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.
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Quantum matrix diagonalization visualized
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November 2019
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November 01 2019
Quantum matrix diagonalization visualized
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Kevin Randles;
Kevin Randles
a)
Department of Physics, Weber State University
, Ogden, Utah 84408-2508
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Daniel V. Schroeder;
Daniel V. Schroeder
b)
Department of Physics, Weber State University
, Ogden, Utah 84408-2508
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Bruce R. Thomas
Bruce R. Thomas
c)
Department of Physics and Astronomy, Carleton College
, Northfield, Minnesota 55057
Search for other works by this author on:
Kevin Randles
a)
Department of Physics, Weber State University
, Ogden, Utah 84408-2508
Daniel V. Schroeder
b)
Department of Physics, Weber State University
, Ogden, Utah 84408-2508
Bruce R. Thomas
c)
Department of Physics and Astronomy, Carleton College
, Northfield, Minnesota 55057a)
Electronic mail: [email protected]
b)
Electronic mail: [email protected]
c)
Electronic mail: [email protected]
Am. J. Phys. 87, 857–861 (2019)
Article history
Received:
May 30 2019
Accepted:
September 06 2019
Citation
Kevin Randles, Daniel V. Schroeder, Bruce R. Thomas; Quantum matrix diagonalization visualized. Am. J. Phys. 1 November 2019; 87 (11): 857–861. https://doi.org/10.1119/10.0000014
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