text: # Wigner function of $|n\rangle\langle m|$
  
  For $$n=m$$, the Wigner function is given by, $$W_n(\alpha) = \frac{2}{\pi} (-1)^n \exp(-2 |\alpha|^2) L_n(4 |\alpha|^2),$$
  
  And for $$n \neq m$$, it is, $$f_{mn}=\sqrt{\frac{m!}{n!}} e^{i(m-n) \arctan\left( p/x\right) } \frac{\left( -1\right)^{m}}{\pi\hbar}\left ( \frac{x^{2}+p^{2}}{\hbar/2} \right ) ^ {\left( n-m\right) /2} L_{m}^{n-m}\left( \frac{x^{2}+p^{2}}{\hbar/2}\right) e^{-\left( x^{2}+p^{2}\right) /\hbar}$$ and in this case, there will be always a $$e^{i(m-n) \arctan\left( p/x\right)}$$ term. But the Wigner function is a real valued function. How to reconcile this?
  
  • What is $f_{mn}$?
  – Alex
  Oct 27, 2023 at 13:53
  • Linked. Note this operator is not Hermitian. Oct 27, 2023 at 16:03
  
  In general, Wigner transforms of Hermitian operators are real, $$g(x,p)= \hbar \int \!\! dy ~ e^{-iyp} \langle x+\hbar y/2| \hat G |x-\hbar y /2 \rangle , \implies \\ g(x,p)^*= \hbar \int \!\! dy ~ e^{iyp} \langle x+\hbar y/2| \hat G |x-\hbar y /2 \rangle ^* \\ =\hbar \int \!\! dy ~ e^{iyp} \langle x-\hbar y/2| \hat G ^\dagger |x+\hbar y /2 \rangle ,$$ so $$g=g^*$$ for $$\hat G= \hat G^\dagger$$, like WF diagonal elements, otherwise not. $$|m\rangle \langle n|$$ is not hermitian.
explanation: The text extract discusses the Wigner function for quantum states, specifically for the cases where \( n = m \) and \( n \neq m \). It presents the mathematical expressions for these cases and raises a question about the apparent contradiction between the complex term \( e^{i(m-n) \arctan(p/x)} \) and the requirement that the Wigner function be real-valued. The extract also includes a comment from another user asking for clarification on the term \( f_{mn} \) and a response that provides additional context and explanation.
  
  The document contains a question and an answer to that question.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Wigner function of $|n\rangle\langle m|$
  
  For $$n=m$$, the Wigner function is given by, $$W_n(\alpha) = \frac{2}{\pi} (-1)^n \exp(-2 |\alpha|^2) L_n(4 |\alpha|^2),$$
  
  And for $$n \neq m$$, it is, $$f_{mn}=\sqrt{\frac{m!}{n!}} e^{i(m-n) \arctan\left( p/x\right) } \frac{\left( -1\right)^{m}}{\pi\hbar}\left ( \frac{x^{2}+p^{2}}{\hbar/2} \right ) ^ {\left( n-m\right) /2} L_{m}^{n-m}\left( \frac{x^{2}+p^{2}}{\hbar/2}\right) e^{-\left( x^{2}+p^{2}\right) /\hbar}$$ and in this case, there will be always a $$e^{i(m-n) \arctan\left( p/x\right)}$$ term. But the Wigner function is a real valued function. How to reconcile this?
  
  • What is $f_{mn}$?
  – Alex
  Oct 27, 2023 at 13:53
  • Linked. Note this operator is not Hermitian. Oct 27, 2023 at 16:03
  
  In general, Wigner transforms of Hermitian operators are real, $$g(x,p)= \hbar \int \!\! dy ~ e^{-iyp} \langle x+\hbar y/2| \hat G |x-\hbar y /2 \rangle , \implies \\ g(x,p)^*= \hbar \int \!\! dy ~ e^{iyp} \langle x+\hbar y/2| \hat G |x-\hbar y /2 \rangle ^* \\ =\hbar \int \!\! dy ~ e^{iyp} \langle x-\hbar y/2| \hat G ^\dagger |x+\hbar y /2 \rangle ,$$ so $$g=g^*$$ for $$\hat G= \hat G^\dagger$$, like WF diagonal elements, otherwise not. $$|m\rangle \langle n|$$ is not hermitian.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract discusses the Wigner function for quantum states, specifically for the cases where \( n = m \) and \( n \neq m \). It presents the mathematical expressions for these cases and raises a question about the apparent contradiction between the complex term \( e^{i(m-n) \arctan(p/x)} \) and the requirement that the Wigner function be real-valued. The extract also includes a comment from another user asking for clarification on the term \( f_{mn} \) and a response that provides additional context and explanation.
  
  The document contains a question and an answer to that question.
  
  Label: question-and-answer