This repository contains an applet that simulates the dynamics of a quantum particle inside a reshapable 1D time-independent potential of finite extent. The time evolution of this quantum mechanical particle is given by |ψ(t)> = U(t)|ψ(0)>, where |ψ> is the wavefunction vector ket of the particle and U(t) is the unitary time evolution matrix. The matrix U(t) can be found on a computer by using the Crank-Nicolson method. A derivation of this method is contained in Exercise 9.8 of Mark Newmann's Computational Physics for the free particle case. It is straightfoward to generalize this to a particle bounded in any energy-preserving potential.

The following modules are used: Numpy for numerical computation, Sympy for symbolic computation, Matplotlib for plotting, and Tkinter for implementing the GUI.

Make sure that you have at least Python 3.6.5 with Numpy, Matplotlib and Sympy installed. You can install these by running pip3 install numpy matplotlib sympy. You may also optionally install Numba, which speeds up the performance of this applet. To install Numba, run pip3 install numba.

The code for this applet is obtained by cloning this repository or downloading it. Ensure that all source code are in the same directory. Now run Applet.py - you should get something like:

The main window displays the wavefunction ψ(x) with the outline of the potential V(x) in the background. At the top left corner of the window is an info box, which displays the Hamiltonian H and the boundaries of our quantum system. At the bottom of the window is the x-axis, which extends from -0.5 to 0.5. Note that this scale and all other units of measurement are not in metric units, but in natural units, where all fundamental constants are set to one.

You can disturb and reshape the wavefunction by clicking on it. You can also use the mouse to reshape the potential V(x). To do this click on the Mouse drop-down menu and select Reshape Potential V(x), then draw a new shape for the potential:

Now the wavefunction ψ(x) is not physically observable, but its probability density *ψ(x)ψ(x) is, which you can switch to by clicking View Probability Distribution. To measure observables, click on the Position, Momentum, and Energy buttons. To view the expectation values and standard deviation of observables, right click and select Toggle Expectation Values. To cycle between different stationary states, press the up and down keys, or right click and select either Higher Stationary State or Lower Stationary State. You can type in a new wavefunction ψ(x) and potential V(x) in Enter Wavefunction ψ(x) and Enter Potential V(x) text boxes. You can also select a preset potential in the Choose Preset Potential V(x) drop-down menu. To clear the wavefunction, press the Clear Wavefunction button. To alter the speed of the simulation, move the Animation Speed slider. To close the application, press QUIT.

If you just want a plain Matplotlib animation, you can either run Animate.py or import it separately, either through the command line or in a new file. If you import it, make sure that you are running Matplotlib in interactive mode, and that you create an instance of the animation object QM_1D_Animation. You can then measure observables by using its methods measure_energy(), measure_position(), and measure_momentum(). To switch between a view of the wavefunction itself or its probability density, use the methods display_probability() or display_wavefunction(). To change the wavefunction or potential, use set_wavefunction([new wavefunction]) and set_unitary([new_potential]), where new_wavefunction and new_potential can either be a string or a Numpy array.

The actual physics are contained in the classes Wavefunction_1D and Unitary_Operator_1D, which represent the wavefunction and unitary operator. These are in the module QM_1D_TI.

Newman, M. (2013). Partial differential equations. In Computational Physics, chapter 9. CreateSpace Independent Publishing Platform.