200
7 Bounded Quantum Systems
7.3 Symmetries and Degeneracy
The spectrum of a Hamiltonian matrix can have degeneracies. However, it is far
more likely to have them if one or more symmetries exist than if no symmetries
exist (von Neumann and Wigner 1929).
Let us consider the case of a real symmetric Hamiltonian matrix for which
one or more symmetries exist. These symmetries, in effect, block-diagonalize the
Hamiltonian. The spectrum from a given block of the Hamiltonian forms a pure
sequence. The spectrum from the Hamiltonian as a whole is a superposition of pure
sequences and is said to form a mixed sequence. It is only necessary to vary one
parameter to obtain a degeneracy between eigenvalues from different blocks of the
Hamiltonian. This is easy to see for the special case of a 4 × 4 Hamiltonian of the
form
¯
H =
⎛
⎜
⎜
⎝
e 1 − a b
0
0
b e 1 + a 0
0
0
0 e 2 − c d
0
0
d e 2 + c
⎞
⎟
⎟
⎠ .
(7.8)
The spectrum is given by the four eigenvalues E 1± = e 1 ±
√
a 2 + b 2 and E 2± =
e 2 ±
√
c 2 + d 2 . Degeneracies between these eigenvalues can be obtained by varying
only one parameter. Thus, degeneracies are far more likely to occur for Hamiltonian
systems with symmetries than those without symmetries.
Accidental degeneracies can exist in the absence of symmetries and can be
identified by the behavior of their wave functions in parameter space in the
neighborhood of the degeneracies (Teller 1937; Berry 1981). For simplicity, let us
consider a 2 × 2 Hamiltonian matrix
¯
H =
e 0 − a b
b e 0 + a
.
(7.9)
The eigenvalues are E ± = e 0 ±
√
a 2 + b 2 . Each eigenvalue forms a sheet
parametrized by a and b, as shown in Fig. 7.1. The sheets touch at only one
point (in this case at a = 0, b = 0, and E ± = e 0 ). It is useful to look at the
eigenvectors. Without loss of generality, we can set e 0 = 0. Then, the eigenvectors
are defined by the equation ¯
H |ψ ± = E ± |ψ ± , where E ± = ±
√
a 2 + b 2 and
|ψ ± =
cos(χ ± )
sin(χ ± )
with
cos(χ ± ) =
(E ± − a)
[(E ± − a) 2 + b 2 ]
1
2
and sin(χ ± ) =
−b
[(E ± − a) 2 + b 2 ]
1
2
.
7 Bounded Quantum Systems
7.3 Symmetries and Degeneracy
The spectrum of a Hamiltonian matrix can have degeneracies. However, it is far
more likely to have them if one or more symmetries exist than if no symmetries
exist (von Neumann and Wigner 1929).
Let us consider the case of a real symmetric Hamiltonian matrix for which
one or more symmetries exist. These symmetries, in effect, block-diagonalize the
Hamiltonian. The spectrum from a given block of the Hamiltonian forms a pure
sequence. The spectrum from the Hamiltonian as a whole is a superposition of pure
sequences and is said to form a mixed sequence. It is only necessary to vary one
parameter to obtain a degeneracy between eigenvalues from different blocks of the
Hamiltonian. This is easy to see for the special case of a 4 × 4 Hamiltonian of the
form
¯
H =
⎛
⎜
⎜
⎝
e 1 − a b
0
0
b e 1 + a 0
0
0
0 e 2 − c d
0
0
d e 2 + c
⎞
⎟
⎟
⎠ .
(7.8)
The spectrum is given by the four eigenvalues E 1± = e 1 ±
√
a 2 + b 2 and E 2± =
e 2 ±
√
c 2 + d 2 . Degeneracies between these eigenvalues can be obtained by varying
only one parameter. Thus, degeneracies are far more likely to occur for Hamiltonian
systems with symmetries than those without symmetries.
Accidental degeneracies can exist in the absence of symmetries and can be
identified by the behavior of their wave functions in parameter space in the
neighborhood of the degeneracies (Teller 1937; Berry 1981). For simplicity, let us
consider a 2 × 2 Hamiltonian matrix
¯
H =
e 0 − a b
b e 0 + a
.
(7.9)
The eigenvalues are E ± = e 0 ±
√
a 2 + b 2 . Each eigenvalue forms a sheet
parametrized by a and b, as shown in Fig. 7.1. The sheets touch at only one
point (in this case at a = 0, b = 0, and E ± = e 0 ). It is useful to look at the
eigenvectors. Without loss of generality, we can set e 0 = 0. Then, the eigenvectors
are defined by the equation ¯
H |ψ ± = E ± |ψ ± , where E ± = ±
√
a 2 + b 2 and
|ψ ± =
cos(χ ± )
sin(χ ± )
with
cos(χ ± ) =
(E ± − a)
[(E ± − a) 2 + b 2 ]
1
2
and sin(χ ± ) =
−b
[(E ± − a) 2 + b 2 ]
1
2
.