Why does the finite square well problem require numerical solutions?

[MGWorden]

Homework Statement

Introduction to Quantum Mech Griffiths 4.9for the problem of a finite square well

v(r) =

{-Vo r<= a (i will call section I)

​

0 r>a (section II)

Homework Equations

After I find uI and uII

uI = A sin(K r)
uII = De^-kr

and then set the boundary condition

uI = uII at a equation 1
uI' = uII' at a equation 2

next i know I divide equation 2 by equation 1

The Attempt at a Solution

then i set Kr = z and

zo = sqrt(2mVo) a / h

and then i graph it with cot(z) and sqrt[(zo/z)^2-1] and look for the intersections.

But I don't know why I do this, or what the graph means.

It tells me something about the bound states but I don't know what or why.

My teacher said it is used to solve the problem numerically, but why does it need to be solved numerically?

So any information on what this step in the solving process is for would be great

 

[vela]

Homework Statement

Introduction to Quantum Mech Griffiths 4.9


for the problem of a finite square well

v(r) =

{-Vo r<= a (i will call section I)

​

0 r>a (section II)

Homework Equations

After I find uI and uII

uI = A sin(K r)
uII = De^-kr

and then set the boundary condition

uI = uII at a equation 1
uI' = uII' at a equation 2

next i know I divide equation 2 by equation 1

What's the equation you finally end up with? It would help us to see that and also any equations relating k and K to the energy of the state and the depth of the well.

The Attempt at a Solution

then i set Kr = z and

zo = sqrt(2mVo) a / h

and then i graph it with cot(z) and sqrt[(zo/z)^2-1] and look for the intersections.

But I don't know why I do this, or what the graph means.

It tells me something about the bound states but I don't know what or why.

My teacher said it is used to solve the problem numerically, but why does it need to be solved numerically?

So any information on what this step in the solving process is for would be great

When you match the solutions at the boundary, you should get an equation you can't solve analytically, so you have to do it numerically.