Eigenstates of the Quantum Harmonic Oscillator Using Spectral Methods

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Consider Schrödinger's differential equation, , where
. This problem describes the quantum harmonic oscillator (i.e., the quantum-mechanical analog of the classical harmonic oscillator). Values of the eigenvalues
can be determined analytically and are
when
. The corresponding eigenfunctions are given by
, where
are the Hermite polynomials associated with the eigenvalue
. This Demonstration approximates values of the eigenvalues numerically using spectral methods. When the number of grid points is large, the numerical values match their analytical counterparts perfectly. In addition, the first five eigenfunctions (analytical solutions in blue and numerical solution in red dots) are plotted for the choice of 100 grid points.
Contributed by: Housam Binous (April 2013)
After work by: L. N. Trefethen
Open content licensed under CC BY-NC-SA
Snapshots
Details
The eigenvalues of the quantum harmonic oscillator are found as follows:
1. First form the vector ,
, where
,
is the number of grid points, and
is an arbitrarily large value (e.g.,
).
2. Then, build the matrix , where
is the Toeplitz matrix of
with the components
and
for
.
3. Finally, find the eigenvalues and eigenvectors of matrix .
Reference
[1] L. N. Trefethen, Spectral Methods in MATLAB, Philadelphia: SIAM, 2000.
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