Coupled Quantum Harmonic Oscillators

view mode

function

wavefunction

PDF

3D

2D

wavefunction configuration

C

00

θ

00

C

01

θ

01

C

02

θ

02

C

10

θ

10

C

11

θ

11

C

12

θ

12

C

20

θ

20

C

21

θ

21

C

22

θ

22

ω

2

time

0

Re(Ψ(

x

1

,

x

2

,t))

Im(Ψ(

x

1

,

x

2

,t)), t = 0

This Demonstration models a coupled system of quantum harmonic oscillators with two electron masses in SI units; two particles have displacements

x

1

and

x

2

from their equilibrium points. The outer springs have an angular frequency

ω=1

and the inner spring an angular frequency

ω

2

, which can be varied. Thus, the potential energy term of the Hamiltonian is

1

2

m

e

2

ω

2

x

1

+

1

2

m

e

2

ω

2

(

x

2

-

x

1

)

+

1

2

m

e

2

ω

2

x

2

.

The corresponding Schrödinger equation can be solved with the substitutions

x

+

=

x

1

+

x

2

and

x

-

=

x

2

-

x

1

2

(which are the normal mode coordinates), which reduces the problem to a two-dimensional harmonic oscillator. The energy eigenstates are then

E

+

=m+

1

2

ℏω

and

E

-

=n+

1

2

ℏ

2

ω

+2

2

ω

2

and the wavefunction is

Ψ(

x

+

,

x

-

,t)=

∑

m,n

C

mn

iθ

mn



i(

E

+m

+

E

-n

)

ℏ

t

e

〈

x

+

,

x

-

+m,-n〉

.

This Demonstration plots

Ψ(

x

1

,

x

2

,t)

(substituting back the regular displacements) and its modulus squared (which is the PDF of the displacements) for states

(0,0)

to

(2,2)

and their linear combinations, in which you can vary both magnitudes

C

mn

and phases

θ

mn

. Some plots are significantly slower to display, especially the ones with more complex wavefunctions.