This Exercise Set is still a work in progress. It has not yet been submitted for peer review.

Discretized Time Dependent Schrödinger Equation in 1D

Developed by K. Roos

In this Exercise Set, students are guided in discretizing and computationally solving the time-dependent Schrödinger equation in 1D. The discretization process involves substituting finite difference equations for partial derivatives, and producing numerical solutions at discrete points in space and time. The numerical scheme implemented is an *explicit* method that is generally applicable to parabolic partial differential equations. Quantum mechanical scattering and tunneling is studied for various potential energy configurations, and transmission and reflection probabilities are calculated from the numerical solutions.
Subject Areas Modern Physics and Quantum Mechanics
Levels Beyond the First Year and Advanced
Learning Objectives
Students who complete these exercises will be able to: - articulate the parametric dependence of a traveling Gaussian wave packet (**Exercises 1-3**); - discretize the 1D time-dependent Schrödinger equation (**Exercises 4-8**); - identify artifacts in the output of computational models involving the *explicit numerical scheme* for solving a parabolic partial differential equation (**Exercises 4,6 and 7**); - implement a special *explicit numerical scheme*, that conserves probability density, to solve the discretized 1D time-dependent Schrödinger equation (**Exercises 5-7**); - numerically solve the 1D tdse with many different (nearly any!) potential energy configurations (**Exercises 5-14**); and - computationally determine reflection and transmission coefficients for a variety of quantum mechanical scattering and tunneling events (**Exercises 8-14**).
Time to Complete 300+ min
#### Exercise 1: Interactive Demo: Gaussian Wave Packet Use the interactive demo, Gaussian Wave Packet.cdf, to explore how various values of the parameters, $x_{c}$, $w$, and $k_{0}$ affect the shape and position of a wave packet defined by the equation $\Psi=\exp\left[-\frac{\left(x-x_{c}\right)^{2}}{w^{2}}\right]\exp\left(ik_{0}x\right)$. You should vary each of these parameters systematically, and explain in detail what effect each has on the wave packet. Note that when $k_{0}$ has a nonzero value, the wave packet is referred to as a “traveling Gaussian.” When $k_{0}=0$, the wave packet is a “stationary Gaussian.” The free Wolfram CDF player is needed to run the interactive demo, and can be downloaded here: http://www.wolfram.com/cdf-player/. #### Exercise 2: Programming Warmup Write and run a computer program that creates a specified number $N$ of discrete spatial elements and assigns to each a value that coincides with a *traveling* Gaussian ($k_{0}>0$) wave packet, given by the equation $\Psi=\exp\left[-\frac{\left(x-x_{c}\right)^{2}}{w^{2}}\right]\exp\left(ik_{0}x\right)$. Let the $N$ elements be spread over an absolute distance of 2 (units arbitrary). Set up the program so that the mean wave number $k_{0}$ can be specified. Plot the real and imaginary parts of the wave function as well as the probability density as a function of position for different combinations of the parameters, $x_{c}$, $w$ and $k_{0}$. A template for this exercise, in the platform that you and your instructor have chosen to work in, can be found in the Code section. This exercise will provide you with an opportunity to produce a program, compile it (if necessary), and run it to get used to generating results computationally. This exercise should also aid in becoming familiar with Gaussian wave packets, and the dependence of the discrete form on the various parameters. Don't forget to try different values of $N$. What happens if $N$ is too small? Can $N$ be too big? #### Exercise 3: Stationary Gaussian Wave Packet Demo As a prelude to studying the dynamics produced by the Forward Difference Time Algorithm described in detail in the Theory section, use the interactive demo, "Wave packet for a Free Particle," to observe how a free quantum particle described by a stationary Gaussian wave packet should behave as time increases. This interactive demo can be found online at http://www.demonstrations.wolfram.com/WavepacketForAFreeParticle/. The free Wolfram CDF player is needed to run the interactive demo, and can be downloaded here: http://www.wolfram.com/cdf-player/. Note: the notation used in the interactive demo is slightly different than that we are using here. In the demo, $\langle p\rangle$ represents the average momentum of the wave packet, and corresponds to the $k_{0}$ we have been using. $\Delta p$ is the uncertainty in the particle's momentum, and is inversely proportional (via Heisenberg Uncertainty) to our Gaussian width $w$. Also in the demo, the Gaussian wave packet is initially centered at position $x_{c}=0$ at time $t=0$. To observe what happens in the demo for a stationary Gaussian wave packet, set $\langle p\rangle$ equal to 0 using the slider bar, set $\Delta p$ equal to about 0.35, and make sure the probability density button $P(x,t)$ is highlighted for the ``function'' choice. The time evolution of this stationary wave packet can then be studied by changing time via the time slider bar. Comment on the consistency of the observed wave packet behavior with the Uncertainty Principle. #### Exercise 4: Failure of the Forward Difference Time Algorithm Study the dynamics of a free quantum particle modeled with a stationary ($k_{0}=0$) Gaussian wave packet either by directly programming the Forward Difference Time Algorithm described in the Theory section, or by making use of the provided coded implementation of the Forward Difference Time Algorithm. Start with parameter values of $\Delta t=2\times10^{-8}$ and $\Delta x=10^{-3}$, and for the Gaussian wave packet, $x_{c}=1.5$ and $w=0.05.