Courses on undergraduate quantum mechanics usually focus on solutions of the Schrödinger equation for several simple one-dimensional examples. When the notion of a Hilbert space is introduced, only academic examples are used, such as the matrix representation of Dirac’s raising and lowering operators or the angular momentum operators. We introduce some of the same one-dimensional examples as matrix diagonalization problems, with a basis that consists of the infinite set of square well eigenfunctions. Undergraduate students are well equipped to handle such problems in familiar contexts. We pay special attention to the one-dimensional harmonic oscillator. This paper should equip students to obtain the low lying bound states of any one-dimensional short range potential.
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At least one of my students simply programmed MATLAB to evaluate Eq. (9) numerically for an arbitrary potential. Though not to my personal taste, many students find such a numerical evaluation much easier than slogging through analytical integrals. The numerical approach allows one to solve for bound states of any one-dimensional short range potential.
14.
The analytical solution given by Eq. (18) provides the solution for a finite square well in an infinite space. Results will therefore differ from the numerical results as the width of the finite square well approaches the width of the infinite square well. The Appendix provides an analytical solution for the -function potential in an infinite square well, which differs from the textbook solution of an infinite square well in infinite space.
15.
These difficulties are well known. For example, bosons in a harmonic trap experience a short range boson-boson interaction. If we try to expand the many-particle wave function in terms of the eigenstates of the harmonic oscillator potential, convergence difficulties arise because of the disparity in length scales. Our problem illustrates this difficulty at the single particle level. I am indebted to Doerte Blume and Pierre-Nicholas Roy for discussions on this point.
16.
Though often referred to as “Heisenberg matrix mechanics,” it is now clear that Born should be credited with this technique. See, for example,
Nancy Thorndike
Greenspan
, The End of the Certain World: The Life and Science of Max Born
(Basic Books
, Cambridge, MA
, 2005
).© 2009 American Association of Physics Teachers.
2009
American Association of Physics Teachers
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