Second Quantization is a body of physical analysis in quantum field theory that focuses on the occupation numbers of states rather than the physical states of particular particles. It talks of operations which create and which annihilate a particle in a field. When this is applied to the quantum analysis of a harmonic oscillator it is seen that what the term particle refers to second quantization is entirely different from what particle refers to in other contexts. Here is what the probability density function looks like for a particular level of energy.

The number of peaks of the probability density function is 2n+1, where n is the principal quantum number.. What an increase in n adds to the solution is roughly depicted below.

It is n which the creation and annihilation operators change. The details will be given later. First the mathematical background will be given.

Mathematical Background

Let V be a vector space of complex elements. A function that maps V onto V is called a transformation. A linear transformation is called an operator. Operators can be added and substracted. The concatenation of two operators is defined and will be referred to as the multiplication of operators. This multiplication of operators is associative (AB)C=A(BC) but not communicative AB≠BA,.

The complex conjugate of an element u of V is denoted as u*. If an operator A maps v to u then the operator that maps v to u* is denoted as A* and is called the adjoint of A..

If V is finite dimensional then an operator is just a matrix. When V consists of functions it is infinite dimensional.

Let A and B be operators on V. The commutator [A, B] is defined as

[A, B] = AB − BA

This leads to

AB = BA + [A, B]
BA = AB + [B, A] = AB − [A, B]

Note that the commutator is anti-symmetric, AB=−BA.

Some General Relationships for
Operators on a Function Space

Let A, B and C be operators. Then for the commutator [AB, C]

[AB, C] = A[B, C] − [A, C]B

Proof:

[AB, C] = (AB)C − C(AB)
and from the associativity
of operator applications
[AB, C] = A(BC) − (CA)B
and from subtracting and
adding A(CB) on the right
[AB, C] = A(BC) − A(CB) + A(CB) − (CA)B
and hence
[AB, C] = A[B, C] + (AC)B − (CA)B
and finally
[AB, C] = A[B, C] + [A, C]B

Now let A be any operator and A* its adjoint (conjugate operation). Applying the above indentity to [A*A, A] gives

[A*A, A] = A*[A, A] + [A*, A]A
but [A, A]=0, so
[A*A, A] = [A*, A]A
and
since commutation
is antisymmetric
[A*A, A] = −[A, A*]A

Likewise

[A*A, A*] = A*[A, A*] + [A*, A*]A = A*[A, A*]

Canonical Quantization

Canonical quantization is defined as

[A, A*] = I

where I is the identity operator. If A satisfies the canonical quantization condition then the above two relationships reduce to

[A*A, A] = −A
and
[A*A, A*] = A*

These can also be expressed as

(A*A)A = A(A*A)−A = A(A*A−I)
and
(A*A)A* = A*(A*A)+A* = A*(A*A+I)

An Eigenvector for an Operator
and its Eigenvalue

For and operator A, any element v of V such that

Av = αv

is called an eigenvector of A and α is its eigenvalue.

The Notation of P.A.M. Dirac

Dirac had a brilliant idea for labeling vectors. Instead of using an arbitrary letter to denote a vector, he suggested that it should be labeled by its eigenvalue. Thus if v is an eigenvector of A with the eigenvalue β then v is denoted as |β>. The symbology |..> Is part of what is known as Dirac's bra...ket notation.

Let an eigenvalue of A*A be denoted as α. Expressed in the Dirac notation this means

A*A|α> = α|α>

Not only is |α> an eigenfunction of A*A, but A|α> is also an eigenfunction of A*A because

(A*A)A|α> = A(A*A−I)|α> = A{α|α> − |α>} = A(α−1)|α>
thus
(A*A)(A|α>) = (α−1)(A|α>)

So (A|α>) is also an eigenfunction of (A*A) but with an eigenvalue that is 1 less than that of |α>.

A reapplication of the above would show that Aⁿ|α> for n over some range is an eigenfunction of A*A with (α−n) as its eigenvalue. Therefore the eigenfunction of Aⁿ|α> is denoted as |(α−n)>.

Similarly A*|α> is an eigenfunction of A*A with an eigenvalue of (α+1) and therefore (A*)ⁿ|α> is an eigenfunction of A*A with an eigenvalue of (α+n). It is represented as |(α+n)>.

