Question

S8. Write the general form of space dependent Schrödinger equation (along z direction) for a particle into a region of space where the potential energy, U is 4 times larger than the total energy, E. Name the involved physical quantities. Enumerate the properties of the wave function.

Answer 1

I'm not sure I understand. The Schrodinger Equation depends on position. There is time-dependent and time-independent forms. "Space-dependent" is redundant.

The general time-independent, 1-dimensional (along z direction) Schrodinger Equation is as follows:\[-\frac{\hbar}{2m}\frac{d^{2}\psi (z)}{dz^{2}}+U(z)\psi(z)=E\psi(z)\]The solution to this partial differential equation is possible only if the potential function U(z) is known. In this case, it appears that your potential function is a constant: U(z) = U = 4E. We can rewrite the equation now as:\[-\frac{\hbar}{2m}\frac{d^{2}\psi (z)}{dz^{2}}+4E\psi(z)=E\psi(z)\]This leaves us with:\[-\frac{\hbar}{2m}\frac{d^{2}\psi (z)}{dz^{2}}=-3E\psi(z)\]
h_bar is Planck's constant divided by 2*pi
m is mass
psi(z) is the wavefunction (time-independent)
E is the total energy

If this is not useful, please clarify the question and I'll be more than happy to revise this accordingly.

Answer 2

That's part of why I put this exercise here, I haven't been able to find anything related to the space-dependency of the Schrödinger equation, so I didn't know which form of the equation I should use. Is this really all that is to it? Thank you for answering!

Answer 3

I'm not sure really. I've never heard of a space-dependent equation, because I though the S.E. was always dependent on space.

If you're looking for space-dependent and it doesn't say anything about time, then there is a chance that you may be looking for the general form that includes time-dependence:\[-\frac{\hbar}{2m}\frac{\partial^{2}\psi(z,t)}{\partial z^{2}}+V(z,t)\psi(z,t)=i\hbar\frac{\partial\psi(z,t)}{\partial t}\]

Answer 4

But again I'm not sure.. maybe saying space-dependent means that you don't want an equation dealing with time, just space.

Answer 5

This will have to do for now. Once I find out the answer, I'll make sure to post it here. Again, thanks a lot for your help!

Answer 6

Yes... it is redundant but only because we're so used to seeing the equations in flavors of time dependence or independence. But as far as the math is concerned.. it doesn't give a hoot what you call it or how you interpret it, correct? All the equation knows is that there are 4 dimensions, call them what you like. (We of course call them 3 space and 1 time but... again, the math doesn't care)