Page 48 - Science at the nanoscale

9:2
RPS: PSP0007 - Science-at-Nanoscale
June 9, 2009
Brief Review of Quantum Mechanics
38
2
where ∇ represents the Laplacian given, in Cartesian coordi-
nates, by
2
2
2
d ψ
d ψ
d ψ
2
(3.13)
∇ ψ =
+
+
2
2
2
dx
dy
dz
and V represents the potential energy term.
The time-independent Schr¨odinger equation is given as follows
2
h
2
(3.14)
−
∇ ψ + Vψ = Eψ
2
8mπ
where E represents the total energy of the particle.
For a one-dimensional system, Eq. 3.14 becomes
2
2
h
d ψ
(3.15)
+ Vψ = Eψ
−
2
2
dx
8mπ
Hence, if the potential energy V of a physical system is known,
one can make use of Eq. 3.15 to determine the corresponding
wavefunction. Thus the functional form for the wavefunction de-
pends on the potential energy V.
3.2.2
Particle in a Potential Box
A simple problem that is commonly discussed and relevant to
nanoscience is the case of a particle of mass m, trapped in a
potential box. Consider the one-dimensional potential box with a
width L as illustrated in Fig. 3.2. Our main task is to find the wave- ch03
functions that would describe the properties of a particle trapped
inside such a potential box. In this case, the particle is restricted to
move only in the region 0 < x < L where the potential energy is
equal to zero. In the regions x < 0 and x > L, the potential energy
increases sharply to infinity such that it is impossible to find the
particle in these regions. i.e. ψ(x, t) = 0 in these regions.
The next task would be to find the wavefunction for the particle
in the region 0 < x < L, where potential energy V = 0. Using
Eq. 3.15 with V = 0, we have
2
h 2 d ψ
− 2 2 = Eψ (3.16)
8mπ dx