$ Use 3000 space cells for the calculation; the center of the Gaussian wave packet will be located in the center of the 3000 cells (at position $x_{c}=1.5)$ at the initial time $t=0$. Take the calculation out to at least $10^{6}$ time steps. Do you notice any "strange" behavior in the model, and if so, at roughly what time step do you first notice it. Is the behavior improved by going to smaller values of $\Delta t$? Describe how the observed behavior differs from that which is expected for a stationary Gaussian wave packet, i.e. from that which you observed in the interactive demo of Exercise 3. #### Exercise 5: Practice Leapfrog Implementation for Small System Write out the finite difference leapfrog equations for $\operatorname{Im}\Psi$ and $\operatorname{Re}\Psi$ for a free particle propagating in a small 5 spatial-cell system for general odd and even time steps. Next, use $\Delta t=10^{-3}$ and $\Delta x=10^{-1}$ and the finite difference equations you just wrote out to fill in all the values of the table below. ![](images/DTDSE/ex53.png "") Use of a spreadsheet can greatly facilitate the calculations required for filling in the values in the tables. If you choose to employ a spreadsheet for these calculations, the above table is conveniently provided in a downloadable Microsoft Excel file, Leapfrog\_small\_system.xlsx. #### Exercise 6: Stationary Wave Packet Study the dynamics of a free quantum particle modeled with a stationary ($k_{0}=0$) Gaussian wave packet either by directly programming the Leapfrog Algorithm described in the Theory section, or by making use of the provided coded implementation of the Leapfrog Algorithm. Start with parameter values of $\Delta t=10^{-8}$ and $\Delta x=10^{-3}$, and for the Gaussian wave packet, $x_{c}=1.5$ and $w=0.05.$ Use 3000 space cells for the calculation; the center of the Gaussian wave packet will be located in the center of the 3000 cells (at position $x_{c}=1.5)$ at the initial time $t=$0. Take the calculation out to at least $10^{6}$ time steps. Does this algorithm produce the proper behavior of a Gaussian wave packet with time? What happens at different time steps? #### Exercise 7: Traveling Wave Packet Study the propagation of a free quantum particle modeled with a traveling ($k_{0}\neq0$) Gaussian wave packet either by directly programming the Leapfrog Algorithm described in Theory section, or by making use of the provided coded implementation of the Leapfrog Algorithm. Use values for $\Delta t$ and $\Delta x$ such that stable solutions are produced $\left(s'\leq0.5\right)$. Employ a total number of space steps such that the particle can move freely between positions 0 and 3. Calculate the probability density and plot it for various times. Once the expected free particle behavior has been verified for your model, and it is obvious that there are no unphysical artifacts present, you are urged to play with the model by methodically varying the parameters to develop an intuitive sense of how the propagation of the particle is affected in each case. In particular, what is the effect of changing the value of $k_{0}$? What is the energy of the free particle, in the units we have chosen ($\hbar=m=1$), in each case? Confirm that your free particles have the correct velocity by directly determining the group velocity from the program output. Comment on what happens when the wave packet gets close to the spatial boundaries, and the validity of the boundary conditions used. #### Exercise 8: Probability Conservation Calculate the area under the probability density curves generated with the Leapfrog algorithm for a free traveling Gaussian particle at a few different times and different values of $k_{0}$, and comment on the results. Because of the discrete nature of our model, you can envision the total area under the probability density to consist of the sum of the individual areas of a set of rectangles. A rectangle is centered on the positions of each spatial element, and its area is the product of the probability density associated with its position and its width $\Delta x$. To calculate the total area under the probability density curve, which, if the wave function were properly normalized, is tantamount to calculating the probability of finding the particle anywhere in our 1D space, it is only necessary to loop thorough all the spatial cells and keep a running sum of the contributions from all the individual rectangles. The figure shows a graphical representation of the approximation of the area under a Gaussian by a series of rectangles, showing three of the total number $N$ of rectangles needed to approximate the area under the entire curve. Each rectangle is centered about a discrete spatial position, and has the width of the space step $\Delta x$. The height of each rectangle is the value of $\left|\Psi(j)\right|^{2}$ associated with the particular position $j$. The area under the curve is approximated by the sum of the areas of all $N$ individual rectangles. In equation form, $area=\textstyle \sum_{j=1}^{N}\Psi\left(j\right)\Delta x$. ![](images/DTDSE/set2Ex6.png "") #### Exercise 9: Interactive Demo: Gaussian Wave Packet Scattering at a Step Use the interactive demo, "Gaussian Wave Packet: Step Scattering" (written by Jose Ignacio Fernández Palop) from Open Source Physics website: http://www.opensourcephysics.org/items/detail.cfm?ID=10535 to study the interaction of a free Gaussian wave packet with a potential energy step. Double clicking the ejs\_qm\_gaussian\_step.jar file, once downloaded, will run the program if Java is installed. In