Page 224 - Principles and Applications of NanoMEMS Physics

214 Appendix A

eigenvalues, both discrete and/or continuous, each one being accompanied
by a respective eigenfunction. Thus, the set of wavefunctions associated with
an operator are said to span a space, called state space. When a particle is not
in a stationary state, it is in a dynamic state. A particle is in a dynamic state
when it is between stationary states, and the dynamic state is embodied by a
superposition of stationary states. During these circumstances, the particle’s
state is found as a solution to the time-dependent Schrödinger’s equation,
∂ψ
Hψ = i= , (A.2)
ˆ
t ∂
where = is Planck’s constant and t is time. If there are n stationary states,
then the solution to (A.2) is expressed as,
ψ = c () t ψ + c () t ψ + ... + c () t ψ , (A.3)
1 1 2 2 n n

where the wavefunctions ψ correspond to respective stationary states with
i
2
energies E , and for a normalized state c represents the probability that,
i i
upon measuring the state of the particle, it will be found in state i. A state is
normalized when its inner product ( ,ψψ ) 1= . Thus, for a normalized state
¦ c i 2 = 1. But this is the norm of ψ , therefore, the norm of the state
i
vector remains constant, i.e., does not depend on time. Two proportional
′
state vectors, say, ψ ′ and ψ , where ψ = ce θ i ψ , represent the same
physical state, but in general, the superposition of states possessing
expansion coefficients with relative phases, such as
′ ′
ψ = c e 1 θ i ψ + c e 2 θ i ψ does not.
1 1 2 2
The state of a particle deprived of interaction with its environment, will
evolve according to the solution to (A.2), which, expressed in Dirac’s ket
notation, is given by,
ψ
ψ () t = U ( ,tt ) ( ) , (A.4)
t
0 0
ˆ
where, when H is time-independent, U is the evolution operator
t
i
− = ³ Hd t′
−
t
/
U ( ,tt 0 ) e= 0 t = e − iH ( t 0 ) = . (A.5)
A system whose Hamiltonian is time-independent exhibits energy
conservation over time, i.e., the total energy is a constant of the motion.
Clearly, U † U = UU † = 1. This means that U conserves the norm of the