Page 225 - Principles and Applications of NanoMEMS Physics

A. QUANTUM MECHANICS PRIMER 215

states it acts upon, i.e., it is a unitary operator. Also, since
U ( tt, ) U= − 1 (t , , this means the system is reversible. When the system
) t
0 0
is disturbed by (or coupled to) the environment, as a result of which its
energy is modified, then its evolution is modified, the norm is no longer
conserved, the system becomes irreversible, and the state is said to
decohere.

A.3 Harmonic Oscillator and Quantization

In the simplest case of a particle of mass m and constant total energy
(Hamiltonian), H, performing an oscillatory motion in a potential
2
dq ·
1 §
V (q ) = kq 2 2 , with kinetic energy T = m¨ ¸ = p 2 2 m ,
2 © dt ¹
Schrödinger’s equation is given by,
1
−
ˆ
2
H ψ [ = (qT ˆ ) + (qV ˆ ) ] ψ = ª ˆ p 2 + (qV ˆ ) º » ψ = ª = 2 d 2 2 + m ω ˆ q 2 º = εψ, (A.6)
»
«
«
¬ 2m ¼ ¬ 2m ˆ q d 2 ¼
where the first and second terms represent kinetic and potential energy
operators, respectively, and are expressed in terms of momentum,
p = − i= d dq , and position q ˆ , operators, with ω defined by, ω = k m .
ˆ
As conjugate operators, p ˆ and q ˆ obey a commutation relation, namely,
=
p ˆ ˆ
q,
[ p ˆˆ ] q − p ˆˆ q = = i , which indicates that the order in which they are
applied is important. ƫ is Planck’s constant .6( 626× 10 − 34 J − sec) divided
2
by π . Furthermore, as conjugate operators, they also obey an uncertainty
relation, namely, ∆q ˆˆ ∆p ≥ = , which gives the uncertainty in their values. A
state prepared such that, say, ( ) ˆ 2 < 2 / = , is called a squeezed state. Such
∆q
a state lowers the uncertainty in one operator at the expense of that in the
other [183].
To repeat ourselves, solving Schrödinger’s operator equation,
ˆ
ε
H ψ >= ψ > , entails finding the eigenvalues, ε , giving the possible
energies (frequencies) of the particle, and their corresponding eigenvectors,
ψ , giving the wavefunctions that describe propagation in the system. For
example, when the particle in question refers to atoms, separated by a
distance a, undergoing longitudinal vibration modes in a monatomic linear
chain (MLC), described by the Hamiltonian,