Localizing an Electron in Space (Worksheet)

Why?

Different linear combinations of momentum eigenfunctions are used in this activity to produce wavefunctions for an electron that extend over different expanses of space. As more momentum components are introduced, the uncertainty in the momentum becomes larger, but the uncertainty in the position of the electron becomes smaller. The results demonstrate the Heisenberg Uncertainty Principle and its connections to the wave properties of matter.

Objectives:

1. To represent a mathematical function in terms of a Fourier series.
2. To interpret the amplitudes of the Fourier components as probability amplitudes for the momentum of an electron.

Performance Criteria

1. Quality of the graphs for different Fourier series illustrating various degrees of localization.
2. Clarity of a table and statement summarizing the relationship between the uncertainty in the position and the uncertainty in the momentum of the electron.

Information

Eigenfunctions of the momentum operator in one dimension have the form \(e^{ikx}\). These functions are not localized in space but extend from \(x = -\infty to +\infty\). To describe an electron that is localized in space, such as one in the hydrogen atom, we can use linear combinations of these functions. Some theoreticians actually use such linear combinations to represent atomic orbitals in electronic structure calculations.

To have the convenience of using real rather than imaginary functions, we first take linear combinations of \(e^{ikx}\) and \(e^{-ikx}\) to produce the trigonometric functions cos(kx) and sin(kx). The linear combination of sines and cosines that is used to represent another function is called a Fourier series. The function that we will describe by a Fourier series is \(h (x) = e ^{-|x|}\), which is a one-dimensional analog of the 1s atomic orbital. We use this function to describe an electron that is localized around a proton in a one-dimensional hydrogen atom. The proton is located at x = 0, and x is the distance of the electron from the proton measured in units of the Bohr radius, 0.0529 nm. We want to represent the function h(x) by the Fourier series f(x) over some distance L extending from x = -L/2 to +L/2. Since we are measuring distance in multiples of the Bohr radius, take L=20, so our space extends from x= -10 to +10 Bohr units, or equivalently from -0.529 nm to +0.529 nm, from the proton.

Each term in the Fourier series contains a wavelength, e.g. \(\cos (kx) = \cos (\frac {2\pi x}{\lambda})\). Since the cosine and sine functions are periodic, the Fourier series also will be periodic, so the distance L must match the repetition period or wavelength of the cosine and sine functions in the Fourier series. By match, we mean the distance must be an integer multiple of the wavelengths, i.e. nλ=L where n is an integer. Substituting λ=L/n changes the argument of the trigonometric functions from kx into 2πnx/L.

The coefficients or amplitudes in the Fourier series f(x) are given below and are derived in many math books on calculus, differential equations, engineering mathematics, and mathematical physics. In practice, only as many terms, N, are included in the sums as are necessary to represent the function to the degree of accuracy needed. The equations needed for this activity are:

\(h (x) = e ^{-|x|}\)

\(f(x) = \frac {1}{2} a_0 + \sum _{n=1}^{\infty} a_n \cos (\frac {2 \pi nx}{L}) + \sum _{n=1}^{\infty} b_n \sin (\frac {2 \pi nx}{L})\)

\(a_n = \frac {2}{L} \int \limits ^{L/2}_{-L/2} h(x) \cos (\frac {2 \pi nx}{L}) dx\)

\(b_n = \frac {2}{L} \int \limits ^{L/2}_{-L/2} h(x) \sin (\frac {2 \pi nx}{L}) dx\)

Tasks

1. In the interval \(-\frac {L}{2} \le x \le +\frac {L}{2}\) represent the function h(x) by the Fourier series f(x). See Mathcad hints and the Mathcad worksheet that are provided below.
2. Graph h(x) and f(x) in the interval \(-\frac {L}{2} \le x \le +\frac {L}{2}\) for different values of L, length of the box, and N, the upper limit for your Fourier expansion. L = 20 and N = 10 should give reasonable results. Describe what happens as you vary N and L.
3. Construct a table for L = 20 and N = 10 giving the values for the coefficients, the a’s and the b’s. Explain why some values are large, some are small, and others are zero.
4. Construct two tables for L = 20, one for N=1 and the other for N=5, giving the full width of f(x) at half its maximum, giving the possible values for the momentum of the electron described by f(x), and also giving the probability that the particle has each of those momentum values.
5. Relate what you have done in this activity to the Heisenberg Uncertainty Principle.

Mathcad Hints

1. Use a range variable for x: e.g., type x:-L/2,(-L+0.1)/2;L/2
2. Use a global variable for N and L so they can be placed directly above your graph: type N~1 L~6
3. Use a square bracket to enter the subscripts on the a’s and b’s: type a[0 a[n and b[n
4. Mathcad performs the integration and sums based on the equations you enter. What could be easier and more fun?

The Mathcad worksheet for this activity

Acrobat File:ElectronLocalizationM.pdf
Mathcad 11 file: Mathcad 11 file: ElectronLocalizationMC11.mcd

Mathcad 2000 file: Mathcad 2000 file: ElectronLocalizationMC2000.mcd

Fourier Series

For those wishing to explore and develop skills using the Fourier series the Mathcad document Introduction to Fourier Series by Theresa Julia Zielinski would be useful. This document provides exercises that serve as a foundation for future work in quantum mechanics. The study of Fourier series gives a basis for understanding linear combinations of atomic orbitals, basis functions, and and those expressions of solutions to the Schrodinger Equation that are written as expansions using complete orthonormal sets of functions. Zielinski, T. J. J. Chem. Educ. 2008, 85, 1708.

Acrobat file: fourier11.pdf

Mathcad 11 file: fourier11.mcd