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  1
Assume for simplicity the initial state ψ(t = 0 ) =   , and denote the state
  0
 at()
of the system at time t by ψ() =   :
t
 bt()
a. Find numerically at which frequency ω the magnitude of b(t) is
maximum.
b. Once you have determined the optimal ω, go back and examine
what strategy you should adopt in the choice of Ω to ensure
⊥
maximum resolution.
c. Verify your numerical answers with the analytical solution of this
problem, which is given by:

Ω 2
2
bt() = ⊥ sin (ω t ˜ )
2
˜ ω 2
where ˜ ω = ( −Ω ω / )2 2 + Ω 2 ⊥ .
2

8.10 Special Classes of Matrices*

8.10.1 Hermitian Matrices
Hermitian matrices of finite or infinite dimensions (operators) play a key role
in quantum mechanics, the primary tool for understanding and solving phys-
ical problems at the atomic and subatomic scales. In this section, we define
these matrices and find key properties of their eigenvalues and eigenvectors.

DEFINITION The Hermitian adjoint of a matrix M, denoted by M is equal to
†
the complex conjugate of its transpose:

M = M T (8.71)
†

For example, in complex vector spaces, the bra-vector will be the Hermitian
adjoint of the corresponding ket-vector:

v = ( v ) † (8.72)

© 2001 by CRC Press LLC