Quantum Mechanics - Prof. Eric R. Bittner - University of Houston

V(hartree)
-20
-40
-60
-80
-100
-120
r(Bohr)
0.02 0.04 0.06 0.08 0.1
1. Calculate this effect on the ground state energy of the H atom using first order perturbation
theory with
2. Explain this choice for H ′ .
H ′ =
� e 2
e3 − for r ≤ R
r R
0 otherwise
3. Expand your results in powers of R/ao ≪ 1. (Be careful!)
4. Evaluate numerically your result for R = 1 fm and R = 100 fm.
5. Give the fractional shift of the energy of the ground state.
(7.243)
6. A more rigorous approach is to take into account the fact that the nucleus has a homogeneous
charge distribution. In this case, the potential energy experienced by the electron goes
as
V (r) = − Ze2
r
when r > R and
V (r) = − Ze2
r
� �� �
1 r 2
+ 2
2R R
R
� �
− 3 − 1
r
for r ≤ R. What is the perturbation in this case? Calculate the energy shift for the H (1s)
220