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WOLFRAM|DEMONSTRATIONS PROJECT

Quantum Theory of the Damped Harmonic Oscillator

​
ω
1
γ
0.2
n
0
1
2
t
2
​The quantum theory of the damped harmonic oscillator has been considered a simple model for a dissipative system, usually coupled to another oscillator that can absorb energy or to a continuous heat bath [1–3]. This Demonstration treats a quantum damped oscillator as an isolated nonconservative system, which is represented by a time-dependent Schrödinger equation. It is conjectured that spontaneous transition to a lower state will occur when the energy is reduced to that of the lower state, and this recurs sequentially, down to the ground state, which asymptotically disappears as the energy approaches zero. Within this model, the obtained result is an exact solution of the time-dependent Schrödinger equation.
As derived in the Details below, the time-dependent wavefunction is given by
ψ
n
(x,t)=
1
n
2
n!
1/4
ω
π
-(ω-iγ)
2
x
2
e
H
n
(
ω
x)
-i(n+1/2)ωt
e
-γt/2
e
,
n=0,1,2,…
,
where
H
n
is an Hermite polynomial, and
ω=
2
ω
0
-
2
γ
,
where
ω
0
is the natural frequency of the undamped oscillator and
γ
is the damping constant. Atomic units
ℏ=m=1
are used. The real part of the expectation value of the Hamiltonian is assumed for the time-dependent energy, which gives
ϵ
n
(t)≈n+
1
2
ω
-γt
e
.
The graphic shows the probability density
ρ
n
(x,t)=
*
ψ
n
(x,t)
ψ
n
(x,t)
and energy
ϵ
n
(t)
as functions of
t
for
n=0,1,2
. The inset shows the energy, as downward transitions occur, asymptotically decreasing to 0.
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