Hi, I've got a few questions about Schrödinger's formulation of QM, mostly about how to interpret the results:(adsbygoogle = window.adsbygoogle || []).push({});

1) How do I choose boundary conditions? I know that I should always normalize the wavefunction and make it continuous but some times you need to make the derivative continuous as well. I get a feeling that the first derivative should always be continuous since there is a Laplacian in the equation, otherwise it would be infinite. But if I remember correctly, the solution of the dirac's delta well does not have a continuous derivative.

2) What do I get when I apply an operator? let's say there is a magnitude [itex]f(\textbf{r},t)[/itex]then [itex]\hat{f}\Psi = f(\textbf{r},t)\Psi[/itex]. Now, this object [itex]f(\textbf{r},t)\Psi[/itex] that somehow resembles a wavefunction, what is it? I know that doing this returns an expected value for the magnitude, in one dimention [itex]<f>= \int_{-\infty}^{+\infty} \Psi^*f(x,t)\Psi dx[/itex]. But, without integrating, is there anything I can know from the result of the operator [itex]f(\textbf{r},t)\Psi[/itex]. It looks as though it is the probability distribution of the magnitude f, but I'm not quite sure how to interpret that, since the distribution is in terms of x, and not of the values of the magnitude.

3) What role does the Fourier Transform play in all this? I gather that you can move from momentum to position, transforming the distributions, but I'm not sure how to do it. For any given wavefunction, can the momentum distribution be obtained from squaring [itex] \int_{-\infty}^{+\infty}\Psi (x,t)e^{-i2\pi f x} dx [/itex] ? If so, from the previous question, does this equal [itex]\hat{p}\Psi = -i\hbar\frac{\partial}{\partial x} \Psi[/itex] ? Again, I get the feeling this can't be right just because the variables don't make sense, the Fourier transform will not be a function of x.

4) What role do initial conditions play in this theory? Since it is a differential equation, initial conditions always tend to change the problem a lot. Other that boundary conditions, how are initial conditions in time taken into account? How do you measure them without destroying the system? Or do you just assume every possible solution at the same time until the system is observed?

5) In the finite potential barrier problem, when the particle has less energy than the barrier, tunneling occurs. As weird as tunneling is, I don't really have a problem with it. What is bugging me is the fact that there is a non-zero probability of finding the particleinsidethe barrier. Never mind how small it is, it's not zero. Wouldn't that violate the conservation of energy? Say you have a particle emitter that emits particles of a certain ammount of energy, within a certain uncertainty. Say the barrier is 10 or 100 times bigger than that, there is still a chance of finding particles within the barrier, but that would imply they have at least the potential energy of the barrier, which is absurd because we know it started with way less than that... you get the idea

6) Finally, kind of a big question that perhaps should go in a separate thread, but I don't want to spam the forum with newbie questions :). Why aren't the solutions to the Hydrogen atom symmetric across the angular coordinates? In electromagnetism, when solving for a point charge or charged sphere, I loved the fact that you could dispose of most of the Laplacian because it totally made sense: the problem is symmetric, thereshouldn'tbe an angular dependency... there's no physical reason for it to happen. But the most compelling argument, as I see it, is this: How does the atomknowwhich directions I chose in my coordinate system? The problem is symmetric so there is no way of relating my particular system of coordinates to a set of directions in an actual physical system, because there aren't any "irregularities" to let you distinguish a particular direction from the other ones, they are all in equal standing.

I know they are a ton of questions but this is a really hard topic! Thank you a LOT in advance

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# Schrödinger theory, bunch of questions

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