From Classical Mechanics to Quantum Field Theory

12 From Classical Mechanics to Quantum Field Theory. A Tutorial

Example 1.2.4. Bosonic systems.
Bosonic creation/annihilationoperators are defined as two operatorsa, a†onH
such that:

[a, a†]=I. (1.28)

Notice that this relation implies that the operatoraand its adjointa†cannot be
bounded, henceHhas to be infinite dimensional. It is useful to define the number
operatorN≡a†a, which is self-adjoint and satisfies: [N,a]=a,[N,a†]=a†.
Then it is an exercise to show[8; 15]thatNis such that:

• its spectrum is composed of all integers:σ(N)={ 0 , 1 , 2 ,···};


• its eigenvector| 0 〉corresponding to the lowest eigenvalueλ 0 =0issuch
  that:a| 0 〉=0;


• its eigenvector |n〉corresponding to then−theigenvalue λn =nis
  given by:

|n〉=

√^1

n!

(a†)n|ψ 0 〉. (1.29)

Also, the following relations hold:

a†|ψn〉=

√

n+1|ψn+1〉, (1.30)

a|ψn〉=

√

n|ψn− 1 〉, (1.31)

showing thata(a†) moves from one eigenstate to the previous (next) one, acting
as ladder operators. The complete set of normalized eigenstates{|n〉}∞n=0provide
a basis forH, which is called Fock basis.
It is interesting to look for the coordinate representation of (1.28). OnH=
L^2 (R)={ψ(x)}, one has[ 15 ]^8 :

a=√^1

2

(

x+ d

dx

)

, (1.32)

a†=√^1

2

(

x− d

dx

)

, (1.33)

while:

| 0 〉=C 0 e−x

(^2) / 2
, (1.34)
|n〉=Cne−x
(^2) / 2
Hn(x), (1.35)
whereHn(x) is the n-th Hermite polynomial, andCnnormalization constants.
(^8) See also sect. 3.2.1 of the third part of this volume.