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## Homework Statement

The wave function for a particle of mass m moving in the potential

V =

{ ∞ for x=0

{ 0 for 0 < x ≤ a

{ V

_{0}for x ≥ a

is

ψ(x)

{ Asin(kx) for 0 < x ≤ a

{ Ce

^{-Kx}for x ≥ a

with

k = √[(2mE/h

^{2})]

K =√[((2m(V

_{0}-E)/h

^{2})]

where h is h-bar in both equations, and remains so throughout this thread.

(a) Apply the boundary conditions at x = a and obtain the transcendental equation which determines the bound state energies E.

(b) If

√[(2mV

_{0}a

^{2})/h

^{2}] = 3pi

determine the allowed bound state energies. Express your answers in the form of a numerical factor multiplying the dimensional factor (h

^{2}/(2ma

^{2})).

(c) Given that the normalization factor for the wave function corresponding to the n

^{th}energy E

_{n}is

A

_{n}= √[2/a]*√[(K

_{n}a)/(1+(K

_{n}*a))]

normalize the wave function for the lowest energy state.

(d) What is the probability that a measurement of the position of a particle in the ground state will give a result ≥ a?

## Homework Equations

## The Attempt at a Solution

I have completed parts (a) and (b) and I believe they are correct, here is a brief outline of what I did.

At x = a: ψ(a)

Asin(ka) = Ce

^{-Ka}(1)

dψ/dx

kAcos(kx) = -KCe

^{-Ka}(2)

Dividing (2) by (1):

kcot(ka) = -K

For the transcendental: z = ka, z

_{0}= a√[(2mV

_{0})/h

^{2}]

∴ -cot(z) = √[(z

_{0}/z)

^{2}-1]

Now setting z

_{0}= 3pi (part b) and graphing -cot(z) and √[(z

_{0}/z)

^{2}-1] on the same plot, I find that intersections occur just below z

_{n}= npi

z = ka = a√[(2mE/h

^{2})] ≈ npi

∴ Solving for E

_{n}= (n

^{2}pi

^{2}h

^{2})/(2ma

^{2})

This is where I hit a road block. I believe the lowest energy state is E

_{1}because it is just below pi (or 1pi), but I'm not sure what I'm supposed to do with this. I want to plug this into K

_{n}, but that creates quite the mess. As for the second portion of ψ(x), I can just normalize that and solve for C using K

_{n}, and I think I have a handle on part (d) as well. Any hints on part (c) or just checking my work so far would be much appreciated.