• Time Evolution
• Stationary States

Estimated Time: 20 minutes

Students are asked to create states of the infinite potential well, composed both of single energy eigenstates and linear combinations of energy eigenstates, and animate the probability densities ${\left| {\psi \left( {x,t} \right)} \right|^2}$ to see how they change with time.

We provide students with a Mathematica template that has useful commands (animate, complex conjugate, plot etc.,) but it has no information about physics. We ask students to generate these animations from scratch, to encourage them to think about it means to superpose and whether the amplitudes or intensities should add. But here is an example that shows a typical result (more elaborate than what students would produce in the class).

• Enough familiarity with Mathematica/Maple to be able to animate functions.
• The energy eigenstates of the infinite potential well.

• Computers with Maple

In a brief mini-lecture, we remind students that the solutions of the (time-dependent) Schrödinger equation are linear combinations of energy eigenstates with an complex exponential coefficient that contains all the time dependence. We also remind then that each term in the superposition gets a coefficient that corresponds to the eigen-energy of that state. Our students have seen this result derived in the Spins paradigm, three weeks prior.

1. Students need to be encouraged to try states that contain only one energy eigenstate, as well as states made up of a linear combination of states. When students see that the probability densities of single eigenstates don't change with time, we introduce the term “stationary” state.
2. This activity can lead to an interesting discussion of time scale; students often do not attend to the time scale that is relevant for their animation. Many groups try to use values of $\hbar$, $m_{e}$ and $L$ (the well width) in SI units, but then try to animate their results over a timescale of seconds, when the relevant timescale is in fact of order $10^{ - 20} s$. Other groups choose units where $\hbar$, $m_e$ and $L$ are 1, but don't realize that this fortuitously makes a time scale of seconds appropriate.
3. A common problem is to use a single ${e^{ - iEt/\hbar }}$ factor instead of a separate one for each term in the superposition.
4. Students should try various superposition states (odd- and even-parity states; odd/odd, even/even etc. The electron probability density is not static, but changes with time. Depending on the particular superposition chose, it may “slosh” back and forth in the well (i.e. $\left\langle x \right\rangle $ is time dependent) or it may “breathe” ($\left\langle x \right\rangle $ is time independent, but the profile changes shape with time.) A single eigenstate should also be tried to demonstrate the idea of “a stationary state”.
5. This time-dependent distribution of a superposition of 2 states can be discussed in the context of elementary radiation theory. An accelerating charged particle radiates, and the particular frequency of radiation in this case is the energy difference between the two states. In this respect, the square well is a model of an atom.

The wrap-up for this activity is typically short, with groups briefly reporting their findings in general terms.

The time evolution of two state systems can lead into a nice discussion of phenomena, like spontaneous emission/absorption (see Chapter 9 of Griffith's “Introduction to Quantum Mechanics”, 2nd Ed.)