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.)