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.
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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 'Product of two boosts and then two inverse boosts is rotation matrix'

So is there some elegant way to do this or am I just supposed to follow my nose and sub the Taylor expansions for terms in the two boost matrices under the assumption ##v,w\ll 1##, then do three ugly matrix multiplications and get some horrifying kludge for ##R## and show that the product of ##R## and its transpose is the identity matrix with det(R)=1? Without loss of generality I made ##\mathbf{v}## point along the x-axis and since ##\mathbf{v}\cdot\mathbf{w} = 0## I set ##w_1 = 0## to...
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