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In almost all references for some two-level quantum system (for example, this Wikipedia article), arbitrary eigenstates are defined in spherical coordinates, and can be written separately as follows (or some equivalent variation) involving $\phi$ and $\theta$:

\begin{align} | \Phi_-(\theta,\phi) \rangle &= (\sin\frac{\theta}{2}e^{-i\phi},-\cos\frac{\theta}{2})^T \\ | \Phi_+(\theta,\phi) \rangle &= (\cos\frac{\theta}{2}e^{-i\phi},\sin\frac{\theta}{2})^T \end{align}

Here, the space assumed is one of the Bloch sphere. However, a professor casually mentioned that one can rewrite the above in terms of momentum space coordinates (usually denoted $k_x$ and $k_y$, living in a 2D space). How do I reformulate the above in terms of those coordinates, i.e. $| \Phi_-(k_x,k_y) \rangle,| \Phi_+(k_x,k_y) \rangle$? I understand that this formulation will not have the much-coveted spherical symmetry, but I am curious as to how we can change the underlying coordinate system in this case.

My suspicion is that the metric of the sphere (that includes its Jacobian terms, 1 and $\sin\theta$) is somehow related. Going off this, my first attempt is to simply make the following substitutions:

\begin{align} \theta &= \tan^{-1}\left(\frac{y}{x}\right) \\ \phi &= \cos^{-1}\left(\frac{z}{r}\right) \\ r &= \sqrt{x^2+y^2+z^2} \end{align}

However, I wanted clarification on what I am trying to do here - I believe I am confused. Are $x,y,z$spatial coordinates and not necessarily momentum? Or is $(x,y,z) \equiv (k_x,k_y,k_z)$? Or do I have to bring in the momentum operators $\hat{p}=-i \bar{h}\frac{\partial}{\partial \hat{x}}$?

Other posts on this site talk about taking Fourier transforms from position space to momentum space, but that's for the Schrodinger equation. Is this applicable?

Most readers will realize by now that I am confused, and that I do not know how to proceed from here. Any references, clarifications or solutions would greatly be appreciated!

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  • $\begingroup$ That's pretty odd. Maybe he was thinking of a stereographic projection? I'm not sure how that would be very useful, though. $\endgroup$ – Emilio Pisanty Mar 22 at 17:55
  • $\begingroup$ @EmilioPisanty We were discussing calculating Berry curvature, etc for this 2-state system and he suggested that perhaps some issues might arise if I used $\phi$ and $\theta$ instead of momentum space coordinates. He mentioned ‘metric’, and that maybe I should do those calculations over momentum space. I have done these calculations for other 2-state models like the Haldane model, but I’m not sure how to bring $k$ in here. $\endgroup$ – TribalChief Mar 22 at 21:19
  • $\begingroup$ The Haldane model is not a two-state system. Perhaps your core confusion is the distinction between two-state and two-band systems? $\endgroup$ – Emilio Pisanty Mar 22 at 23:58
  • $\begingroup$ @EmilioPisanty, good point. I confused myself there, but maybe this mistake is not too relevant to my problem? Let me try again: I am trying to find a way to calculate the non-Abelian Berry curvature (a 2x2 matrix) of the Bloch sphere case in terms of $k_x, k_y$ instead of $\phi, \theta$. I am guessing the geometry of the problem somehow matters (via the metric of the sphere). The professor said that if I use momentum space coordinates, the metric won’t matter. So, is there a natural way for me to change the parameters of the problem? $\endgroup$ – TribalChief Mar 23 at 0:11
  • $\begingroup$ That sounds like you're still in the same confusion. A non-abelian Berry curvature is a matrix-valued function of parameter space, i.e. typically momentum. There's some crucial bit of context which you're not reporting (say, an explicit definition of the Bloch-sphere coordinates in terms of momentum, which is extremely common) and which we cannot reconstruct. I don't think this question is answerable as it stands. $\endgroup$ – Emilio Pisanty Mar 23 at 9:57

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