| course |
course_year |
question_number |
tags |
title |
year |
Quantum Mechanics |
IB |
71 |
IB |
2010 |
Quantum Mechanics |
|
Paper 3, Section II, D |
2010 |
A $\pi^{-}$(a particle of the same charge as the electron but 270 times more massive) is bound in the Coulomb potential of a proton. Assuming that the wave function has the form $c e^{-r / a}$, where $c$ and $a$ are constants, determine the normalized wave function of the lowest energy state of the $\pi^{-}$, assuming it to be an $S$-wave (i.e. the state with $l=0$ ). (You should treat the proton as fixed in space.)
Calculate the probability of finding the $\pi^{-}$inside a sphere of radius $R$ in terms of the ratio $\mu=R / a$, and show that this probability is given by $4 \mu^{3} / 3+O\left(\mu^{4}\right)$ if $\mu$ is very small. Would the result be larger or smaller if the $\pi^{-}$were in a $P$-wave $(l=1)$ state? Justify your answer very briefly.
[Hint: in spherical polar coordinates,
$$\left.\nabla^{2} \psi(\mathbf{r})=\frac{1}{r} \frac{\partial^{2}}{\partial r^{2}}(r \psi)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial \psi}{\partial \theta}\right)+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial^{2} \psi}{\partial \phi^{2}}\right]$$