From Classical Mechanics to Quantum Field Theory

196 From Classical Mechanics to Quantum Field Theory. A Tutorial

In the harmonic oscillator, the fundamental observables are the positionxand
the momentump. Since by the quantization prescription ˆxand ˆpdo not commute,
the space of states has to be infinite dimensional, because if dimH=n<∞,
tr [x, p]=0= itrI=in. Moreover, any other operatorOthat commutes with
both fundamental operators [ˆp, O]=[ˆx, O] = 0 has to be proportional to the
identity O=cI. Now, since the projectorPto the subspace spanned by the
stationary states|n〉commutes with ˆxand ˆp([ˆp,P]=[ˆx,P] = 0), it follows that
P=cI. This implies that the subspace spanned by the vectors|n〉is complete,
i.e. does coincide with the whole Hilbert spaceH.
Although the principles of quantum mechanics are identical for all quantum
system and all separable Hilbert spaces are isomorphic, different systems can be
distinguished by their algebra of observables. In the particular case of the har-
monic oscillator, the position ˆxand momentum ˆpoperators belong to the algebra
of observables, which does not only imply that the Hilbert space is infinite dimen-
sional, but also that it can be identified with the space of square integrable func-
tionsL^2 (R) of the position (Schr ̈odinger representation) or momentum (Heisenberg
representation).
In the Schr ̈odinger representation the ground state reads

〈x| 0 〉=

√

ω

π

e−^12 ωx

2

.

The position and momentum operator are given by

ˆxψ(x)=xψ(x); pψˆ(x)=−i∂xψ(x).

The Hamiltonian reads

Hˆ=−^1

2 m

d^2

dx^2

+

1

2

mω^2 x^2 ,

and the creation and destruction operators are

a=

1

√

2 mω

(

d

dx

+mωx

)

; a†=

1

√

2 mω

(

−

d

dx

+mωx

)

.

It is easy to check that the excited states Eq. (3.8) are given by

〈x|n〉=Hn(x)e−

(^12) ωx 2
,
in terms of the Hermite polynomials
Hn(x)=e
(^12) ωx 2 1
√
(n+1)!
(a†)ne−
(^12) ωx 2
.
The generalization for multidimensional harmonic oscillators is straight-
forward. If we have n-harmonic oscillators of frequencies ωi and masses