Hydrogenic systems

Problem:

The ground state wavefunction for a hydrogen like atom is  ,

where   is the reduced mass m»me = mass of the electron.

(a)  What is the ground state wavefunction of tritium?
(b)  What is the ground state wavefunction of 3He+ ?
(c)  An electron is in the ground state of tritium.  A nuclear reaction instantaneously changes the nucleus to 3He+.  Assume the beta particle and the neutrino are immediately removed from the system.  Calculate the probability that the electron remains in the ground state of 3He+ .

Solution:

For a hydrogen atom we have

.

Defining

,

we write

.

For a hydrogenic atom we have

.

.

Defining

,

we write

.

The same equations have the same solutions.  To find the eigenfunctions and eigenvalues of the Hamiltonian of a hydrogenic atom we therefore replace in the eigenfunctions of the Hamiltonian of the hydrogen atom

,

and in the eigenvalues of the Hamiltonian of the hydrogen atom we replace

.

(a)  tritium: The reduced mass is

,

assuming the neutron mass equals the proton mass.  The nuclear charge is Z=1.  Since a0'»a0 the ground state wavefunction of the tritium is the ground state wavefunction of the hydrogen atom.

.

(b) 3He+: The reduced mass is

,

assuming the neutron mass equals the proton mass.  The nuclear charge is Z=2.  Since a0'»a0/2 the ground state wavefunction of the 3He+ is

.

(c) 

Problem:

A negative K meson with mass m=1000 electron masses is captured into a circular Bohr orbit around a lead nucleus (Z=82).  Assume it starts with principal quantum number n=10 and then cascades down through n=9,8,7,... etc.

(a)  What is the energy of the photon emitted in the n=10 to n=9 transition?
(b)  What is the approximate radius of the lead nucleus if no further quanta are observed after the n=4 to n=3 transition (because of nuclear absorption of the K meson)?

Solution:

We have a hydrogen like system with Z=82:

(a)  .

.

.

(b)  The radius of Bohr orbits is .

.

This is the approximate radius of the lead nucleus.

,

for Z=1.

Problem:

An electron in the n=2, l=1 m=0 state of a hydrogen atom has a wavefunction y(r,q,f) proportional to , where a0 is the Bohr radius.

(a)  Find the normalized wavefunction y(r,q,f).
(b)  Find the probability per unit length of finding the electron a distance R from the nucleus.
(c)  Find the most probable distance RMP of the electron from the nucleus.
(d)  Find the average distance <R> from the nucleus.

Hint: .

Solution:

(a)  .

.

. .

.

(b)  .

.

(c)  

(d)   0.

Find the momentum space wavefunction for the electron in the 1s state of the hydrogen atom.
Solution:

The 1s wavefunction is .

Therefore .

Symmetry demands that .

Let us choose a convenient direction for p. Let .  Then

                                      

.

.