PH427: Periodic Systems (Winter 2016)

Worksheet 2: Periodic quantum wells lead to band structure

Written By: Ethan Minot

Updated Feb 20, 2016( Matt Graham)

This worksheet is based on the PhET Simulation from UC Boulder “Band Structure”

(Note: If you are interested in superpositions of eigenfunctions, see “Quantum Bound States”)

Worksheet overview:

Part 1: A new toy, let’s play!

Part 2:Check that the simulation is giving physically reasonable results

Part 3:Use the simulation to explore the spread in energies as the number of wells is increased

Part 4:Compare exact eigenstates to LCAO states. Plot a dispersion relation

Part 1:A new toy, let’s play!

1. Starting from a single quantum well, watch the energy levelsmultiply as you add more quantum wells (unclick the magnifying glass check box for now)

2. Starting from a double quantum well, play with the barrier thickness and width of the wells.
(use configure potential)

3. Watch the real and imaginary parts of a wavefunction evolve in time. Why do some states evolve faster than others? (Hint: Think time evolution operator exp(-i E t/hbar)) !

4. Apply an electric field to a system of 10 potential wells. What happens to the probability density of the lowest energy electron?

Part 2: Use the simulation to explore the spread in energies as the number of wells is increased

1. Reset the simulation so that you have the default potential well dimensions.

2. Vary the number of potential wells from 2 to 10. Fill in the chart below

Number of potentials wells / Spread in the lowest group of eigenenergies
Units:______/
(“Spread” refers to the energy difference between the lowest state in the magnifying glass and the highest state in the magnifying glass)

2. As the number of potential wells becomes very large, the spread in the lowest energy band will

a) increase without bound.

b) approach ______eV

Part 3:Compare exact eigenstates to LCAO states. Plot a dispersion relation

Set the number of potential wells to 10

1. Each eigenstate can be approximated by a linear combination of atomic orbitals state (LCAO state). The form of the LCAO state is dictated by boundary conditions (either periodic or fixed). In this case, the boundary conditions are fixed and the LCAO states must have nodes at either end. The real part of the LCAO states at time zero are given by

,

where n is an index referring to the nth potential well (n= 1, 2, 3 … 10). The constant A is a normalization factor. Fill in the chart below for the 10 lowest energy states of the system.

Eigenenergy
Units:______/ Wavenumber, k
Units:______/ (OPTIONAL: Sketch of k(x, t =0) )

2. Plot these eigenenergiesas a function of wavenumber (use the graph paper provided). Don’t forget to title your graph, label your axes, and indicate the units of measurement.

3. Try to dream up (brainstorm) a fitting function thataccurately describes the mathematical relationship between eigenenergy and k. What is your best guess?

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