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Advanced Physics Homework Help
How can I determine the values of α and β for a canonical transformation?
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[QUOTE="Lambda96, post: 6845270, member: 696125"] [B]Homework Statement:[/B] See Screenshot [B]Relevant Equations:[/B] none Hi, unfortunately, I have problems with the following task [ATTACH type="full" alt="Bildschirmfoto 2023-01-21 um 17.53.00.png"]320810[/ATTACH] I first tried to calculate ##JIJ^T##. $$\left( \begin{array}{rrr} \frac{\partial q'_i}{\partial q_j} & \frac{\partial q'_i}{\partial P_j} \\\frac{\partial P'_i}{\partial q_j} & \frac{\partial P'_i}{\partial P_j} \\ \end{array}\right)\left( \begin{array}{rrr} 0 & \textbf{1} \\\ -{\textbf{1}} & 0 \\ \end{array}\right)\left( \begin{array}{rrr} \frac{\partial q'_i}{\partial q_j} & \frac{\partial P'_i}{\partial P_j} \\\frac{\partial P'_i}{\partial q_j} & \frac{\partial q'_i}{\partial P_j} \\ \end{array}\right)=\left( \begin{array}{rrr} \frac{\partial q'_i}{\partial q_j} \textbf{1} \frac{\partial q'_i}{\partial P_j} -\frac{\partial q'_i}{\partial P_j} \textbf{1} \frac{\partial q'_i}{\partial q_j} & \frac{\partial q'_i}{\partial q_j} \textbf{1} \frac{\partial P_i}{\partial P_j} -\frac{\partial q'_i}{\partial P_j} \textbf{1} \frac{\partial P_i}{\partial q_j} \\ \frac{\partial P_i}{\partial q_j} \textbf{1} \frac{\partial q'_i}{\partial P_j} -\frac{\partial P_i}{\partial P_j} \textbf{1} \frac{\partial q'_i}{\partial q_j} & \frac{\partial P_i}{\partial q_j} \textbf{1} \frac{\partial P_i}{\partial P_j} -\frac{\partial P_i}{\partial P_j} \textbf{1} \frac{\partial P_i}{\partial q_j} \\ \end{array}\right)$$ ##\textbf{1}## is supposed to be the unit matrix, but unfortunately I don't know how to write it with latex, which is why I have represented it in my calculation like this Then I took the following relation from my professor, I hope you can read it well. [ATTACH type="full" alt="Bildschirmfoto 2023-01-21 um 18.38.51.png"]320815[/ATTACH] and get the following $$\left( \begin{array}{rrr} \Bigl\{ q'_i,q'_i \Bigr\} & \Bigl\{ q'_i,P_i \Bigr\} \\ \Bigl\{ P_i,q'_i \Bigr\} & \Bigl\{ P_i,P_i \Bigr\} \\ \end{array}\right)$$ Unfortunately, I am not getting anywhere now, because in order to show which values ##\alpha## and ##\beta## must assume in order for it to be a canonical transformation, I would have to get the symplectic unit matrix again, but with my calculation I would only get numbers as entries in the matrix and not unit matrices, as it should be. My professor's script contains the following formulation [ATTACH type="full" alt="Bildschirmfoto 2023-01-21 um 18.58.05.png"]320816[/ATTACH] [/QUOTE]
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How can I determine the values of α and β for a canonical transformation?
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