Abstract
In this chapter we put together all the techniques studied so far: internal-spin degrees of freedom (Chap. 3) and spatial (motional) degrees of freedom (Chap. 4) are combined with the tensor-product formalism (Chap. 2). We arrive at a complete numerical description of interacting spin-ful particles moving through space. To showcase these powerful tools, we study Rashba coupling as well as the Jaynes–Cummings model.
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Remember that \(\mathbbm {1}=|\uparrow \rangle \langle \uparrow |+|\downarrow \rangle \langle \downarrow |\) and \(\hat{S}_z=\frac{1}{2}|\uparrow \rangle \langle \uparrow |-\frac{1}{2}|\downarrow \rangle \langle \downarrow |\).
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Naturally, the following calculations would be simpler if we had represented the ground state in the position basis; however, we use this opportunity to show how to calculate in the momentum basis.
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In a harmonic oscillator of mass m and angular frequency \(\omega \), we usually introduce the position operator \(\hat{x}=\sqrt{\frac{\hbar }{m\omega }}\hat{X}\) and the momentum operator \(\hat{p}=\sqrt{\hbar m \omega }\hat{P}\). Here we restrict our attention to the dimensionless quadratures \(\hat{X}\) and \(\hat{P}\).
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Schmied, R. (2020). Combining Spatial Motion and Spin. In: Using Mathematica for Quantum Mechanics. Springer, Singapore. https://doi.org/10.1007/978-981-13-7588-0_5
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DOI: https://doi.org/10.1007/978-981-13-7588-0_5
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