Time Evolution of Quantum-Mechanical Harmonic Oscillator with Time-Dependent Frequency
Time Evolution of Quantum-Mechanical Harmonic Oscillator with Time-Dependent Frequency
The harmonic oscillator, described by the Schrödinger equation
iℏ=-ψ(x,t)+ψ(x,t)
∂ψ(x,t)
∂t
2
ℏ
2m
2
∂
∂
2
x
2
ω
2
x
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)+ψ(x,t)
∂ψ(x,t)
∂t
2
ℏ
2m
2
∂
∂
2
x
2
ω(t)
2
x
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 and for various time-dependent frequencies of the functional form
2
ψ(x,t)
arg(ψ(x,t))
ω(t)=+t+cos(t)+cos(t)
c
0
c
1
c
c,1
ω
1
c
c,2
ω
2
usingtheinitialwavefunction(movingharmonicoscillatorgroundstate)
ψ(x,0)=
1/4
2
π
-
2
x
e
ikx
e
("Calculating ..." sometimes appears when the Demonstration cannot compute a solution.)