- #1

Yitzach

- 61

- 0

[itex]\hat{p}=\frac{mv}{\sqrt{1-(\frac{v}{c})^2}}[/itex]6.47

[itex]p=-i\hbar\nabla\Rightarrow v=-\frac{i\hbar}{m}\nabla[/itex]cover

[itex]\hat{p}=\frac{-i\hbar\nabla}{\sqrt{1-(\frac{-i\hbar\nabla}{mc})^2}}[/itex]

[itex]\hat{p}=\frac{-i\hbar\nabla}{\sqrt{1+(\frac{\hbar}{mc})^2\nabla^2}}[/itex]

Use:

[itex]\int\Psi*\frac{-i\hbar\nabla}{\sqrt{1+(\frac{\hbar}{mc})^2\nabla^2}}\Psi dx=\left<p\right>[/itex]

[itex]\int\Psi*\frac{-i\hbar\nabla\Psi}{\sqrt{1+(\frac{\hbar}{mc})^2\nabla^2\Psi}} dx=\left<p\right>[/itex]

This doesn't look quite intelligent to me. Does anyone have a better solution to the problem? It would be wonderful if you could provide the eigenfuction for it as well. If this is it, the next trick will be to solve the following equation or an approximation of it:

[itex]\frac{-i\hbar\nabla\Psi}{\sqrt{1+(\frac{\hbar}{mc})^2\nabla^2\Psi}}=p\Psi[/itex]