Quantum Calisthenics

Time Development of A Simple Gaussian (note the time label)

Note that a packet which starts as an ordinary Gaussian at time t=0 began as a shrinking generalized Gaussian at earlier times and expands as a generalized Gaussian at later times.  The red curve is the real part of the packet, the blue curve is the imaginary part of the packet and the black curves are the envelope, defined as the absolute value of the packet.  The first picture is the starting Gaussian at time t=0, the animation below is what this Gaussian looks like at earlier and later times.

Gaussian evolving under free particle Hamiltonian

NOTE:  A generalized Gaussian is much more complicated than the usual Gaussian.  This is evident when we look at the non-trivial behavior of the real and imaginary part of the wavefunction.

Bucket Brigade versus Exact Evolution for Free Hamiltonian

Bucket Brigade vs Exact Evolution for Gaussian: Non-Zero Momentum and Position Free Hamiltonian

NOTE:  The additional complexity of the generalized shifted Gaussian is even more pronounced than that for the stationary packet.  This is why using generalized Gaussians as intermediate states is so important.

Bucket Brigade for Evolution In The Harmonic Oscillator Potential For A Narrow Packet

NOTE: The phase difference between  the real and imaginary parts of the packets in the exact and approximate solution are actually an error in the exact plot.  The reason has to do with how Maple evaluates the square root of a complex number whose phase is bigger than 180 degrees.

Bucket Brigade For Harmonic Evolution For A Packet With Non-vanishing Momentum

A Non-Perturbative Problem:  The Anharmonic Oscillator

The following pictures show the parameters defining the generalized Gaussian states obtained by propagating states starting at the center of the potential and moving right or left using the effective Hamiltonian. 

Bucket Brigade Approximation To The Time Evolution of Anharmonic Oscillator

Tunneling In A Double Well Potential

This picture shows the two possible stationary Gaussian packets in this double well potential.  As stated in the paper each Gaussian is offset from the minimum of the potential in order that the expectation value of the derivative of the potential vanish.  The width of each Gaussian is determined by the expectation value of the second derivative of the potential at this point.  The arrows are drawn to emphasize the offset of the centers of the packets from the minima of the potential.

Instanton Solution

The two curves below show the classical positions and momenta of the generalized Gaussian packets chosen from the instanton solution. This solution is determined by assuming that the Gaussian's have a fixed g equal to that of the stationary packets.  The assumption is that the momentum is determined from the classical position x(t) and then the curve is determined by the condition that the expectation value of exp(-t H) is maximized. 

This picture shows 22 states which allows us to compute the ground state energy and the splitting between the two lowest states to a few parts in 10^-7 .

Tunneling Between the Wells

The picture on the left shows the double well in closeup so that the Gaussian packets are visible.  The red and green packets indicate the two original stationary packets and the time-dependent black, red and blue curves show how the packet initially on the left evolves in time.  The black curve, as always is the absolute value of the packet as a function of time and the blue and red curves are the real and imaginary parts of the wave-function.  On the right hand side we see the behavior of the approximate solution.  The two wells and initial pictures of the starting packets are not shown.

The Exact and Approximate Solution Superimposed On One Another

This animation shows the exact and approximate solution superimposed upon one another to give a better feeling about the error in the calculation.