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Time Evolution of Quantum-Mechanical Harmonic Oscillator with Time-Dependent Frequency

​
show
absolute value squared
phase
total evolution time
domain size
wave vector k
time-dependent frequency:
ω(t) = 1 + 1.2t + 0cos(t) + 0cos(2t)
c
0
c
t
c
1
ω
1
c
2
ω
2
The harmonic oscillator, described by the Schrödinger equation
iℏ
∂ψ(x,t)
∂t
=-
2
ℏ
2m
2
∂
ψ(x,t)
∂
2
x
+
2
ω
2
x
ψ(x,t)
is a central textbook example in quantum mechanics. Its time evolution can be easily given in closed form. More generally, the time evolution of a harmonic oscillator with a time-dependent frequency
ω(t)
iℏ
∂ψ(x,t)
∂t
=-
2
ℏ
2m
2
∂
ψ(x,t)
∂
2
x
+
2
ω(t)
2
x
ψ(x,t)
can also be given in quadratures. This allows the efficient solution of the Schrödinger equation as a system of just three coupled nonlinear ordinary differential equations.
This Demonstration lets you see
2
ψ(x,t)
and
arg(ψ(x,t))
for various time-dependent frequencies of the functional form
ω(t)=
c
0
+
c
1
t+
c
c,1
cos(
ω
1
t)+
c
c,2
cos(
ω
2
t)
usingtheinitialwavefunction(movingharmonicoscillatorgroundstate)
ψ(x,0)=
1/4
2
π
-
2
x
e
ikx
e
.
("Calculating ..." sometimes appears when the Demonstration cannot compute a solution.)
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