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Illustrations of Quantization
in Quantum Field Theory

In a previous study it was found that harmonic oscillator systems there exists an operator A such that its product with its conjugate A*, A*A, is equal something closely related to the Hamiltonian H^ of the system then any eigenfunction Z of H^ with an eigenvalue of α is also an eigenfunction of A*A but with an eigenvalue of α−½.

In the quantum analysis of particles quantization means the properties of particles, such as energy and momentum, can take on only discrete values. In quantum field theory quantization has an entirely different meaning. It means that certain constructs function to keep track of the numbers of things in the system. This is referred to as the second quantization in quantum theory.

In Schrödinger's wave mechanics the Hamiltonian operator for a system is constructed from its Hamiltonian function by replacing the momentum p with −ih∇, where i is the square root of negative one and h is Planck's constant divided by 2π. (An operator is just a function that takes a function over space as an input and returns a function over space as an output. Usually the operator for a variable is denoted by a hat mark over the symbol for the variable. Such notation is not availabe in HTML so the operator corresponding to a variable is indicated by the capitalized form of its symbol. In the case of a variable that is already a capital the operator is indicated by the use of "^" as a superscript. )

For example, the Hamiltonian function for a particle attached to a spring (a harmonic oscillator) is ½p²/m + ½kx², where m is the mass of the particle, k is the stiffness coefficient of the spring and x is deviation from equilibrium. For this one dimensional system ∇ is (∂/∂x) and ∇² is (∂²/∂x²). Thus the Hamiltonian operator for a haromonic oscillator is

H^ = −½(∂²/∂x²)/m + ½kX²

A harmonic oscillator oscillates with a frequency ω equal to (k/m)½. Thus the Hamiltonian operator for a harmonic oscillator may also be represented as

H^ = −½h(∂²/∂x²)/m + mω²X²

In wave mechanics the time independent wave function ψ(z) satisfies the equation

H^ψ(z) = Eψ(z)

where z is the vector of the coordinates of spatial location, and E is energy. This means that ψ(z) is an eigenfunction of the operator H^ and E is an eigenvalue. The wave function ψ(z) is such that |ψ(z)|² is the probability density of finding the particle at point z.

The energy of a harmonic oscillator is quantized to [(2n+1)/2]hω where n is an integer called the principal quantum number. The solution to the time-independent Schrödinger equation for a one dimensional harmonic oscillator with principal quantum number n is

ψ(y, n) = (α/π)¼Hn(y)exp(−y²/2)/(2nn!)½

where α is mω/h, y=(α)½x and Hn(y) is the n-th Hermite polynomial.

In particular for n=0 the Hermite polynomial is just 1 and hence

ψ(y) = (α/π)¼exp(−y²/2)

Now define the following two operators

A = β(X + γP)
and its conjugate
A* = β(X − γP)

where β is equal to (mω/2)½ and γ is equal to i/(mω). The operator X is just multiplication by x and the operator P is −ih(∂/∂x).

In the previous study it is shown that if L is an eigenfunction of H^ with an eigenvalue of λ then L is also an eigenfunction of A*A but with than eigenvalue of λ+½. Furthermore the eigenvalue of A*A is a nonnegative integer. If Q is an eigenfunction of A*A with an eigenvalue of m then A*Q is an eigenfunction of A*A with an eigenvalue of (m+1). There is thus a sequence of eigenfunctions {Qm for m=0, 1, …, M} of A*A of the form

Qm = (A*)mQ0

The operator A*A is sometimes called the number operator becauses when applied to any eigenfunction of H^ it gives an integer. That integer represents a count of something.

Applying the operator X to ψ gives

xψ(y) = (α/π)¼x·exp(−y²/2)

Applying the operator P to ψ gives

Pψ(x) = −ih(∂ψ/∂x)

But ψ(y) is an exponential function of y. Therefore

(∂ψ/∂x) = ψ(y)[−(2y/2)(∂y/∂x)] = −y(α)½ψ(y)

Therefore

Pψ(x) = −ih[−y(α)½ψ(y)]
Pψ(x) = ih(α)½y·ψ(y)

This means that

(i/(mω))Pψ = −h(α)½y·ψ(y)
and
Xψ − (i/(mω))Pψ = x·ψ + h(α)3/2x·ψ(y)
Xψ − (i/(mω))Pψ = [1 + h(α)3/2]xψ(y)

The eigenfunction for A*(A*A) is then

(mω/2)½[Xψ − (i/(mω))Pψ ]
= (mω/2)½[1 + h(α)3/2]xψ(y)

The functional form is given by xψ(y)=xψ(α½x). This is the same shape as the wave function for a principal quantum number of 2. The shapes of the wave functions and probability density functions for n=1 and n=2 are shown below.

The Hermite polynomials satisfy a recursion relation of the form

Hn+1(x) = xH(x) + H'n(x)

From this and from the analysis above it is seen that the sequence of eigenfunctions of A*A for m=0, 1, …, M correspond to the solutions to the Schrödinger equations for the harmonic oscillators for principal quantum numbers n=1, 2, …. Thus what the number operator A*A counts is the number of energy units. The wave functions and probability density functions could be construed to be those for n simple particles.

In a system of units such that the speed of light in a vacuum is unity and h (h-bar) (Planck's constant divided by 2π) is also equal to unity the eigenvalues for the eigenfunctions of H^ are integers plus ½. P.A.M. Dirac proposed a brilliant bit of notation. He suggested that the eigenfunctions be denoted in the form |n> and are thus labeled by values closely related to their eigenvalues. This means that the time independent Schrödinger equation

H^ |n> = (n+½)|n>


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