Schrodinger's Equation: Find E for l=0

[Nusc]

Homework Statement

$$\frac{1}{r}\frac{d}{dr}(r^2\frac{d}{dr}\Psi (r)) + { \frac{2m}{\hbar^2}[E-V(r)] - \frac{l(l+1)}{r^2}}\Psi (r) = 0$$
$$V(r) = -Vo r\leq a$$
$$0 r > a$$Use
$$\Psi (r) = \Xi (r)/r$$

Questions asks to Find E l = 0, do I solve the general equation first or should I make l = 0 right away?

Homework Equations

The Attempt at a Solution

 

[pam]

It's much easier to set l=0 first.

 

[Nusc]

Inside the dot I get

$$\Xi (r) = C*Cos[l*r]+D*Sin[l*r]$$

Outside the Dot I get

$$\Xi (r) = A*Exp[-k*r]+B[Exp[k*r]$$
Taking r to infinity yields:

$$\Xi (r) = A*Exp[-k*r]$$What other boundary conditions can I apply?

 

[Nusc]

I took
$$\Xi (a)$$
and
$$\frac{d\Xi}{dr}$$ at r = a

How would I solve these equations?
$$A*Exp[-k*a] = C*Cos[l*a]+D*Sin[l*a]$$
$$-kAExp[-k*a]=-C*l*Sin[l*a]+D*l*Cos[l*a]$$

 

[Nusc]

$$\frac{h^2\pi^2}{8ma^2} < E + V < \frac{h^2\pi^2}{2ma^2}$$

Now that's the energy for l=0. how would one want to estimtae the min radius to ensure at least one bound e-state?

 

[Nusc]

NO bound states occur if a^2 < (pi^2 h^2 )/(8*m*Vo)

take positive root.

that's the minimum radius . Is that correct? if not I can't continue.

Now given this:

$$\frac{h^2\pi^2}{8ma^2} < E + V < \frac{h^2\pi^2}{2ma^2}$$

letting V = 0.5 eV and m = 0.04*9.11x10^-31,
how can one find the colour of light in this circle?

 

[malawi_glenn]

what do you mean by: "Inside the dot I get " in post #3

This Boundary condition is good too:
[tex}] \Xi (r=0) = 0 [/tex]

Is the question to find $$E{l=0}$$?

What is your "k" in post #3 ?

Is it an atomic physics problem? You are speaking about bound e-states

 

[Nusc]

Take post #6 as a given.

how would one find the minimum radius

 

[malawi_glenn]

This was posted to me via private message T07:54

"question asks, estimate the min radii required of an InAs spherical quantum dot embedded in a GaAs matrix in order to ensure that there will be at least one bound electron state, or at least two bound states in the dot.

Choose V = 0.5 eV and effective mass m* = 0.04m in the dot and barrier layers. Does the separation in the energies between the two bound states tell you about the colour of light the dot would emit when excited?

If not, why not? What else would you need to do in order to obtain this information?

So since I found the bound states.
What do I need to solve for to answer the remaining questions?

I'm not sure about my minimum radius as I posted in the thread, is it correct?"