Download Problem Set 1

Atomic and Molecular Physics January -May 2010
Problem Sheet I
5th January 2010
due 15th January 2010
1. According the Bohr atom model what is the speed of an electron for the ground
state of Hydrogen atom?
2.Consider the absorption or emission of photon of energy hν by an atom initially at
rest.After the transstion the atom has momentum P . If M is the mass of the atom find
the frequency of the photon absorbed or emitted in terms of ν0 where hν0 is the energy
difference between the levels of transition and M including the recoil effect.
3.Find the average value of r and r2 for the ground state of the hydrogen atom.
4. An electron is in the n = 2, l = 1 state of the hydrogen atom. Write down the
passible wave functions of the electron including its intrinsic spin. The wave function
~ J~ and Jz where J~ is the total angular momentum operator
must be eigenfunctions of J.
and Jz is its z-component. ( You have a total of six wave functions)
5. Assuming the spin-orbit interaction hamiltonian is given by
Hsl =
~ S
~
Ze2 L.
8π0 m2 c2 r3
. Find the energy splitting between j = l + 1/2 and j = l − 1/2 levels of the hydrogen
atom to the first order in Hsl .
( Please learn how to obtain the above hamiltonian- see for example Modern Quantum Mechanics Sakurai)
(For Hydrogen atom there are two more effects of the same order which have to be
included )
6. Consider a two electron system.Let
!
α=
1
0
!
β=
0
1
be the spin up and down wave function for a single electron .(SZ is diagonal) Write
down all the possible spin wave functions of the system in terms of the single particle
wave function such that the wave funstions are eigenstates of the total spin and its
z-component in terms of α and β.
7. The relativistic formula for the kinetic energy is
(m2 c4 + c2 p2 )1/2 − mc2 =
p4
p2
−
+ ...
2m 8m3 c2
Find the shift in energy due to the p4 term for the hydrogen atom levels using first order
perturbation.
8. Convert electron volt/(molucule or atom) to Kilo Joules per gram mole, a unit
used by chemists.
9. Consider black body radiation at 10000 K. (a) find the energy density. (b) the
fraction of the energy in the range 5000 K − 15000 K.
10. Calculate the energy difference of levels in electron volts for deuterium atom and
compare it with hydrogen atom. In particular calculate the difference in wave numbers
the transition for the Balmer series.