In practice, the position-space wave function is used much more often than the momentum-space wave function. The potential entering the relevant equation (Schrödinger, Dirac, etc.) determines in which basis the description is easiest. For the harmonic oscillator, and enter symmetrically, so there it does not matter which description one uses. The same equation (modulo constants) results. From this, with a little bit of afterthought, it follows that solutions to the wave equation of the harmonic oscillator are eigenfunctions of the Fourier transform in .The Fourier transform viewed as a unitary operator on the space has eigenvalues . The eigenvectors are "Hermite functions", i.e. Hermite polynomials multiplied by a Gaussian function. See for a description of the Fourier transform as a unitary transformation. For eigenvalues and eigenvalues, refer to Problem 27 Ch. 9.
in practice the position-space wave function is used much more often than the momentum-space wave function the potential entering the relevant equation (schrödinger dirac etc) determines in which basis the description is easiest for the harmonic oscillator and enter symmetrically so there it does not matter which description one uses the same equation (modulo constants) results from this with a little bit of afterthought it follows that solutions to the wave equation of the harmonic oscillator are eigenfunctions of the fourier transform in the fourier transform viewed as a unitary operator on the space has eigenvalues  the eigenvectors are "hermite functions" ie hermite polynomials multiplied by a gaussian function see for a description of the fourier transform as a unitary transformation for eigenvalues and eigenvalues refer to problem 27 ch 9