I present a Fourier transform approach to the problem of finding the stationary states of a quantum harmonic oscillator. The simplicity of the method may make it a desirable substitute for the rather cumbersome polynomial approach to the problem which is commonly used in the standard graduate quantum mechanics textbooks.

L. Schiff, Quantum Mechanics (McGraw–Hill, New York, 1968), 3rd ed.
C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley-Interscience, New York, 1977), Vol. I.
A. Sommerfeld, Wave Mechanics (Academic, New York, 1929).
In reality, of course, the “magical” termination of the series solution for half-integer values of the energy is directly related to the normalization condition for the wave functions. However, this connection often is obscure to the students because it involves rather complicated mathematics.
J. J. Sakurai, Modern Quantum Mechanics (Addison–Wesley, Reading, MA, 1994).
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