 Welcome to Quantum Field Theory 4, Second Quantization. If there is a second quantization, then presumably there is a first quantization. That is where we start with the description of a system in terms of classical particles and through quantization arrive at a description in terms of waves. This is the standard quantum mechanics that we developed throughout the quantum mechanics series. In the case of a hydrogen atom, a positively charged proton is assumed heavy enough that its position can be considered fixed at the origin. A much lighter negatively charged electron of mass m orbits the proton with position x and momentum p. We write the classical Hamiltonian, the expression for the total energy equal to the kinetic energy plus potential energy in terms of the components of electron, momentum and position. Then we transition to quantum mechanics by replacing these components with operators. The resulting Hamiltonian operator is used in the Schrödinger equation for the electron wave function psi, which is a function of position and time. Solutions of particular interest are the so-called stationary states of definite energy. For these states the wave function separates into a spatial factor psi k and a time factor e to the minus i omega k t. The spatial factors psi k of x satisfy the time independent Schrödinger equation with e k equals h bar omega k. These solutions are the hydrogen atom orbitals that we discussed in the quantum mechanics series. As we also established there, each orbital can contain one spin up and one spin down electron. Though a complete quantum state of an electron in a hydrogen atom would be specified by an orbital and a spin state. We'll assume the subscript k indexes the so-called spin orbitals. More general quantum states can be represented as a superposition of the stationary states, each with some coefficient bk. The superposition can, of course, represent individual stationary states. If bm equals 1 and all other bk's are 0, then the electron is in state m. If bn equals 1 and all other bk's are 0, then the electron is in state n. If both bm and bn are non-zero, then the superposition contains different frequency components and the wave function is not stationary, but its magnitude squared varies with time. This was one reason that Schrödinger thought his equation could restore determinism to atomic physics. If the magnitude squared of the wave function could be interpreted as the actual distribution of matter and charge, then this oscillation of electronic charge would create electromagnetic radiation that would carry away some of the atom's energy, causing it to continuously transition from the higher to the lower energy state in a completely deterministic manner. The normalizing condition, sum of magnitude squared bk equals 1, would represent the conservation of matter and charge. During a radiative transition, the bk values would vary continuously while maintaining this condition. However, careful observation has shown that electromagnetic energy is not radiated continuously, but rather in discrete energy quanta, we call photons. The creation of photons is not described by a deterministic process, but rather a probabilistic one. Therefore, the magnitude squared bk must be interpreted as the probability an electron is in the kth spin orbital, and radiative transitions must occur in quantum jumps. At one moment an atom is in a high energy state, the next moment it's in a low energy state, and a photon has been created. As we discussed in the first video of this series, one goal of quantum field theory is to represent this quantum jump as the destruction of a high energy electron and the simultaneous creation of a low energy electron and a photon. Second quantization is the process where we start with a wave description of a physical system, and through quantization arrive at a description in terms of quantum particles. This is the process we have developed in this series for the electromagnetic field. In video three we saw that if a radiation mode of the classical electromagnetic field is described by a vector potential, a k alpha, expressed in terms of time factors e to the minus i omega t and its complex conjugate, then in the Heisenberg picture this becomes a quantum field operator formed by replacing the coefficients with photon destruction and creation operators. Now we want to apply this idea to electrons. Our approach will be to start with a classical electron. Apply first quantization to obtain a wave function, then apply second quantization to the wave function to arrive at a quantum field description of the electron. We dealt with first quantization in the quantum mechanic series, which we have just briefly reviewed. Now we'll treat the wave function psi like a classical field expanded in its modes. For electrons, this wave function should be a solution of the Dirac equation, the fully relativistic description of the electron, but that involves a lot of complicated bookkeeping. It's easier to work with solutions of the non-relativistic Schrödinger equation. This is simpler yet still quite instructive and useful in its own right for describing the non-relativistic low energy limit, such as atomic radiation, and it provides a useful foundation that we can extend to the relativistic case in the future. We start by writing the wave function as a general superposition of the spin orbitals. Then we apply the formalism we developed for photons. The coefficients of the e to the minus i omega t terms become destruction operators and the wave function becomes a field operator and that's basically it. Well, there is a wrinkle having to do with the destruction operator that we'll come to shortly and figuring out what this expression actually means will take some consideration. But yes, we have just produced a quantum field theory of the electron. In quantum