Download PHY4605–Introduction to Quantum Mechanics II Spring 1997 Problem Set 4 Jan. 31, 2005

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Transcript
PHY4605–Introduction to Quantum Mechanics II
Spring 1997
Problem Set 4
Jan. 31, 2005
Due: Feb. 7, 2005
Reading: Griffiths, Ch. 6
1. Finite extent of proton.
The binding energy and ground state of an electron in an H-atom are normally
obtained under the assumption that the proton is a fixed point charge, such
that the potential is V0 (r) = −e2 /(4π²0 r) all the way to r = 0. In this approximation, the wave function is ψ(r) ∝ exp(−r/a0 ), where a0 is the Bohr radius.
In a slightly better model, the charge of the proton is distributed uniformly
◦
throughout a sphere of radius r0 , of order 1 Fermi ≡ 10−5 A . The charge
density is explicitly
( 3e
r < r0
3
(1)
ρ(r) = 4πr0
= 0 r > r0
(a) To see how this distributed charge affects the ground state energy, use
classical physics (Gauss’s law) to find the classical potential V1 (r) associated with the charge density above, and define the perturbation to the
point-proton model to be δV (r) = V1 (r) − V0 (r). Use 1st-order perturbation theory to find the correction δE0 to the ground state energy due to
the finite size of the proton. Obtain a numerical value for the fractional
correction δE0 /E0 . (Hint: you can simplify the calculation by noting that
r0 ¿ a0 !)
(b) Explain qualitatively why corrections to the energy levels in hydrogen due
to the finite width of the proton charge distribution are larger in the s-wave
states (orbital angular momentum ` = 0) than for states with ` > 0.
2. Bead on wire loop with a “dimple”. Griffiths Problem 6-7
1