Infinite string at rest for t<0, has instantaneous transverse blow at t=0 which gives initial velocity of V \delta ( x - x_{0} ) for a constant V. Derive the position of string for later time.
I thought that this would be y_{tt} = c^{2} y_{xx} with y_{t} (x, 0) = V \delta ( x - x_{0} ) ...
I have a result for the commutator of \left[ H , \ \mathbf{p} \times \mathbf{L} - \mathbf{L} \times \mathbf{p} \right] which is almost certainly true (the question was "show that ..." so unless there is an error then this is fine) and thus I am calculating the last part.
The reason I am...
The question is just to calculate the commutator \left[ H , \ \mathbf{K} \right] where \mathbf{K} = \frac{1}{2m} \left( \mathbf{p} \times \mathbf{L} - \mathbf{L} \times \mathbf{p} \right) + U(r) \mathbf{r} .
The reason I wanted to do [H,Ur] was because I wanted to consider \mathbf{r} =...
My attempt to calculate \left[ H , \ U(r) r \right] is as follows. Clearly we are just having to calculate \nabla ^{2} \left( U r \psi \right) - Ur \nabla ^{2} \psi , we drop the 'r' because U is just U(r) so we can absorb it into U (later do a substitution).
Then note that \nabla ^{2}...
Indeed. I did the differentiation and got -2xr^{-3} however this could be incorrect. I was just trying to see if taking a specific example would help me with the longer question.
I need to calculate \left[ H , \ U(r) \mathbf{r} \right] in general case of U and am somewhat unsure about...
OK, I think that \mathbf{0} was what I wanted the answer to be. This is all from a larger question.
I have shown that \left[ H , \ \mathbf{p} \times \mathbf{L} - \mathbf{L} \times \mathbf{p} \right] = \mathbf{r} \left( \mathbf{G} \cdot \mathbf{p} + \mathbf{p} \cdot \mathbf{G} \right) -...
OK, but surely if we have that H = \frac{- \hbar ^{2}}{2m} \nabla ^{2} + U then it is clear that the 'U' part will commute with things like x/r, then clearly one is considering something like \nabla ^{2} \left( \frac{x \psi}{r} \right) - \frac{x}{r} \nabla ^{2} \left( \psi \right) ? This is...
Thanks for the response. I am a bit unsure about what you mean by the 'cyclic permutation' and this whole idea of using 'r' has been troubling me from the start of this question.
Just used that \mathbf{p} = -i \hbar \nabla and H = \frac{- \hbar ^{2}}{2m} \nabla ^{2} + U
Hence H \mathbf{p} \psi - \mathbf{p} H \psi can be written but note that the \mathbf{p} \dot \mathbf{p} part of H will commute with \mathbf{p} , hence only consider U \mathbf{p} \psi -...
I have a potential of -1/r and I need to compute \left[H , \ \mathbf{p} \right] .
I got the result of i \hbar \left( \frac{1}{r^{2}}, \ 0 , \ 0 \right) .
Am I wrong about this?
I need to calculate \left[ H , \ \frac{1}{r} \mathbf{r} \right] .
My initial idea is to use \left[ H, \ U \mathbf{r} \right] = \left[ H, \ U \right] \mathbf{r} + U \left[ H , \ \mathbf{r} \right] .
Then clearly \left[H , \ U \right] \psi = \frac{ - \hbar ^{2} }{2m} \left( \nabla ^{2}...