- #1

doggydan42

- 170

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## Homework Statement

Consider the standard square well potential

$$V(x) =

\begin{cases}

-V_0 & |x| \leq a \\

0 & |x| > a

\end{cases}

$$

With ##V_0 > 0##, and the wavefunctions for an even state

$$\psi(x) =

\begin{cases}

\frac{1}{\sqrt{a}}cos(kx) & |x| \leq a \\

\frac{A}{\sqrt{a}}e^{-\kappa |x|} & |x| > a

\end{cases}

$$

where we have included the ##\frac{1}{\sqrt{a}}## prefactor to have consistent units for ##\psi##, and A is a constant required by continuity at x = a.

For the finite square well, recall that

$$\eta = ka, \xi = \kappa a, k^2 = \frac{2m(V_0-|E|)}{\hbar^2}, \kappa^2 = \frac{2m|E|}{\hbar^2}$$

and

$$\xi^2 + \eta^2 = z_0^2, \xi = \eta tan(\eta)$$

We want to have a better understanding of the limit as ##V_0 \rightarrow \infty##, and understand why the discontinuity in ##\psi'## in the infinite well does not give trouble. Keeping m and a constant as we let ##V_0## grow large is the same thing as letting ##z_0## grow large, where

$$z_0^2 = \frac{2ma^2V_0}{\hbar^2}$$

Part A:

Consider the ground state of the potential. In the limit of large ##z_0##, compute the approximate values of ##\eta## and ##\xi## to leading order in ##z_0##. Your answers should have no trigonometric functions in them. (Hint: You will need to approximate the relations between ##\xi## and ##\eta##. Think about the range you expect ##\eta## to lie in.)

Write your answer in terms of ##z_0##.

##\eta = ##

##\xi = ##

##A = ##

## Homework Equations

These are given in the problem statement.

## The Attempt at a Solution

My main issue was with the approximations.

I was thinking of how to approximate the tangent.

I was also thinking of possibly using the average of the range ##\eta## should lie in, which I believe is 0 to ##z_0##, but that does not seem to be right.

What would be the best way to go about approximating ##\eta## and ##\xi##?

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