Heisenberg Picture

In the SchroÐdinger approach to quantum mechanics, operators O

` are time-independent and the wave-

function of the system carries all the time-dependent information. The expectation value of O`

obeys the

Ehrenfest relation

i Ñd XO

`\

d t= ZBO

`, HF^

Buried in this is the time-dependent wavefunction

ΨHtL\ = ã-ä H tÑ ΨH0L] = UHtL ΨH0L]

If we make this substitution into an expectation value we get

XΨHtL O ΨHtL\ = YΨH0L U-1HtL OUHtL ΨH0L] º XΨH0L OHtL ΨH0L\

where in this Heisenberg picture the (time-independent) SchroÐdinger operator O becomes time-depen-

dent:-

OHtL = U-1HtL OUHtL

The Heisenberg operator OHtL obeys an equation similar to the Ehrenfest equation, but without the

expectation values:

(1)ä Ñ

d O

`HtL

d t

= BO`

HtL, HF

Such an equation is understood to operate on a time-independent wavefunction Ψ0\ = ΨH0L\.

An example of the usefulness of this is to discover the operator corresponding to the vector potential Ax

of a single mode of the electromagnetic field, given the electric field operator

ExHtL = EIaHtL + a† HtLM sinHk zL = -

d AxHtLd t

Comparing this to Eq. (1) gives

(2)d AxHtL

d t

=-ä

Ñ

@AxHtL, HD = -EIaHtL + a† HtLM sinHk zL

Since @a, HD = Ñ Ω a, Aa†, HE = -Ñ Ω a

†, we get

d aHtL

d t= -ä Ω aHtL so aHtL = a ã

-ä Ω t, a

† HtL = a†

ãä Ω t

so

ExHtL = EIa ã-ä Ω t

+ a†

ãä Ω tM sinHk zL = -

d AxHtLd t

so

AxHtL =-ä E

ΩIa ã

-ä Ω t- a

†ã

ä Ω tM sinHk zL

so the correct Schrodinger vector potential operator is

Ax = Ax H0L =-ä E

ΩIa - a

† M sinHk zL