Now it turns out that if we use the information that the particle is approximately at a particular location, we must give up the idea that the particle has a definite energy. The resulting probability density then does depend on time: the middle of the probability density (which in the above we connect with the idea of "a" position) is seen to move much as classical mechanics says its mythical point particles would move.
Let's start by seeing how this would all work classically: we say the particle starts approximately at z_{0} but with some uncertainty: z_{0}. We release the particle on average with zero initial velocity, but because of Heisenberg uncertainty relation, there must be some spread of velocity v_{0}, which is sometime up and sometimes down. If we knew exactly where the particle was starting from and its speed, we'd know exactly where it was:
z(t)=z_{0}+v_{0}t+½at^{2}:
But because of the initial uncertainty in the velocity, there is growing uncertainty in position:
z(t)= [(z_{0})^{2}+ (v_{0}t)^{2}]^{½}
Thus the packet of probability is becoming wider as the central value of the probability follows the usual classical relation:
z(t)=z_{0}+½at^{2}
In our dimensionless variables this reads:
z'(t')=z'_{0}-t'^{2}
Our particular application will have z'_{0}=121.5, z'_{0}=0.873, v'_{0}=1.50.
Here is the behavior of the classical probability distributions:
Here are the quantum mechanical results for t'=0,1,2,3,4
The last plot shows the average position of the wavefunction vs time. All of the above look much like the classical result. Note that the "broadening" of the wavefunction is simply the result of our uncertainty in the initial velocity. Since the motion is periodic, one period later the particle must be back where it started, so the wavefunction "narrows" on the upward bounce, and looks like the below one period (t'=22) later:
For reasons that should be clear after the next page, over several periods the wave packet will broaden out and eventually the probability will be distributed throughout (and a bit beyond) the classically allowed region.
Here is some data on the location of the particle:
Time | Height |
---|---|
0 | 121.242 |
1 | 120.245 |
2 | 117.257 |
4 | 105.309 |
6 | 85.411 |
8 | 57.603 |