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I've been recently studying the Harmonic Oscillator in one-dimensional space, more specifically I've been reading Introductory Nuclear Physics by Kenneth S. Krane.

The eigenfunctions of the Harmonic Oscillator in 1D are the Hermite polynomials. However, the spin degree of freedom is not mentioned in the solution. I assume if the spin degree of freedom were included, you'd simply have the eigenfunctions include one particle with spin up and one with spin down. And, assume the wavefunction is a product of a spatial and spin wavefunction whose product is antisymmetric when studying fermions.

How exactly would the spin degree of freedom be characterized in the wavefunction with the Harmonic Oscillator in 1D (in first quantization)?

Additionally, I know that solutions to the Schrodinger equation can be solved via non-analytically techniques like Variational Monte Carlo (VMC). If the spin degree of freedom (which is discrete) is included, how can an expectation value be calculated by sampling over continuous and discrete variables simultaneously? Does one have to fix the spins in some configuration, or are configurations purely sampled in the position degrees of freedom with spin imposed by the wavefunction in some way?

Apologizes for the vague/silly question, and thanks for any help in advance!

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