Electron One Split Energy Probability Density Function

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SUMMARY

The discussion focuses on calculating the Energy Probability Density Function using the Fourier transform of a wave function. The participant successfully converted the coordinate space wave function to momentum space but encountered issues with the normalization constant. It is crucial to integrate the square of the magnitude of the wave function to determine the normalization constant accurately. The final goal is to find the probability density for an infinitesimal energy range from E to E + dE without integrating over a broader interval.

PREREQUISITES
  • Understanding of Fourier transforms in quantum mechanics
  • Knowledge of wave functions in both coordinate and momentum space
  • Familiarity with probability density functions in quantum physics
  • Experience with normalization constants in wave function analysis
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  • Study the process of converting wave functions between coordinate and momentum space
  • Learn about normalization techniques for quantum wave functions
  • Research the implications of probability density functions in quantum mechanics
  • Explore advanced topics in quantum mechanics related to energy intervals and their significance
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Students and professionals in quantum mechanics, particularly those working on wave function analysis and energy probability density calculations.

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Homework Statement


https://www.physicsforums.com/attachment.php?attachmentid=69371&d=1399142463

2. The attempt at a solution
I am working on the last problem now.
Here is what I have got so far. Basically I have converted the coordinate space wave function to a momentum space wave function. Then one can associate an Energy interval to a momentum space interval and integrate the absolute value of momentum space wave function squared in that interval to find the probability density function as a function of Energy. But somehow I am getting a weird result.
View attachment Übung 19.pdf
 
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When finding the normalization constant, c, make sure you integrate the square of the magnitude of the wave function.

You are asked to find the probability density for an infinitesimal range E to E + dE. So, you don't need to integrate over an interval.

Your Fourier transform to momentum space looks good to me.
 
Thanks. Yeah you're right about the normalization constant, really sloppy on my side.
Thanks for your hint.
 

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