Exercise 6

Quantenmechanik 2 - SS 2013
Organisation Übungen:
Prof. Dr. O. Philipsen
Dr. Wolfgang Unger
Goethe-Universität Frankfurt
Raum 2.105
[email protected]
Exercise 6
23.05.13, due on 30.05.13 in the lecture
14) Exchange Interaction(3+3+3+1 points)
For the Helium atom we found the energy
(0)
E1n = E1n + Knl ± Anl
with the signs
+ corresponding to parahelium (symmetric spatial wave function) and
− corresponding to orthohelium (anti-symmetric spatial wave function).
The exchange energy Anl , although of purely quantum mechanical nature, can be better understood
by evaluating the mean distance square.
Assume two particles in one spatial dimension. The particles are in the states a and b, described by
the wave functions ψa (x) and ψb (x) which are orthogonal to each other. Compute the mean distance
square
d2 = h(x1 − x2 )2 i
for (a) distinguishable particles, (b) Bosons and (c) Fermions
and (d) compare, for which cases the particles are closer together or further apart.
15) Harmonic Forces between two Particles (6+4 points)
Consider two spin 1/2 particles in 3 dimensions, interacting by harmonic forces, governed by the
Hamiltonian
1 p21
1 p22
1
2
2
H=
+ αr1 +
+ αr2 + κ(~r1 − ~r2 )2
2 m
2 m
2
a) Solve for the ground state energy E and the wave function ψ(r1 , r2 ) exactly, by rewriting the
Hamiltonian in terms of the variables
~ = √1 (~r1 + ~r2 ) and ~r = √1 (~r1 − ~r2 ).
R
2
2
Recall from QM 1 the solution to the harmonic oscillator in three dimensions.
b) Derive the Hartree-Fock equation for the original problem (in variables ~r1 , ~r2 ).
Vladimir Aleksandrovich Fock
(22 December 1898 - 27 December 1974)
Russian mathematical physicist who made seminal contributions to quantum mechanics and the general theory of relativity. Fock became progressively deaf at a young age because of injuries sustained during military
service in World War I. In 1922 he graduated from Petrograd University
(Saint Petersburg State University), and he taught there from 1924, becoming a professor in 1932. The Hartree-Fock equation, improved by him
in 1930, became a basic approximation method for calculations involving
multielectron atoms in quantum chemistry. He also introduced the Fock
representation (1928) for a quantum oscillator, particularly important in
quantum field theory; the Fock space with varying dimensions (1932) to legitimize the second quantization (many-body formalism); and the method of
Fock functionals (1934) for treating systems with an indeterminate number
of particles in quantum electrodynamics.
In 1926 Fock and several other physicists independently proposed a relativistic generalization of the
Austrian physicist Erwin Schrödinger’s wave equation. Fock was arrested on phony political charges in
1937, during the Great Purge, but he was quickly released thanks to the Russian physicist Pyotr Kapitsa,
who appealed directly to Joseph Stalin. In 1939 Fock solved the problem of the motion of ponderable bodies
(objects with appreciable mass) in Albert Einstein’s general relativity by using harmonic coordinates. In
1939 Fock was elected a full member of the U.S.S.R. Academy of Sciences (he had been a corresponding
member since 1932). Fock also developed philosophical interpretations of relativity and quantum mechanics
that he considered consistent with Marxism.
[From Encyclopaedia Britannica Online Academic Edition.]