From Classical Mechanics to Quantum Field Theory

104 From Classical Mechanics to Quantum Field Theory. A Tutorial

withD(H 0 ):=S(R). Above,x^2 is the multiplicative operator andm, ω >0are
constants.
To go on, we start by defining a triple of operators,a, a+,N:S(R)→L^2 (R,dx)
as

a+:=

√

mω

2 

(

x−



mω

d

dx

)

,a:=

√

mω

2 

(

x+



mω

d

dx

)

, N:=a+a.

aanda+are called creation and annihilation operators. These operators have
the same domain which is also invariant: a(S(R))⊂S(R),a+(S(R))⊂S(R),
N(S(R))⊂S(R). It is also easy to see, using integration by parts thata+⊂a†
and thatN(callednumber operator) is Hermitian an also symmetric because
S(R)isdenseinL^2 (R,dx). By direct computation, exploiting the given definitions
one immediately sees that

H 0 =

(

a†a+

1

2 I

)

=

(

N+

1

2 I

)

.

Finally, we have the commutation relationsonS(R)

[a, a†]|S(R)=I|S(R). (2.38)

Supposing that there existsψ 0 ∈S(R) such that

||ψ 0 ||=1,aψ 0 = 0 (2.39)

starting form (2.38) and using an inductive procedure on the vectors

ψn:=

√^1

n 1

(a†)nψ 0 (2.40)

it is quite easy to prove that (e.g., see[ 5 ]for details), forn, m=0, 1 , 2 ,...,the
relations hold

aψn=

√

nψn− 1 ,a†ψn=

√

n+1ψn+1, 〈ψn,ψm〉=δnm. (2.41)

Finally, theψnare eigenvectors ofH 0 (andN)since

H 0 ψn=ω

(

a†aψn+

1

2

ψn

)

=ω

(

a†

√

nψn− 1 +

1

2

ψn

)

=ω

(

ψn+1+^1

2

ψn

)

=ω

(

n+^1

2

)

ψn. (2.42)

As a consequence,{ψn}n∈Nis an orthonormal set of vectors. This set is actually
a Hilbertian basis because (1), a solution inS(R) of (2.39) exist (an is unique):

ψ 0 (x)=

1

π^1 /^4

√

s

e−

2 xs^22

,s:=

√



mω

,