Probability density for observable with continuous Spectrum

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SUMMARY

The discussion focuses on calculating the probability density for an electron's kinetic energy using the wave function in an infinite-dimensional Hilbert space. The kinetic energy operator is defined as p²/2m, and the participant considers applying the Born rule to determine the probability density in the interval [E, E+dE]. Challenges arise due to the degeneracy of the eigenstates of the kinetic energy operator, prompting the suggestion to calculate the momentum space wave function, ψ(p), as a potential solution.

PREREQUISITES
  • Understanding of quantum mechanics principles, specifically wave functions and operators.
  • Familiarity with the Born rule in quantum mechanics.
  • Knowledge of Hilbert spaces and their properties in quantum systems.
  • Basic concepts of kinetic energy operators in quantum mechanics.
NEXT STEPS
  • Research the application of the Born rule in infinite-dimensional Hilbert spaces.
  • Learn how to calculate the momentum space wave function, ψ(p), for quantum states.
  • Study the properties of eigenstates of kinetic energy operators in quantum mechanics.
  • Explore probability density functions for continuous spectra in quantum systems.
USEFUL FOR

Quantum physicists, students studying quantum mechanics, and researchers focusing on wave functions and probability densities in quantum systems.

Naarogaut
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Misplaced Homework Thread
I'm given a wave function for an electron which is given as:
1684834653817.png

For an electron in this state the kinetic energy is being measured, where the kinetic energy operator is p^2/2m. How can I find the probability (density) that an electron is found to have kinetic energy in the interval [E, E+dE]? I was thinking about using the born rule, but I am struggeling to use it for the infinite dimensional Hilbert space, since the eigenstates of the kinetic energy operator degenerate as far as I can tell...
 
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How about calculating ##\psi(p)##?
 
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