Coupled Quantum Harmonic Oscillators
Coupled Quantum Harmonic Oscillators
This Demonstration models a coupled system of quantum harmonic oscillators with two electron masses in SI units; two particles have displacements and from their equilibrium points. The outer springs have an angular frequency and the inner spring an angular frequency , which can be varied. Thus, the potential energy term of the Hamiltonian is ++.
x
1
x
2
ω=1
ω
2
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 =+ and = (which are the normal mode coordinates), which reduces the problem to a twodimensional harmonic oscillator. The energy eigenstates are then =m+ℏω and =n+ℏ+2 and the wavefunction is .
x
+
x
1
x
2
2
x

x
2
x
1
2
E
+
1
2
E

1
2
2
ω
2
ω
2
Ψ(,,t)=t〈,+m,n〉
x
+
x

∑
m,n
C
mn
iθ
mn
i(+)
E
+m
E
n
ℏ
e
x
+
x

This Demonstration plots (substituting back the regular displacements) and its modulus squared (which is the PDF of the displacements) for states to and their linear combinations, in which you can vary both magnitudes and phases . Some plots are significantly slower to display, especially the ones with more complex wavefunctions.
Ψ(,,t)
x
1
x
2
(0,0)
(2,2)
C
mn
θ
mn