Page 56 - Science at the nanoscale

9:2
RPS: PSP0007 - Science-at-Nanoscale
June 9, 2009
Brief Review of Quantum Mechanics
46
Since the particle can travel in the forward as well as backward
directions, we can express the wavefunction as
ψ I (x) = Fe
+ Ge
where the first term represents the incident wave while the second
term represents the reflected wave. F and G are coefficients that
can be determined using the boundary conditions.
For region II, the potential has a finite height of V o such that
V o > E, where E is the energy of the particle. The Schr¨odinger
equation can be expressed as
2
2
h
d ψ II
(3.37)
−
= (E − V o )ψ II
2
dx
8mπ
We can re-write the above equation as
2
d ψ II
2
(3.38)
= κ ψ II
dx
2
2
= 8mπ (Vo − E)/h , and hence we can express the
where κ
wavefunction in region II as
−κx
(3.39)
ψ II (x) = He
To determine the coefficients F, G and H in Eqs. (3.36) and
(3.39), ψ(x) and dψ(x)/dx must be continuous at the boundary
points x = 0. We have
(3.40)
ψ I (0) = ψ II (0)
dψ I
(3.41)
dx 2 2 2 = ikx dψ II −ikx (3.36) ch03
dx
and thus
F + G = H ik(F − G) = −κH
For region I
ik + κ
ψ I (x) = F e ikx + e −ikx (3.42)
ik − κ
For region II
2ik −κx
ψ II (x) = F e (3.43)
ik − κ