WOLFRAM NOTEBOOK

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
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
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.

Details

The plots are
• 3D wavefunction plots with their real and complex parts
• wavefunction real and complex density plots
• 3D PDF plots
• cross sectional plots of the PDF plot at its expectation value (with the cross section planes orthogonal to the
x
1
and
x
2
axes)
• PDFs of the
x
1
and
x
2
displacements
• density plots of the
x
1
and
x
2
displacements over time

References

[1] R. M. McDermott and I. H. Redmount, "Coupled Classical and Quantum Oscillators." arxiv.org/abs/quant-ph/0403184v2.

External Links

Permanent Citation

Humberto Munoz Bauza, Rachael M. McDermott, Ian H. Redmount
​
​"Coupled Quantum Harmonic Oscillators"​
​http://demonstrations.wolfram.com/CoupledQuantumHarmonicOscillators/​
​Wolfram Demonstrations Project​
​Published: September 20, 2013
Wolfram Cloud

You are using a browser not supported by the Wolfram Cloud

Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.


I understand and wish to continue anyway »

You are using a browser not supported by the Wolfram Cloud. Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.