- #1

Xyius

- 508

- 4

## Homework Statement

*In a spherical well in which..*

[tex]V=

\begin{cases}

0,\text{for }0 \le r < R \\

∞, \text{for } r > R

\end{cases}

[/tex]

the s-wave eigenstates are

[tex]\phi_n(r)=\frac{A}{r}\sin\left( \frac{n\pi r}{R} \right)[/tex]

where A is a normalization constant. If a particle is in the ground state and R suddenly

increases to ##R'=R/\epsilon##, where ##0 \le \epsilon \le 1##

(i) Compute and plot the probability of ﬁnding the particle in the new ground state as a

function of ##\epsilon##.

(ii) Compute and plot the probability of ﬁnding the particle in a new state with quantum

number n as a function of ##\epsilon## and n (this will be a 3D plot). What are the most important

features you see in your plot?

(iii) Sum all probabilities in (ii) for = 1=2 to show that they add to 1.

Note: The relations

[tex]\sum_{n=0}^{∞}\frac{1}{(2n+1)^2}=\frac{\pi^2}{8}[/tex]

[tex]\sum_{n=0}^{∞}\frac{1}{(2n+3)^2(2n-1)^2}=\frac{\pi^2}{64}[/tex]

may be useful here.

[tex]V=

\begin{cases}

0,\text{for }0 \le r < R \\

∞, \text{for } r > R

\end{cases}

[/tex]

the s-wave eigenstates are

[tex]\phi_n(r)=\frac{A}{r}\sin\left( \frac{n\pi r}{R} \right)[/tex]

where A is a normalization constant. If a particle is in the ground state and R suddenly

increases to ##R'=R/\epsilon##, where ##0 \le \epsilon \le 1##

(i) Compute and plot the probability of ﬁnding the particle in the new ground state as a

function of ##\epsilon##.

(ii) Compute and plot the probability of ﬁnding the particle in a new state with quantum

number n as a function of ##\epsilon## and n (this will be a 3D plot). What are the most important

features you see in your plot?

(iii) Sum all probabilities in (ii) for = 1=2 to show that they add to 1.

Note: The relations

[tex]\sum_{n=0}^{∞}\frac{1}{(2n+1)^2}=\frac{\pi^2}{8}[/tex]

[tex]\sum_{n=0}^{∞}\frac{1}{(2n+3)^2(2n-1)^2}=\frac{\pi^2}{64}[/tex]

may be useful here.

## Homework Equations

Obviously the ones given in the problem.

**Also, this problem deals with the sudden approximation. Which says that when the system is changed very rapidly, the state remains unchanged initially and then is allowed to evolve.**

## The Attempt at a Solution

My solution is in the attached images. I know typing it out would have been ideal but it would have been too much and I could give more of my work this way. I scanned them to make sure they were high quality images.

I don't know where my thought process went wrong, but what I am getting for part (iii) is obviously wrong since it diverges at n=2.