Chapter 7. The Eigenvalue Problem

7. The Eigenvalue Problem, December 17, 2009 14
Section 7.2. Degenerate eigenvalues
§ 10 Introduction. The eigenvalues of a matrix A are the roots of the
characteristic polynomial det[A − λI]. There is no law that prevents some
of these roots from being equal. When two or more eigenvalues are equal,
we say that they are degenerate and the corresponding eigenvectors are also
called degenerate. This name suggests a certain moral depravity for these
eigenvalues and I am sure that they find it offensive. I could not find who
introduced this terminology. Mathematiciansaremorepoliteanduseinstead
the terms ‘multiple’ and ‘single’ eigenvalues, which is simpler and friendlier.
One would think that finding operators that have identical eigenvalues
should be as rare as finding by accident polynomials with multiple roots.
But this is not so. Degenerate eigenvalues are fairly frequent and here I use
two examples to suggest why.
§ 11 Particle in a cubic box with infinite potential-energy walls. One of
the simplest models in quantum mechanics is a particle in a rectangular
parallelepiped whose walls are infinitely repulsive. The Hamiltonian is
Ĥ = ˆK x + ˆK y + ˆK z (70)
where
ˆK x = − ¯h2 ∂ 2
(71)
2m ∂x 2
is the operator representing the kinetic energy of the motion in direction x;
ˆK y and ˆK z are defined similarly. In classical mechanics, the total kinetic
energy is
p 2
2m = p2 x
2m + p2 y
2m + p2 z
(72)
2m
where p is the momentum vector and p x is the momentum in the direction
x (and analogously for p y , p z ). The operator ˆK x is the quantum analog of
p 2 x/2m.
The total energy of the particle is discrete and is given by (see H. Metiu,
Physical Chemistry, Quantum Mechanics, page 104)
E n,j,k = ¯h2 π 2
2m
n 2 ¯h 2 π 2 2
j
+ + ¯h2 π 2 2
k
(73)
L x 2m L y 2m L z