Is a Distribution Function a Ratio of Differentials?

AI Thread Summary
The discussion explores the relationship between distribution functions and ratios of differentials, particularly in the context of wavefunction representation in k-space and the Planck distribution. It highlights that the squared-amplitude can be expressed as a ratio of differentials, leading to the conclusion that the squared amplitudes in momentum and wavevector spaces are related by a factor of 1/ħ. The conversation also touches on the transformation of the Planck distribution from frequency to wavelength, confirming that this transformation adheres to the principles of distribution functions as ratios of derivatives. A clarification is made regarding the notation of ħ, correcting an earlier misstatement about its representation. Overall, the thread emphasizes the mathematical relationships inherent in quantum mechanics and distribution functions.
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Homework Statement
Show that ##\vert\tilde{\phi}\vert^{2}=\frac{1}{\sqrt{\hbar}}\vert\phi\vert^{2}## where ##\tilde{\phi}## is a momentum-basis and ##\phi## is a k-space-basis wave packet.
Relevant Equations
##p=k\hbar##
I read on a post here titled 'Understanding Fourier Transform for Wavefunction Representation in K Space' that if one represents the squared-amplitude as a ratio of differentials, the solution is given. Letting the squared-amplitude be ##\phi##.
$$\frac{d\phi}{dp}=\frac{d\phi}{dk}\frac{dk}{dp}$$
Where $$\frac{dk}{dp}=\frac{1}{\hbar}$$
And therefore
$$\vert\tilde{\phi}\vert^{2}=\frac{1}{\hbar}\vert\phi\vert^{2}$$

Additionally, when I represent the Planck distribution with respect to frequency as ##\frac{du}{df}## and multiply by ##\left\vert\frac{df}{d\lambda}\right\vert##, I get the correct expression for the Planck distribution with respect to wavelength. Is this just distribution functions being represented as a ratio of derivatives?
 
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I would say, more simply, that the relation follows because ##p=\hbar k## and ##\int \vert\tilde{\phi}\vert^{2}dp## is a dimensionless probability and so is ##\int \vert\phi\vert^{2} dk##.
 
For OP: In statement of problem, why is it ##\sqrt(\hbar) ## and not just ##\hbar##????
 
My apologies. Yes, it is ##\hbar## that shouldn't be squared.
 
shouldn't be square-rooted.
 
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