Thinking of spin as arising from a change in the phase of a wave function:
Is it possible to see the existence of spin using the quasi-classical wave function [itex]\psi (\vec{r}) = e^{iS/\hbar}[/itex]? If the action being invariant under a rotation gives angular momentum then [itex]\psi[/itex] should remain as [itex]\psi[/itex] yet the quote above seems to be saying that [itex]\psi (\vec{r}) = e^{i(S+\lambda \hbar)/\hbar} = Ae^{iS/\hbar}[/itex] can happen, furthermore it can happen in two ways (which I do not see, e.g. I do not see how this form expresses mixing up components of the wave function).
If there is a nice way to see this, perhaps one can also somehow understand spin, when thought of as arising from Lorentz invariance, if you think of the Lorentz group as being generated by unitary operators [itex]T = \Pi_{\mu} e^{s_{\mu}K_{\mu}} = e^{\sum _{\mu} s_{\mu}K_{\mu}}[/itex] and somehow see this as like the phase of the wave: [itex]T \psi = \Pi_{\mu} e^{s_{\mu}K_{\mu}} \psi = Ae^{iS/\hbar} = \psi (\vec{r})[/itex], if that makes sense?
Quote:
The angular momentum is defined by the change of the phase of the wave function under rotations, which may come from the dependence of the wave function on space, but also from the transformations of the components of the wave function among each other, which is possible even if everything is localized at a point. So even point-like objects may carry an angular momentum in quantum mechanics, the spin. |
Is it possible to see the existence of spin using the quasi-classical wave function [itex]\psi (\vec{r}) = e^{iS/\hbar}[/itex]? If the action being invariant under a rotation gives angular momentum then [itex]\psi[/itex] should remain as [itex]\psi[/itex] yet the quote above seems to be saying that [itex]\psi (\vec{r}) = e^{i(S+\lambda \hbar)/\hbar} = Ae^{iS/\hbar}[/itex] can happen, furthermore it can happen in two ways (which I do not see, e.g. I do not see how this form expresses mixing up components of the wave function).
If there is a nice way to see this, perhaps one can also somehow understand spin, when thought of as arising from Lorentz invariance, if you think of the Lorentz group as being generated by unitary operators [itex]T = \Pi_{\mu} e^{s_{\mu}K_{\mu}} = e^{\sum _{\mu} s_{\mu}K_{\mu}}[/itex] and somehow see this as like the phase of the wave: [itex]T \psi = \Pi_{\mu} e^{s_{\mu}K_{\mu}} \psi = Ae^{iS/\hbar} = \psi (\vec{r})[/itex], if that makes sense?
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