 #1
Mr_Allod
 42
 16
 Homework Statement:
 Imagine the electron of a hydrogen atom has been replaced by a negative pion. Calculate the probability of finding the pion in a small volume near the nucleus of pionic hydrogen and compare it to the probability of finding an electron in the same volume for electronic hydrogen. Assume ##r << a_\pi## where ##a_\pi## is Bohr radius for pionic hydrogen.
 Relevant Equations:

Ground State Wavefunction: ##\Psi = \frac {1}{\sqrt {\pi}} \left (\frac {1}{a_0} \right )^{\frac {3}{2}} \exp \left [ \frac {r}{a_0}\right ]##
Probability: ##P = \int_0^{\infty}\int_0^{\pi}\int_0^{2\pi} \Psi.\Psi^* r^2 \sin(\theta)dr d\theta d\phi##
Hello, I am trying to figure out the right way to approach this. First of all, other than the different Bohr radius value, does the change to a negative pion make any other difference to calculating the probability?
Also what would be the correct way to apply the "small volume"? What I'm thinking is I find the probability between ##r = 0## and ##r = R## where R is some radius much smaller than ##a_\pi## like so:
$$P = \int_0^{R}\int_0^{\pi}\int_0^{2\pi} \Psi.\Psi^* r^2 \sin(\theta)dr d\theta d\phi$$
With ##\Psi.\Psi^* = \frac {1}{\pi a_0^3}\exp \left [ \frac {2r}{a_0}\right ]##
I would then follow a similar approach to finding the probability for the electron. Does this sound reasonable? Or should I be looking at some range of radii ##R_1 < r < R_2## where ##R_2 << a_\pi##?
Also what would be the correct way to apply the "small volume"? What I'm thinking is I find the probability between ##r = 0## and ##r = R## where R is some radius much smaller than ##a_\pi## like so:
$$P = \int_0^{R}\int_0^{\pi}\int_0^{2\pi} \Psi.\Psi^* r^2 \sin(\theta)dr d\theta d\phi$$
With ##\Psi.\Psi^* = \frac {1}{\pi a_0^3}\exp \left [ \frac {2r}{a_0}\right ]##
I would then follow a similar approach to finding the probability for the electron. Does this sound reasonable? Or should I be looking at some range of radii ##R_1 < r < R_2## where ##R_2 << a_\pi##?