the simulation, the type of potential energy step can be chosen, barrier $\left(V>0\right)$ or cliff $\left(V<0\right)$. The propagation of a free Gaussian $\left(V=0\right)$ can also be observed. The Gaussian energy can be varied, as well as the Gaussian width. The simulation displays the time evolution of $\operatorname{Re}\Psi$,$\operatorname{Im}\Psi$, and the probability density $\left|\Psi(j)\right|^{2}$, and the measured reflection and transmission coefficients, $R$ and $T$, respectively, are calculated directly in the simulation. #### Exercise 10: Scattering from a Potential Energy Cliff Study the quantum mechanical scattering of a Gaussian wave packet incident on a region of $-V_{0}$, as shown in the figure below, by either directly programming the Leapfrog Algorithm described in the Theory section, or by making use of the provided coded implementation of the Leapfrog Algorithm. ![](images/DTDSE/pecliffset3ex2.jpg "") You should start with parameters that produce stable solutions, say $\Delta t=10^{-8}$ and $\Delta x=10^{-3},$ and study the scattering of the wave packet as a function of particle energy for a particular value of $V_{0}$, say $V_{0}=60,000$. You are encouraged to methodically explore your scattering model by using a variety of values for $V_{0}$, but $V_{0}=60,000$ is a good starting point. It will be convenient to locate the onset of the cliff at position $x=1.0$, and employ a number of space steps such that the propagation field extends from 0 to 2.0. Explore the scattering for different values of the average wave packet energy $E=k_{0}^{2}/2$ (remember $\hbar=m=1$). Comment on the dynamic behavior, and the interesting structure produced as the particle enters the region of non-zero potential energy. Also, calculate the transmission and reflection probabilities (how?), $R$ and $T$, for each value of $E$. Remember to normalize properly: $R+T=1$. #### Exercise 11: Scattering from a Potential Energy Barrier Explore the scattering of a Gaussian wave packet interacting with the potential energy barrier shown in the figure below. Study the scattering as a function of particle energy $E$ using a range of values for $E>V_{0}$. Calculate the transmission and reflection probabilities, $R$ and $T$, for each value of $E$. ![](images/DTDSE/pestepset3ex3.jpg "") #### Exercise 12: Scattering from a Potential Energy Well Explore the scattering of a Gaussian wave packet interacting with the potential energy well shown in the figure below. Study the scattering as a function of particle energy $E$. Calculate the transmission and reflection probabilities, $R$ and $T$, for each value of $E$. ![](images/DTDSE/set3ex4.png "") #### Exercise 13: Tunneling Through a Potential Energy Barrier Study the quantum mechanical tunneling of a Gaussian wave packet incident on a potential energy barrier of height $V_{0}$ and width $a$, as shown in the figure below, by either directly programming the Leapfrog Algorithm described in the Theory section, or by making use of the provided coded implementation of the Leapfrog Algorithm. ![](images/DTDSE/set4ex1.jpg "") You should start with parameters that produce stable solutions, say $\Delta t=10^{-9}$, $\Delta x=10^{-4},$ $a=0.05$, $V_{0}=60,000$, and Gaussian width $w=0.05$, and study the tunneling of the wave packet as a function of particle energy $E < V_{0}$. You are encouraged to methodically explore the tunneling behavior of your model by using a variety of values for $a$ and $V_{0}$, but $a=0.05$ and $V_{0}=60,000$ are good starting points. It will be convenient to locate the onset of the potential barrier at position $x=1.0$, and employ a number of space steps such that the propagation field extends from 0 to 2.0 ($2\times10^{4}$ space cells). With $2\times10^{4}$ space cells and $\Delta x=10^{-4}$, there will be 500 space cells from position 1.0 to 1.05. The 500 space cells within the barrier will provide sufficient resolution to study what happens within the barrier. It will be interesting to observe the wave function behavior as well as the probability density. #### Exercise 14: Alpha Decay Study the quantum mechanical tunneling of a Gaussian wave packet incident on the potential energy configuration shown in the figure below. This potential energy configuration is a qualitative representation of the potential encountered by an alpha particle in a nucleus. As the particle moves towards the edge of the nucleus, it encounters a steep repulsive energy barrier. For those particles that escape the nucleus through the alpha decay process (quantum mechanical tunneling), a long-range Coulomb repulsion is felt as the particles move away from the nucleus. ![](images/DTDSE/alphaset4ex2.png "") Model the alpha decay process by either directly programming the Leapfrog Algorithm described in the Theory section, or by making use of the provided coded implementation of the Leapfrog Algorithm, and programming the potential energy barrier that includes the long Coulomb "tail." To start, use parameters that are similar to those used, for the Gaussian wave packet, $\Delta t$, and $\Delta x$, in the previous scattering and tunneling exercises. The mean particle energy should be less than the barrier height. Explore your model by varying mean particle energy, the barrier height, and the steepness of the Coulomb tail. Calculate $R$ and $T$. How does $T$ relate to the decay rate and half life of an alpha emitter?

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Credits and Licensing

The instructor materials are ©2017 K. Roos.

The exercises are released under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 license

Creative Commons Attribution-NonCommercial-ShareAlike 4.0 license