130_notes.dvi

Γi→f=

2 πVni^2

̄h

ρf(E)

whereρf(E) is the density of final states. When particles (like photons or electrons) are emitted,
the final state will be a continuum due to the continuum of states available to a free particle. We
will need to carefully compute the density of those states, often known asphase space.

28.3 Examples

28.3.1 Harmonic Oscillator in a Transient E Field

Assume we have an electron in a standard one dimensional harmonic oscillator of frequencyωin its
ground state. An weak electric field is applied for a time intervalT. Calculate the probability to
make a transition to the first (and second) excited state.

The perturbation iseExfor 0< t < Tand zero for other times. We can write this in terms of the
raising an lowering operators.

V=eE

√

̄h

2 mω

(A+A†)

We now use our time dependent perturbation result to compute thetransition probability to the
first excited state.

cn(t) =

1

i ̄h

∫t

0

eiωnit

′

Vni(t′)dt′

c 1 =

1

i ̄h

eE

√

̄h

2 mω

∫T

0

eiωt

′

〈 1 |A+A†| 0 〉dt′

=

eE

i ̄h

√

̄h

2 mω

∫T

0

eiωt

′

dt′

=

eE

i ̄h

√

̄h

2 mω

[

eiωt

′

iω

]T

0

= −

eE

̄hω

√

̄h

2 mω

[

eiωT− 1

]

= −

eE

̄hω

√

̄h

2 mω

eiωT/^2

[

eiωT/^2 −e−iωT/^2

]

= −

eE

̄hω

√

̄h

2 mω

eiωT/^22 isin(ωT/2)

P 1 =

e^2 E^2

̄h^2 ω^2

̄h

2 mω

4 sin^2 (ωT/2)

P 1 =

2 e^2 E^2

m ̄hω^3

sin^2 (ωT/2)