Srednicki — Quantum Field Theory · Problem 7.3

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · path-integral-harmonic-oscillator

Problem

a) Use the Heisenberg equation of motion,

\dot{A} = i[H,A]

, to find explicit expressions for

\dot{Q}

and

\dot{P}

. Solve these to get the Heisenberg-picture operators

Q(t)

and

P(t)

in terms of the Schr\"odinger picture operators

Q

and

P

. b) Write the Schr\"odinger picture operators

Q

and

P

in terms of the creation and annihilation operators

a

and

a^\dagger

, where

H = \hbar\omega(a^\dagger a + \frac{1}{2})

. Then, using your result from part (a), write the Heisenberg-picture operators

Q(t)

and

P(t)

in terms of

a

and

a^\dagger

. c) Using your result from part (b), and

a|0\rangle = \langle 0|a^\dagger = 0

, verify eqs. (7.16) and (7.17).