Recent content by MBPTandDFT

I agree. I would call it an heuristic derivation, and I would say it holds because H doesn't involve (besides doesn't depend on) time at all, and tha's a differential equation in time, if you close your eyes and don't look at the space-laplacian inside H that would prevent you from dividing by...

Let's take ##\hbar=1##. Then you have: ##i\frac{d\psi}{dt}=H\psi## The usual separation of variables technique is: ##i\frac{d\psi}{\psi}=Ht## and integrate: ##i\int_{\psi_0}^{\psi}\frac{d\psi'}{\psi'}=H\int_{t_0}^tt'\longrightarrow\ln\frac{\psi}{\psi_0}=-iH(t-t_0)## So finally...

Finally I got it: a locality in real space is usually transformed to a non-locality in reciprocal space, and this exactly balances the fact that a constant is transformed to a delta: in the end one ends up with a constant Hamiltonian term in both representations. Thanks for your hints!

Your derivation is clear. Just I cannot get what you are proving with it...

Cannot get the conclusion..

No, what I mean is that a constant in real space, ##V(x)=c##, corresponds to the same constant in reciprocal space, ##V(k)=c##. Hence I'm puzzled because a trivial application of Fourier would result in a delta function in reciprocal space, not a constant.

Consider a constant potential in real space c. Its second quantization hamiltonian is ##\hat{H}=\int d^3x \hat{\psi}^{\dagger}(x)c\hat{\psi}(x)##. Now write this same term in momentum space. It is again ##\hat{H}=\sum_{k} \hat{a}^{\dagger}(k)c\hat{a}(k)##, meaning ##v(x)=const## corresponds to...

You are calling two quantities with the same name: the ket |Psi> is the "quantum state", it belongs to the Hilbert space and it is a vector, while the wavefunction Psi(x) is the projection of this state on a basis element (x), and it's a function.

H=p^2/2m+c What's c? It's of course a shift in energy, but can be thought also as a smoother and smoother real-space local potential that becomes a constant all over the space. On the other hand, why couldn't one think about it as a constant potential in reciprocal space? It's a shift in energy...