbers, NR1, NR2, etc. Since photons are bosons, these numbers can be any non-negative integers. The radiation Hamiltonian is the sum over all modes of the photon energy h bar omega k times the radiation mode number operator a hat k alpha plus a hat k alpha minus. The state of the entire system, psi, equals the product of the atom and radiation states psi a and psi r. The total Hamiltonian h hat is the sum of the atom and radiation Hamiltonians, h hat atom and h hat rad. This basically summarizes everything we've done in this series so far. Unfortunately, this result is rather boring because in our current description, these fields are non-interacting. The atom will just sit there in whatever state it is in, and likewise the radiation field. The two fields are invisible to each other. To get the system to do something more interesting, we need an interaction term in the Hamiltonian that describes how the fields interact. We can get an idea of what kinds of expressions must appear in this term by considering the types of quantum jumps it must account for.