Causal Interpretation of the Quantum Harmonic Oscillator
Causal Interpretation of the Quantum Harmonic Oscillator
The harmonic oscillator is an important model in quantum theory that could be described by the Schrödinger equation: , () with . In this Demonstration a causal interpretation of this model is applied. A stable (nondispersive) wave packet can be constructed by a superposition of stationary eigenfunctions of the harmonic oscillator. The solution is a wave packet in the (, ) space where the center of the packet oscillates harmonically between with frequency . From the wavefunction in the eikonal representation , the gradient of the phase function and therefore the equation for the motion could be calculated analytically. The motion is given by , where are the initial starting points. The trajectories of the particles oscillate with the amplitude and frequency and they never cross. In practice, it is impossible to predict or control the quantum trajectories with complete precision. The effective potential is the sum of quantum potential (QP) and potential that leads to the time-dependent quantum force: .
-ψ-iψ+Vψ=0
1
2
2
∂
x
∂
t
h=m=1
V=
1
2
2
ω
2
x
x
t
±
x
0
ω
ψ=ω--t+2x-cos(ωt)
1
2
2
x
-ωt
x
0
x
0
1/4
ω
S
x(t)=cos(ωt)+
x
0
x
P
x
P
x
0
ω
V
F=-(QP+V)=-cos(ωt)
∂
x
x
0
2
ω
On the right side, the graphic shows the squared wavefunction and the trajectories. The left side shows the particles' positions, the squared wavefunction (blue), the quantum potential (red), the potential (black), and the velocity (green). The quantum potential and the velocity are scaled down.