# Potential value , Bound state

1. Nov 27, 2013

### Dassinia

Hi,
1. The problem statement, all variables and given/known data

A particle of mass m has a potential
V(r)= -Vo r<a
0 r>a

Find the minimum value of Vo for which there's a bound state of energy and angular momentum are zero by solving shrodinger equation for E<0 and taking the limit E-> 0

2. Relevant equations

3. The attempt at a solution
I want to know what does l represent in the radial shrodinger equation ? Is it the angular momentum ?
For now, I considered l=0 and solved Shrodinger equation
I got
r<a
u(r)=C*sin(βr) with β=sqrt(2m(Vo+E)/h²)
r>a
u(r)=Bexp(-kr) with k= sqrt(-2mE/h²)

With boundary conditions we have
-β/k=tan(βa)

Where am I supposed to use E->0 ..?
Is it just in β=sqrt(2m(Vo+E)/h²)
so that Vo=β²h²/2m ?

Last edited: Nov 27, 2013
2. Nov 28, 2013

### TSny

You need to let E → 0 in both β and k.

3. Nov 28, 2013

### Dassinia

So we have that k=0 and then ? I dont Know what i have to do then

4. Nov 28, 2013

### TSny

What is the limit of β/k as E → 0? What does that tell you about the quantity βa in tan(βa)?

5. Nov 28, 2013

### Dassinia

We have tan(aβ)= -β/k=-sqrt(-1-Vo/E)
so when E-> 0-
tan(aβ)=-infinity
so βa=-pi/2 ?

Last edited: Nov 28, 2013
6. Nov 28, 2013

### TSny

Yes, except for the sign. β is a positive quantity.

7. Nov 28, 2013

### Dassinia

OK Thank you !