Wavefunction of a space and spin rotated state

AI Thread Summary
The discussion focuses on the mathematical formulation of a space and spin rotation operator, represented as U(R) = e^{-i\textbf{L}\cdot \hat{\textbf{n}}\phi/\hbar}·e^{-i\textbf{S}\cdot \hat{\textbf{n}}\phi/\hbar}. Participants explore the implications of this operator on the wave function, specifically questioning how it transforms under rotation and whether the position state |r⟩ corresponds to angular momentum eigenkets |nlm⟩. It is clarified that |r⟩ and |nlm⟩ are distinct, and the wave function for given quantum numbers n, l, and m can be derived from the ket using the expression ψ_{n,l,m} (r) = ⟨r | nlm⟩. The conversation emphasizes the need for further computation of matrix elements related to these transformations. Understanding these relationships is crucial for advancing the discussion on quantum states and their transformations.
NiRK20
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Homework Statement
Basically, I have the wavefunction of a particle with spin (not necessarly 1/2) given by ##\psi(\textbf{r}, m) = \psi _{m} (\textbf{r}) = \langle\textbf{r}, m|\psi \rangle##. My task is to find the wavefunction of a rotated state ##U(R)|\psi \rangle##, with ##U## being the product of a spacial rotation by a spin rotation.
Relevant Equations
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Since ##U## is a space and spin rotation, it would be

$$U(R) = e^{-i\textbf{L}\cdot \hat{\textbf{n}}\phi/\hbar}\cdot e^{-i\textbf{S}\cdot \hat{\textbf{n}}\phi/\hbar}$$

And, then

$$\psi'(\textbf{r}, m) = \langle\textbf{r}, m|e^{-i\phi(\textbf{L} + \textbf{S}) \cdot \hat{\textbf{n}}/\hbar}|\psi \rangle = \sum_{m'}\int \langle\textbf{r}, m|e^{-i\phi(\textbf{L} + \textbf{S}) \cdot \hat{\textbf{n}}/\hbar}|\textbf{r}', m'\rangle\psi _{m'}(\textbf{r}') d^{3}r'$$

The problem is where to go from here (if this is right until now). There is a way to compute these matrix elements? Does ##|\textbf{r}\rangle## equals to the eigenkets of angular momentum ##|nlm\rangle##?
 
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It is difficult to answer because it depends on what is expected. Can you post the full text of the question?

NiRK20 said:
$$U(R) = e^{-i\textbf{L}\cdot \hat{\textbf{n}}\phi/\hbar}\cdot e^{-i\textbf{S}\cdot \hat{\textbf{n}}\phi/\hbar}$$
How does ##U(R)## become a function of ##\phi## only?

NiRK20 said:
Does ##|\textbf{r}\rangle## equals to the eigenkets of angular momentum ##|nlm\rangle##?
No, they are completely different. This is how you recover the wave function for a given ##n,l,m## from the ket:
$$
\psi_{n,l,m} (\mathbf{r}) = \braket{\mathbf{r} | nlm}
$$
 
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