# Charged Harmonic Oscillator in Electric Field

Charged Harmonic Oscillator in Electric Field

For an electron (mass , charge ) bound by a harmonic potential m and acted upon by a constant external electric field , the Schrödinger equation can be written as

m

-e

1

2

2

ω

2

x

E

-''(x)+m-eEx(x)=(x)

2

ℏ

2m

ψ

n

1

2

2

ω

2

x

ψ

n

E

n

ψ

n

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

V(x)=mx--=m-

1

2

2

2

ω

eE

m

2

ω

2

e

2

E

2m

2

ω

1

2

2

ω

2

ξ

2

e

2

E

2m

2

ω

Introducing the new variable , the Schrödinger equation can be written as

ξ=x-

eE

m

2

ω

-''(ξ)+m(ξ)=+(ξ)=n+ℏω(ξ)

2

ℏ

2m

Ψ

n

1

2

2

ω

2

ξ

Ψ

n

E

n

2

e

2

E

2m

2

ω

Ψ

n

1

2

Ψ

n

n=0,1,2,…

making use of the known solution of the standard harmonic-oscillator problem, expressed in terms of . The perturbed energies are shifted downward by a constant term:

ξ

E

n

1

2

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 and . For simplicity, atomic units, , are used.

ω

E

ℏ=m=e=1

If the electric field is turned on during a time interval that is short compared to the oscillation period , the sudden approximation in perturbation theory can be applied [2]. Accordingly, the transition probability from state to a state is given by . These results can be seen by selecting "show transition probabilities" and the initial state .

Δt

2π/ω

n

m

P(nm)=

2

〈|〉

Ψ

m

ψ

n

n