Relationships between Eigenfunctions

Let |α> be an eigenfunction of A*A. Then A|α> and A*|α> are also an eigenfunction of A*A. Thus

A*|α> = |(α+1)> = C|α> A*|(α-1)> = α|(α-1)> A*A(A|α>)

The Integralness of the Eigenvalues of A*A

The eigenfunction of an operator cannot be the zero function. Therefore there must be an integer m such that A^m|α> is an eigenfunction of A*A but A^m+1 |α> is not. This implies that (α−m) is equal to 0 and hence α is equal to that integer m. Therefore the eigenvalues of A*A are necessarily integers from 0 to some maximum integer m.

What we found above is that the assumption of canonical quantification for the commutator is sufficient to assure the existence of the creator, annihilator and number operator without any reference to the physics of the particles.

A Physical System

A harmonic oscillator is a mass m attached to a spring of stiffness coefficient k. The deviation from equilibrium is denoted as x. The total energy E IS

E - ½mv² + ½kx²

where v is velocity (dx/dt).

The system oscillates sinusoidally at a frequency ω equal to (k/m)^½. This means that the total energy may be expressed as

E = ½mv² + ½mω²x²

When this is expressed in terms of momentum p=mv the result is called the Hamiltonian function for the system; i.e.,

H = ½p²/m + ½kx²

Quantum Analysis

For the quantum theoretic analysis of the harmonic oscillator p and x in the Hamiltonian function must be replaced by their operator representations. The operator representation of the deviation x is very simple; it is just multiplication by x. The momentum operator is

p^ = −ih(∂/∂x)

where i is the imaginary unit, the square root of −1 and h is Planck's constant divided by 2π.

Thus the Hamiltonian operator H^ for a harmonic oscillator is

H^ = ½(p^)²/m + ½mω²(x^)² = −½h²(∂²/x∂²)

Let φ(x) denote the complex-valued function such that its squared value |φ(x)|² is the probability density function at x. It is called the wave function and its values are determined as a solution to the time-independent Schrödinger equation

H^φ(x) = Eφ(x)

where E is the total energy of the oscillator. This equation has solutions only for discrete values of E, which are positive integers times ω. The positive integer for the system is called its principal quantum number and will be denoted as n. This is the first quantization of a harmonic oscillator.

Second Quantization

Consider the commutators of p^ and x^,

[p^, x^]^φ = p^(x^φ) − x^(p^φ)
= (p^x)φ+xp^φ − x(p^φ)
= (p^x)φ = −ihφ
therefore
[p^, x^] = −ih
and
[x^, p^] = ih

Now consider the two operators

α =γ(x^ + βp^)
and
α* = γ(x^ − βp^)

where β=i/(mω) and γ=(mω/(2h))^½.

Now consider [α, α*],

[α, α*] = [γ(x^ + βp^), γ(x^ − βp^)] = γ² [(x^ + βp^), (x^ − βp^)]
[(x^ + βp^), (x^ − βp^)] = [x^, (x^ − βp^)] + [βp^, (x^ − βp^)]
= [x^, x^] −β[x^, p^] + β[p^, x^] − β²[p^, p^]
But [x^, x^]=0, [p^, p^]=0
and [p^, x^]=−[x^, p^] so
[α, α*] = γ² (−2β)[x^, p^] = γ² (−2β) ih

Replacing β and γ by their defined values gives

[α, α*] = (mω/(2h))(−2i/(mω)(ih) = 1^

where 1^ is just the identity operation. This means that α satisfies the canonical quantification condition and therefore α is a creation operator, α* is an annihilation operator and α*α is a number (counting) operator. That is to say, according to the conventional presentation of second quantization, α operating on a field increases the number of particles by one, α* decreases the number of particles in a field by one and α*α counts the number of particles in a field. But what does the probability density function look like for a harmonic oscillator. Here is the solution for principal quantum number n equal to 30.

The number of peaks of the probability density function is 2n+1. What an increase in n adds to the solution is roughly depicted below.

The standard second quantization is, in effect, calling a pair of peaks of the probability density function a particle even though this does not correspond to a particle in the usual sense of the term.

A photon, as a perturbation in an electromagnetic field, would fit the notion of particle as this term is used in second quantization. However, in general, the use of the term particle in second quantization anaysis is misleading, very misleading. Peaks in probability density correspond to states which a particle passes through in its periodic path.

Conclusions

When a physical system is analyzed the eigenvalues correspond to energy quanta which may or may not have any correspondence to particles in the usual sense of that term. For example, consider a harmonic oscillator. Its energy is proportional to a integer n, called its principall quantum number. The number of peaks of the probability density function is 2n+1.