Ladder Operators

This appendix deals with the boson and fermion ladder operators (i.e., annihilation and creation operators). These operators are not only used to find, for example, the eigenspectrum of a the quantum harmonic oscillator (Section 2.6.4); they are also in the second quantization to deal with systems of identical particles, such as, bosons (symmetric particles, e.g., phonons and photons) and fermions (anti-symmetric particles, e.g., electrons). This appendix covers the basics of the second quantization (as compared to the first quantization, of which a brief description is also given here), including the Fock (number) states and the basic forms of the creation and annihilation operators and their commutators (or anti-commutators, in the case of fermion operators). Finally, the use of these operators in creating arbitrary states, the number operators, field operators, and non-interacting and interacting Hamiltonians are discussed. When dealing with bosons (phonon and photon, symmetric) and fermions (electrons, anti-symmetric), we must represent many identical particles. Consider a simple, non-interacting Hamiltonian where each Hamiltonian H i generates the identical eigenspectrum (∣ü 1 〉, ∣ü 2 〉, …, ∣ü j 〉, …) for all N particles. Subscripts on ü denote the state and subscripts on ∣ü j 〉 denote the particle occupying the j th state. For example, consider a two-particle Hamiltonian with particles in the first two states. The boson and fermion states (∣ψ B 〉 and ∣ψ F 〉)are where ∣ü j 〉 i indicates the j th state (ü) of the i th particle. Note that ∣ü 1 〉 1 ∣ü 1 〉 2 is naturally symmetrized, such that no similar anti-symmetric system is possible (i.e., there is no equivalent fermion state since the Pauli exclusion principle holds).