# WKB Approximation with V-Shaped well

## Homework Statement

Good day all!
I'm studying for finals and i'd like to know how to do this problem (its not homework):

"Using the WKB method, find the bound state energies $E_n$ of a particle of mass m in a V-shaped potential well:

$V(x)= \begin{Bmatrix} -V_0 (1- \begin{vmatrix} \frac{x}{a} \end{vmatrix})\: \: -a<x<a\\ 0 \: \: \: if \: \: \: \: \: \: x\notin (-a,a) \end{Bmatrix}$ where $V_0$ and $a$ are positive constants.

What is the energy of the highest bound state?

## Homework Equations

WKB approximation for "potential with no vertical walls":

$\int_{x_1}^{x_2} p(x)dx=(n-\frac{1}{2})\pi \hbar$

Here $x_1,x_2$ are the classical "turning points" i.e to find them I set $p(x)=0$ and solve for x

## The Attempt at a Solution

Well, if its a bound state, then $E < V$, so the wave function is "in" the well.
$$p(x)=\frac{\sqrt{2m(V-E)}}{\hbar}=\frac{\sqrt{2m(-V_0(1-|\frac{x}{a}|)-E))}}{\hbar}$$
Thus the integral is:

$\int_{x_1}^{x_2} \frac{\sqrt{2m(V-E)}}{\hbar}=\int_{x_1}^{x_2} \frac{\sqrt{2m(-V_0(1-|\frac{x}{a}|)-E))}}{\hbar} dx=(n-\frac{1}{2})\pi \hbar$

So a few question arise:

Is my p(x) term correct? In this case is it: $$p(x)=\frac{\sqrt{2m(V-E)}}{\hbar}$$ OR $$p(x)=\frac{\sqrt{2m(E-V)}}{\hbar}$$?

Also, the well is symmetrical, so I can just to the integral:

$$2 \int_{0}^{x_2} \frac{\sqrt{2m(-V_0(1-|\frac{x}{a}|)-E))}}{\hbar} dx$$ So my question is: what is my turning point?
If I look just at the positive x values, I can drop the absolute value sign in the potential and I get:

$V(x)= \frac {V_0}{a}x-V_0$ for $0<x<a$. Thus my momentum becomes:

$\frac{\sqrt{2m(\frac {V_0}{a}x-V_0-E))}}{\hbar}$. Setting this equal to zero to find the left most turning point gives:

$\frac{\sqrt{2m(\frac {V_0}{a}x-V_0-E))}}{\hbar} =0 \rightarrow x=a$

Is this turning point correct? Am I on the right track? Thanks!

TSny
Homework Helper
Gold Member
the wave function is "in" the well.
$$p(x)=\frac{\sqrt{2m(V-E)}}{\hbar}=\frac{\sqrt{2m(-V_0(1-|\frac{x}{a}|)-E))}}{\hbar}$$
If ##p(x)## represents the momentum, should you be dividing by ##\hbar##?
Is my p(x) term correct? In this case is it: $$p(x)=\frac{\sqrt{2m(V-E)}}{\hbar}$$ OR $$p(x)=\frac{\sqrt{2m(E-V)}}{\hbar}$$?##
It should be the expression that follows from solving the energy equation ##\frac{p^2}{2m} + V = E## for ##p##.

So my question is: what is my turning point?
$\frac{\sqrt{2m(\frac {V_0}{a}x-V_0-E))}}{\hbar} =0 \rightarrow x=a$
Is this turning point correct?

I don't think you solved for ##x## correctly. But, first, you want to make sure the left side is the correct expression for ##p##. A sketch of V(x) with a line drawn for E can help you see that the turning point cannot be ##x = a## in general.

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