Rigid Box and 3D Schrodinger equation

[budafeet57]

Homework Statement

An electron is confined within a three-dimensional cubic region the size of an atom where L = 200 pm.
a) write a wave equation for the electron
b) wirte a general wave function for the possbile states of the electorn. List any quantum numbers and their possible values.
c) calculate the energy of the four lowest states
d) calculate the energies and wavelength of photons created during transitions between these states

Homework Equations

and A = (2/L)^(3/2)

The Attempt at a Solution

The note above is from my lecture note. I think they probably work for this question.

a, c)

b) I am confused here.
d)

 

[Simon Bridge]

I think
(a) means the Schrödinger equation ... it will have to be peicewise.
(b) is the wave-function you have written ... if this is long answers you'll have to justify doing $\psi_{k}(x,y,z)=\psi_l(x)\psi_m(y)\psi_n(z)$
(c) ... look for states k=l+m+n with lowest energy.
(d) ... once you have (c) this is just subtraction

I think your reasoning in the later part is OK - from what I can make out - I'd prefer you to show more thinking for the first parts since it just looks like you are copying from lectures. You should try to show that you have understood the lecture to get full marks, and this usually means you have to write sentences as well as equations.

 

[budafeet57]

Hi Simon, thanks for helping me again.

wave equation is the Schrödinger equation? and wave function is the solution?

 

[Simon Bridge]

How else do we account for the first two questions.

The (time independent) SE is a special case of the Helmholtz equation which is the time-independent part of a wave equation.
(Therefore, the statistics described by solutions to the SE will behave like waves.)

I suspect the answer for the wave equation should be the time-dependent SE ... I'd start by writing out $$\left ( \nabla^2+V(\vec{r})-i\hbar\frac{\partial}{\partial t}\right )\Psi(\vec{r},t)=0$$... then get more explicit for $V$.

The next question is asking for $\Psi(\vec{r},t)$ ...

See why I think your prof is seeing if you have understood the lectures?

 

[budafeet57]

Hi Simon, I'll come back and think more after my coming exam.

 

[Simon Bridge]

You may want to examine this:
http://en.wikipedia.org/wiki/Particle_in_a_box
... which goes through the 1D case in about the same steps as your exercize.