Charged Harmonic Oscillator in Electric Field

oscillator frequency ω

1

electric field E

2

show transition probabilities

For an electron (mass

m

, charge

-e

) bound by a harmonic potential

1

2

m

2

ω

2

x

and acted upon by a constant external electric field

E

, the Schrödinger equation can be written as

-

2

ℏ

2m

ψ

n

''(x)+

1

2

m

2

ω

2

x

-eEx

ψ

n

(x)=

E

n

ψ

n

(x)

.

An exact solution can be obtained by completing the square in the potential energy [1]:

V(x)=

1

2

m

2

ω

x-

eE

m

2

ω

-

2

e

2

E

2m

2

ω

=

1

2

m

2

ω

2

ξ

-

2

e

2

E

2m

2

ω

.

Introducing the new variable

ξ=x-

eE

m

2

ω

, the Schrödinger equation can be written as

-

2

ℏ

2m

Ψ

n

''(ξ)+

1

2

m

2

ω

2

ξ

Ψ

n

(ξ)=

E

n

+

2

e

2

E

2m

2

ω

Ψ

n

(ξ)=n+

1

2

ℏω

Ψ

n

(ξ)

,

n=0,1,2,…

,

making use of the known solution of the standard harmonic-oscillator problem, expressed in terms of

ξ

. The perturbed energies are shifted downward by a constant term:

E

n

=n+

1

2

ℏω-

2

e

2

E

2m

2

ω

.

The graphic shows the potential energy and energy levels for the unperturbed (in black) and perturbed (in red) oscillator, for selected values of

ω

and

E

. For simplicity, atomic units,

ℏ=m=e=1

, are used.

If the electric field is turned on during a time interval

Δt

that is short compared to the oscillation period

2π/ω

, the sudden approximation in perturbation theory can be applied [2]. Accordingly, the transition probability from state

n

to a state

m

is given by

P(nm)=

2

〈

Ψ

m

|

ψ

n

〉

. These results can be seen by selecting "show transition probabilities" and the initial state

n

.