Probability density for observable with continuous Spectrum

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To find the probability density for an electron's kinetic energy in the interval [E, E+dE], the wave function must be analyzed using the kinetic energy operator p^2/2m. The Born rule can be applied, but challenges arise due to the infinite dimensional Hilbert space and the degeneracy of the kinetic energy operator's eigenstates. A suggested approach is to calculate the momentum space wave function, denoted as ψ(p), to facilitate the probability density determination. This method may help in overcoming the difficulties associated with the continuous spectrum of kinetic energy. Understanding these concepts is crucial for accurate probability calculations in quantum mechanics.
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)##?
 
Thread 'Help with Time-Independent Perturbation Theory "Good" States Proof'

Thread 'Help with Time-Independent Perturbation Theory "Good" States Proof'

(Disclaimer: this is not a HW question. I am self-studying, and this felt like the type of question I've seen in this forum. If there is somewhere better for me to share this doubt, please let me know and I'll transfer it right away.) I am currently reviewing Chapter 7 of Introduction to QM by Griffiths. I have been stuck for an hour or so trying to understand the last paragraph of this proof (pls check the attached file). It claims that we can express Ψ_{γ}(0) as a linear combination of...
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