This material is to establish the quantization of a field and the particles it contains. Let ψ(r, t) be the wavefunction for a field where r stands for the set of coordinates of a point. The dynamics of the field are given by the time-dependent Schrödinger equation

−(ħ/i)(∂ψ/∂t) = − (ħ²/(2m))∇²ψ + V(r)ψ

where ħ is Planck's constant divided by 2π.

Consider ψ(r) expressed as a linear combination of a set of function {φ_n(r); n=0, 1, …, M}; i.e.,

ψ(r, t) = Σ_nb_n(t)φ_n(r)

The set of functions to be used are the solutions to the time-independent Schrödinger equation

− (ħ²/(2m))∇²φ_n(r) + V(r)φ_n(r) = E_nφ_n(r)

The solutions are orthogonal and can therefore be made into an orthonormal set; i.e.,

∫φ_n(r) φ_m(r)dr = δ_nm

where δ_nm=1 if n=m and is zero otherwise.

When the linear combination is put into the time-dependent equation the result is

−(ħ/i)Σ_n(db_n/dt)φ_n(r) = Σ_nE_nφ_n(r)

This equation can be multiplied by φ_m(r) and integrated over space. Because of the orthonorality of the set of φ_n(r) functions the result is

−(ħ/i)(db_m/dt) = E_mb_m(t)

For particles the expression −(ħ²/(2m))∇²ψ + V(r)ψ is the Hamiltonian function; for a field it is the density of the Hamitonian function. The Hamiltonian for a field is given by its expected value; i.e.,

H = ∫ψ*( − (ħ²/(2m))∇² + V(r))ψdr

where ψ* is the complex conjugate of ψ.

The quantity E(r)ψ can replace −(ħ²/(2m))∇² + V(r))ψ(r) in the integral to give

H = ∫ψ*(r)E(r)ψ(r)dr

If ψ is replaced by the linear combination of the φ_n functions and the integration carried out the result is

H = Σ_nb*_nE_nb_n

This Hamiltonian is in the nature of a sum of the Hamiltonians of a set of harmonic oscillators. Consider the harmonic oscillator for a particular value of n. It is an oscillator with a frequency of E_n/ħ.

Bosons

Let B_n− denote the operator corresponding to b_n and B_n+ the operator corresponding to b_n*. B_n+ is the operator adjoint to B_n−. The reason for this notation is that B_n+ represents the creation of an additional particle in the field and B_n− the annilhilation of a particle in the field.

The Hamiltonian operator is then

H = Σ_nB_n+E_nB_n−

Let the bracket symbol [P, Q] denoted the commutator of two operations, P and Q; i.e.,

[P, Q] = PQ − QP

The set of operators B_n− and B_n+ are required to satisfy the following canonical quantification conditions

[B_n−, B_m−] = 0^
[B_n+, B_m+] = 0^
[B_n−, B_m+] = δ_nm

where 0^ denotes the zero operation; i.e., the operation that generates the zero function.

From these it follows that

[B_n−, H^] = E_nB_n−
and
[B_n+, H^] = E_nB_n+

These conditions along with the previously derived

−(ħ/i)(db_n/dt) = E_nb_n(t)

make the results consistent with the Heisenberg equation for the dynamics of the operators

−(ħ/i)(dB_n−/dt) = [B_n−, H^]
and
−(ħ/i)(dB_n+/dt) = [B_n+, H^]

Eigenfunctions and Eigenvalues

Consider the operator B_n+B_n−. It is so significant for the analysis that it is given a special name and symbol. It is called the number operator and is denoted as N_n

Let L be an eigenfunction of N_n and λ its eigenvalue; i.e.,

N_nL = (B_n+B_n−)L = λL

In the webpage Operator Relations it is shown that (B_n−)L is also an eigenfunctions of N_n but with an eigenvalue which is 1 less than that of L. From this it follows that λ must be an integer and hence the eigenvalues of N_n are the nonnegative integers 0, 1, 2, 3, …. If n is the eigenvalue of N_n then the eigenvalue of (B_n−)L is (n−1). This is why B_n− is the annilhilator operator; it diminishes the number of particles in the field by one.

Similarly it can be shown that (B_n+)L is an eigenfunction of N_n but with an eigenvalue that is one unit more than that of L. Thus B_n+ is a creation operator that creates an additional particle for a field. This is the case of bosons.

Fermions

For the case of fermions it is the anticommutator of two operations that must be considered. The anticommutator of two operations, P and Q, is defined as

{P, Q} = PQ + QP

The operator corresponding to b_n will be denoted as β_n− and that of b_n* as β_n+. The canonical quantification conditions to be satisfied by β_n− and β_n are

{β_n−, β_m−} = 0^
{β_n+, β_m+} = 0^
{β_n−, β_m+} = δ_nm

Note that

{β_n−, β_n+ = 1^
implies that
β_n−β_n+ = 1^ − β_n+β_n−

where 1^ is the identity operator.

Now consider

(β_n+β_n−)(β_n+β_n−) = β_n+(β_n−β_n+)β_n−
= β_n+(1^ − β_n+β_n−)β_n−
= (β_n+β_n−) − β_n+(β_n+β_n−)β_n−
= (β_n+β_n−) − (β_n+β_n+)(β_n−β_n−)

But (β_n+β_n+) and (β_n−β_n−) are both zero. Thus

(β_n+β_n−)(β_n+β_n−) = (β_n+β_n−)

Let L be an eigenfunction of (β_n+β_n−) and λ its eigenvalue. Then

(β_n+β_n−)L = λL
and
(β_n+β_n−)(β_n+β_n−)L = (β_n+β_n−)λL = λ²L
however

(β_n+β_n−)(β_n+β_n−)L =
(β_n+β_n−)L = λL

This means that

λ² = λ
and hence
λ must equal 0 or 1

Thus a fermion state can have at most one particle. This is the Pauli exclusion principle.