Appendix C
Symmetries and the Hamiltonian Matrix
The Hamiltonian operator, ˆ
H , is Hermitian ( ˆ
H = ˆ
H † , where † denotes complex
conjugate transpose), and its matrix representations are Hermitian. This ensures
that its eigenvalues, the allowed energies of the system, are real. In this appendix,
we consider N-particle systems described by Hamiltonians of the form ˆ
H (t) =
ˆ
H ({ ˆ
p α }, { ˆ
q α }, {ˆ s α }, t), where ˆ
p α and ˆ
q α are the momentum and position operators
for the αth particle, ˆ
s α is the spin of the αth particle, and { ˆ
p α }, { ˆ
q α }, and {ˆ s α }
denote the sets of momenta, positions, and spins, respectively, of the N particles.
The Hamiltonian matrix is formed from the numbers obtained by evaluating the
Hamiltonian operator with respect to some chosen complete orthonormal set of basis
states. If the complete set of basis states is denoted as {|a i }, then the (i, j )th element
of the Hamiltonian matrix is H i,j (t) = =a i | ˆ
H (t)|a j .
If symmetries exist, they can simplify the structure of the Hamiltonian matrix. A
symmetry causes the dynamics of the system, and therefore the Hamiltonian, to be
invariant under the corresponding symmetry transformation. In Sect. C.1, we discuss
the types of transformations associated with the various space-time symmetries. In
Sect. C.2, we then show explicitly how the various space-time symmetries affect the
Hamiltonian matrix.
C.1 Space-Time Symmetries
The space-time symmetries give rise to infinitesimal or discrete symmetry transformations. Symmetry transformations are “generated" by linear and antilinear
operators. All of the operators that we deal with in this book are linear except for
the time reversal operator, which is antilinear.
Linear Operators
A linear operator, ˆ
O, when acting on a state | = c 1 |ψ 1 + c 2 |ψ 2 , where c 1 and
c 2 are complex constants, gives
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https://doi.org/10.1007/978-3-030-63534-3
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