From Classical Mechanics to Quantum Field Theory

A Short Course on Quantum Mechanics and Methods of Quantization 15

Notice that, as function of the momentum, the Hamiltonian (1.45) has the same
expression as in the classical case: to obtain it we just need to replace the function
p with the operator pˆ. This can be considered as a first rule of quantization.
The corresponding (continuous) eigenvalues and (generalized) eigenfunctions
are easily found to be:

Ep=

|p |^2

2 m ,ψp(^ x)=e

ıp·x. (1.46)

Example 1.2.7. The 1D harmonic oscillator.
Many interesting dynamical systems are described by a Hamiltonian of the form:

H=^ p

2

2 m

+V( x), (1.47)

that gives the energy of a particle in an external potentialV. In the previous
example, we have seen that, to get the quantum version of a free particle, we need
just to replace the momentum pwith the operator pˆin the classical Hamiltonian
function. In a similar way, one is led to consider the quantum version of (1.47) to
be given by the operator:

H=

pˆ^2

2 m+V(

xˆ). (1.48)

To see to what extent we can use this approach, let us consider the 1D harmonic
oscillator whose classical Hamiltonian is given by

H=

ω

2

(

p^2 +x^2

)

. (1.49)

Here we work with suitable units in which bothxandpare adimensional and
{x, p}=1/. Thus we can consider the quantum Hamiltonian:

Hˆ=ω

2

(ˆp^2 +ˆx^2 ), (1.50)

with [ˆx,pˆ]=I. One may argue, however, that (1.49) can be rewritten in three
different equivalent ways:

H=

ω

2

(p^2 +x^2 )=

ω

2

(x+ip)(x−ip)=

ω

2

(x−ip)(x+ip), (1.51)

sincex, pare commuting functions. At the quantum level, however, this equiv-
alence no longer holds true (see (1.16)) and one would obtain different quantum
Hamiltonians, specifically, differing by some constants. This ambiguity in the pro-
cess of quantization can be resolved by introducing the so-called “symmetrization
postulate”[ 8 ]: to obtain the quantum HamiltonianHˆout of the classical oneH,