Sunday, 4 November 2018

quantum mechanics - Solving the Schrodinger equation for a free particle in momentum space


Time-independent form of the Schrodinger equation states $$\hat H\psi=E\psi$$ For a Hamiltonian in form of $$\hat H=\frac{\hat p^2}{2m}$$ Which indicates a free particle, In the position space is routine and starts with plugging in the momentum operator in position space as $$\hat p=-i\hbar\frac{\partial}{\partial x}$$ And we can obtain eigenvalues and eigenfunctions as $$E=\frac{\hbar^2k^2}{2m}$$ $$\psi^+(x)=e^{ikx}\space\space,\space\space\psi^-(x)=e^{-ikx}$$ $$\psi(x)=A\psi^++B\psi^-$$ I also know we can derive the wavefunction in the momentum space with a Fourier transform. But I want to solve the SE in the momentum space. So $$\hat H\tilde\psi(p)=E\tilde\psi(p)$$ $$\frac{p^2}{2m}\tilde\psi(p)=E\tilde\psi(p)$$ $$\tilde\psi(p)(\frac{p^2}{2m}-E)=0$$ One answer is the same as the previous method $$E=\frac{\hbar^2k^2}{2m}$$ But here $\tilde\psi(p)$ can be any function of $p$. But we know it should be the same as the result of the Fourier transform on $\psi$.


How can we obtain $\tilde\psi(p)$ with this method?



Answer




But here $\tilde{\psi(p)}$ can be any function of $p$.




In $p$ space, $p$ is the variable, not a constant and so, in general


$$p\,f(p) \ne P\,f(p)$$


where $P$ is a constant. Only for the case that $f(p) \propto \delta(p - P)$ can we write, e.g.,


$$p\, \delta(p - P) = P\, \delta(p - P)$$


Can you take it from here?


No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...