From Classical Mechanics to Quantum Field Theory

194 From Classical Mechanics to Quantum Field Theory. A Tutorial

The solution of Eq. (3.4) in terms of the initial Cauchy conditions (x(0),x ̇(0)) is

x(t)=x(0) cosωt+

x ̇(0)

ω

sinωt

=

1

2

[

x(0) +i

x ̇(0)

ω

]

e−iωt+

1

2

[

x(0)−i

x ̇(0)

ω

]

eiωt. (3.5)

Upon Legendre transformation

p=mx, ̇

and the Poisson bracket structure,

{x, x}={p, p}=0; {x, p}= 1 (3.6)

one obtains the Hamiltonian of the harmonic oscillator

H=

1

2 m

p^2 +

1

2

mω^2 x^2 ,

and the corresponding Hamilton equations of motion

x ̇={x, H}=

p

m

, p ̇={p, H}=−mω^2 x,

that are equivalent to Newton’s equations Eq. (3.4).
In field theory, it is very convenient to use the coherent variables

a=

√

mω

2

x+

i

√

2 mω

p; a∗=

√

mω

2

x−

i

√

2 mω

p,

in terms of which time evolution becomes

a(t)=a(0)e−iωt,

which gives Eq. (3.5),

x(t)=

√^1

2 mω

(a(t)+a∗(t))=

√^1

2 mω

(

a(0)e−iωt+a∗(0)eiωt

)

.

The Quantum Harmonic Oscillator
The canonical quantization prescription proceeds by mapping the classical ob-
servables, positionxand momentumpinto selfadjoint operators ˆxand ˆpof a
Hilbert spaceHand replacing the Poisson bracket{·.·}by the operators commu-
tator−i[·,·], i.e.

{x, p}=1⇒[ˆx,pˆ]=iI, (3.7)

where we assume that the Planck constant=1.