- #1
member 545369
Homework Statement
Given ##\hat{x} =i \hbar \partial_p##, find the position operator in the position space. Calculate ##\int_{-\infty}^{\infty} \phi^*(p) \hat{x} \phi(p) dp ## by expanding the momentum wave functions through Fourier transforms. Use ##\delta(z) = \int_{\infty}^{\infty}\exp(izy) dz ##.
Homework Equations
I'll explicitly state them in my attempt
The Attempt at a Solution
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First we focus on the integrand. We move over to position space through Fourier transformations $$\phi^*(p) \hat{x} \phi(p) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{2 \pi \hbar} \exp(-ipx/\hbar) \psi(x) i \hbar \frac{\partial}{\partial p} \left(\exp(ipy/\hbar) \psi^*(y) \right) dx dy $$ my reasoning is that I can move over conjugate of the spatial wave function as it does not explicitly depend on momentum. Upon simplifying a little we get $$ \phi^*(p) \hat{x} \phi(p) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{-y}{2 \pi \hbar} \exp\left(\frac{ip(y-x)}{\hbar}\right) \psi(x)\psi^*(y)dx dy $$ which we integrate with respect to ##p## to get
$$ \int_{-\infty}^{\infty} \phi^*(p) \hat{x} \phi(p) =\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{-y}{2 \pi \hbar} \exp\left(\frac{ip(y-x)}{\hbar}\right) \psi(x)\psi^*(y)dx dy dp$$ utilizing the dirac delta function relation from earlier, we find that $$ \int_{-\infty}^{\infty} \exp\left(\frac{ip(y-x)}{\hbar}\right) dp = \delta\left(\frac{y-x}{\hbar}\right) $$ which, when put back into our integral and integrated with respect to y, simply substitutes ##y \rightarrow x## in the integrand (by the definition of the dirac delta function $$ \int_{-\infty}^{\infty} \phi^*(p) \hat{x} \phi(p) =\int_{-\infty}^{\infty} \frac{-x}{2 \pi \hbar} \psi(x)\psi^*(x)dx$$ which is close, I think? I may be off by a factor of ##-1/2π\hbar##? Ideally, since ##\hat{x} = x## in position space, I should get
$$ \int_{-\infty}^{\infty} x \psi(x)\psi^*(x)dx = \int_{-\infty}^{\infty} \psi(x) \hat{x} \psi^*(x)dx $$
no? I don't know, i feel like I've gone so much through the weeds that either everything is wrong or my brain is fried and I can't see that I've actually gotten the right answer (unlikely). A bit of guidance would be much appreciated!
EDIT: I published this on safari and the formatting seemed off (some sentences were not being displayed). It looks fine on chrome, though. Weird!