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WOLFRAM|DEMONSTRATIONS PROJECT

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
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(nm)=
2
〈
Ψ
m
|
ψ
n
〉
. These results can be seen by selecting "show transition probabilities" and the initial state
n
.
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