mechanics, the expectation value of the Hamiltonian operator is the Hamiltonian operator applied to the quantum state projected onto the quantum state. In terms of our superposition, this is sum over L, BL conjugate, bra psi L, Hamiltonian operator applied to sum over K, dK, ket psi K. The Hamiltonian operator applied to the kth orbital produces the kth orbital energy ek times the orbital. So we get sum over L, sum over K, ek, BL conjugate, bK times the projection of the kth orbital onto the lth orbital. But this projection is zero unless L equals K, in which case it's one. So the expression reduces to the sum over K, ek, bK conjugate times bK, which is sum over K, ek, magnitude squared bK. This simply says that the expected value of the energy is the sum over each orbital of the orbital energy times the probability the electron is in that orbital. Based on our experience with second quantization of the electromagnetic field, we expect that if the coefficient bK becomes the destruction operator, b hat K minus, then the conjugate of bK should become the creation operator, b hat K plus. So the quantum field theory Hamiltonian will be the sum over K, ek, b hat K plus, b hat K minus. The creation operator times the destruction operator is the number operator, n hat K. So finally, the Hamiltonian operator is the sum over K, ek times n hat K. That is, the energy is simply the sum over all atomic orbitals of the orbital energy times the number of electrons in that orbital. The wrinkle referred to previously is that the creation and destruction operators we use for electrons cannot be the same ones we use for photons. They have to be separate fermion operators. For bosons, the destruction operator applied to the zero-photon state, the vacuum state, gives zero. You cannot destroy a photon if there are no photons present to destroy. The creation operator applied to the vacuum state gives the one-photon state. The destruction operator applied to the one-photon state takes us back to the vacuum state. The creation operator applied to the one-photon state produces square root two times the two-photon state, and so on. There is no limit to the number of photons that can be created. For fermions, the destruction operator applied to the vacuum state should also produce zero. The creation operator applied to the vacuum state should give the one-electron state, and the destruction operator applied to the one-electron state should take us back to the vacuum state. But now we come to the central issue. According to the exclusion principle, there cannot be two electrons in the same quantum state. So the creation operator applied to the one-electron state has to produce zero. Therefore, the fermion operators cannot be identical to the boson operators. They must differ in a manner that makes them compatible with the exclusion principle. Now, derivation of boson operators followed logically from a quantum analysis of a classical system, the harmonic oscillator. The creation and destruction operators were linear combinations of the corresponding position and momentum operators, which represent physical observables. But derivation of fermion operators will necessarily be abstract because there is no classical system analogous to a harmonic oscillator that obeys the exclusion principle. So let's see what we can come up with using logic and math. Boson operators satisfy a commutator relation. A hat minus A hat plus minus A hat plus A hat minus equals one. Rearranging, we have A hat minus A hat plus equals one plus A hat plus A hat minus. Therefore, A hat minus A hat plus applied to the one-photon state produces the one-photon state plus A hat plus A hat minus applied to the one-photon state. But A hat plus A hat minus is the number operator, which applied to the one-photon state produces one times the one-photon state. So the final result is two times the one-photon state. If we try to follow the same steps with fermion operators, we have to get zero. Because the very first operation is the creation operator applied to the one-electron state. But by the exclusion principle, we cannot create a second electron in a state that already contains an electron. We will obtain this result if the fermion expression has minus signs where the boson expression had plus signs. Working backwards, we must have B hat minus B hat plus equals one minus B hat plus B hat minus. And working back from there, the fermion operators must satisfy an anti-commutator relation, which we represent with curly braces. The anti-commutator of the destruction and creation operators, B hat minus B hat plus plus B hat plus B hat minus equals one. For bosons, it's trivially true that any operator commutes with itself. The commutator of the destruction operator with itself is A hat minus A hat minus minus A hat minus A hat minus, which is obviously zero. Likewise for the creation operator and indeed any operator. If fermion operators satisfy anti-commutator relations, then the corresponding requirement is that operators should anti-commute with themselves. For the destruction operator, this requires B hat minus B hat minus plus B hat minus B hat minus equals zero. And similarly for the creation operator, this is not trivially true. In fact, it is equivalent to the requirement that two applications of the destruction operator or two applications of the creation operator are zero. Those in turn are requirements of the exclusion principle. Consider two applications of the creation operator. If we start with the vacuum state, one creation operator produces the one electron state. A second creation operator would be an attempt to create two electrons in the same state, which cannot happen. So, the second quantization formalism we developed for photons can be applied to electrons with the modification that commutator relations are replaced with anti-commutator relations.