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Homework Statement
Homework Equations
##\hat{P}= ih d/dx##
The Attempt at a Solution
To actually obtain ##\psi_{p_0}## I guess one can apply the momentum operator on the spatial wavefunction. If we consider a free particle (V=0) we can easily get obtain ##\psi = e^{\pm i kx}##, where ##k= \sqrt{2mE/ \hbar}##.
By now applying momentum operator we get ##\hat{P} \psi= ih \cdot \pm i \sqrt{2mE/ \hbar} ##. This is as far as I get, how do I actually get the eigenfunction of momentum on the required form?
And 2:
How do I show that the completeness relation is satisfied for that 'specific' expression? I know that we can express ##x> = \int dp p><px>,##take ##i \hbar \frac{d}{d_x} f_p(x) = p f_p(x)## and turn into a fourier transformation (including k in the kernel) and get ##p' \tilde{f_p(p')}## which can be solved picking p=p' and any function of p can be converted into an eigenfunction if we transform back to x s.t ##f_p(x) = \int dp e^{ipx \hbar} \tilde{f_p(p)}##, but I have no idea how to use this to confirm the completeness relation for the above expression....
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