November 2019 AJP coverNovember 2019 Issue,

Volume 87, No. 11


Quantum matrix diagonalization visualized featured

We show how to visualize the process of diagonalizing the Hamiltonian matrix to find the energy eigenvalues and eigenvectors of a generic one-dimensional quantum system. Starting in the familiar sine-wave basis of an embedding infinite square well, we display the Hamiltonian matrix graphically with the basis functions alongside. Each step in the diagonalization process consists of selecting a nonzero off-diagonal matrix element and then rotating the two corresponding basis vectors in their own subspace until this element is zero. We provide mathematica code to display the effects of these rotations on both the matrix and the basis functions. As an electronic supplement, we also provide a javascript web app to interactively carry out this process.



Comment on “Confining rigid balls by mimicking quadrupole ion trapping” [Am. J. Phys. 85, 821 (2017)] Alexey V. Borisov, Alexander A. Kilin and Ivan S. Mamaev. DOI: 10.1119/10.0000006


Quantum matrix diagonalization visualized by Kevin Randles, Daniel V. Schroeder and Bruce R. Thomas. DOI: 10.1119/10.0000014

Finite and half-infinite solenoids and the Aharonov-Bohm effect by J. Franklin and Gopal Goel. DOI: 10.1119/1.5124814

Conservation laws and energy transformations in a class of common physics problems by Jonathan Bougie and Asim Gangopadhyaya. DOI: 10.1119/1.5125213

Student and instructor framing in upper-division physics by Deepa N. Chari, Hai D. Nguyen, Dean A. Zollman and Eleanor C. Sayre. DOI: 10.1119/1.5120392

Demonstrating the elliptical orbit of Mars using naked eye data by Kevin Krisciunas. DOI: 10.1119/1.5120020

Physical analysis and optimization of electromagnetic coilgun launch systems by Joshua Zhou and Takyiu Liu. DOI: /10.1119/1.5124975

Why pushing a bell does not produce a sound by Loris Ferrari. DOI: 10.1119/1.5124976

Damped harmonic oscillation: Linear or quadratic drag force? by R. Hauko and R. Repnik. DOI: 10.1119/1.5124978

Diamagnetic behavior in random classical radiation by Timothy H. Boyer. DOI: 10.1119/1.5123158

Bose–Einstein condensation in spherically symmetric traps by Sálvio Jacob Bereta, Lucas Madeira, Vanderlei S. Bagnato and Mônica A. Caracanhas. DOI: 10.1119/1.5125092


Philosophy of Physics: Quantum Theory by Allan Walstad. DOI: 10.1119/1.5121386


American Journal of Physics 87, 940 (2019);

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