Movies of Solutions to the Time Dependent Schrodinger Equation (TDSE)

Here we have a series of MPEG movies designed to illustrate basic physical behaviors of the solutions of the TDSE.

One must be careful in the physical interpretation of these movies. Each movie consists of a series of frames. Each frame is just the probability distribution of the outcome of experiments measuring the position of the particle a fixed time t after the experiment is begun. These distributions are the played in sequence as a function of the time t. Note that the experiments are carried out by preparing systems in indentical quantum states at time t=0, and then waiting and doing no measurement until the position measurement at time t. Thus, we are not looking at the position of a particle subjected to a sequence of measurements of its position!

In the different movies, the systems are begun in different quantum states and then subjected to different external potentials (forces).

Initial State: Gaussian Packet with zero average velocity; External Potential: none (free space). The time evoluation of an (almost) pure state of position in free space. The Heisenberg Uncertainty Principle forces the packet to spread!
Initial State: Gaussian Packet with an initial velocity; External Potential: none (free space). Now we set the packet into motion. This movie illustrates the constant velocity progress of a packet in free space, with the spreading expected from the Heisenberg Uncertainty Principle and the decay in amplitide associated with the conservation of probability. Note that this movie, as are all the ones below, is displayed with periodic booundary conditions, so that when the particles flow off of the right hand side of the screen, they are returned on the left.
Initial State: Gaussian Packet; External Potential: a potential "step" (discontnuous increase in potential near x=12). This movie shows the phenomena of quantum scattering, relative transmission and reflection probabilities, the slowing of the wavepacket in regions of higher potential, and interference of left and right travelling waves.
Initial State: Gaussian Packet with zero average velocity; External Potential: weak SHO (k too small). Here we see that an external potential may keep the packet from spreading too far. In this case, there is a weak spring (Simple Harmonic Potential) binding the particles to the origin.
Initial State: Gaussian Packet with zero average velocity; External Potential: strong SHO (k large). Here we see the result when the SHO potential is too strong and initially constricts the packet. We can then imagine the limiting critical case when the spring constant is just right to produce the stationary state.
Initial State: Coherent State with zero average velocity; External Potential: SHO. Balancing the spring constant perfectly, we get a stationary state of the SHO.
Initial State: Coherent State with a small initial displacement velocity; External Potential: SHO. Giving the stationary state a small push, we get a coherent state of the SHO, one where the packet moves in space and does not spread but shifts its form rigidly back and forth in simple harmonic motion.
Initial State: Gaussian Packet (Coherent State); External Potential: SHO. We can in fact give arbitrary pushes to produce a wide range of coherent states.
Initial State: Gaussian Packet; External Potential: Simple Harmonic Oscillator (SHO). Not all states are coherent, but still Ehrenfest's holds and the average packet position still shows simple harmonic motion.
Initial State: First excited pure state of energy (n=1); External Potential: SHO. Other stationary states exist, they are the pure states of energy.
Initial State: Fifth excited pure state of energy (n=5); External Potential: SHO. As we go to higher stationary states (solutions of the TISE), we find that the probability distribution begins to approach the classical limit, P=const/v and the probability goes up near the classical turning points.
Initial State: Random; External Potential: SHO. A random state placed in the SHO to illustrate the Ehrenfest's Theorem for the simple harmonic motion of the center of the packet holds even when the packet is of a random form.