Chemistry Reference
In-Depth Information
The allowed energy levels are then determined by finding those values
of
E
2
k
2
2
m
for which either of the transcendental equations (1.50) or
(1.51) can be satisfied.
We could have ignored the symmetry properties of the quantum well,
allowing all confined states to have the general form given by eq. (1.45) in
the well, and then found the allowed energy levels by directly solving the
four linear equations in (1.48); that is, requiring that
⎛
⎝
=
/
⎞
⎠
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
A
B
D
e
−
κ
a
/
2
F
e
−
κ
a
/
2
sin
(
ka
/
2
)
cos
(
ka
/
2
)
−
10
0
0
0
0
k
cos
(
ka
/
2
)
−
k
sin
(
ka
/
2
)κ
0
=
(1.52)
−
sin
(
ka
/
2
)
cos
(
ka
/
2
)
0
−
1
k
cos
(
ka
/
2
)
k
sin
(
ka
/
2
)
0
−
κ
Non-trivial solutions of eq. (1.52) are obtained when the determinant of
the 4
4 matrix is zero; it can be explicitly shown that the determinant is
zero when
×
[
(
/
)
−
κ
(
/
)
][
(
/
)
+
κ
(
/
)
]=
k
sin
ka
2
cos
ka
2
k
cos
ka
2
sin
ka
2
0
(1.53)
which, not surprisingly, is just a combination of the separate conditions in
eqs (1.50) and (1.51) for allowed even and odd states. When we multiply
out the two terms in eq. (1.53) and use the standard trigonometric identities
cos
cos
2
sin
2
, we obtain an
alternative transcendental equation which must be satisfied by confined
states in a square well, namely
θ
=
(θ/
2
)
−
(θ/
2
)
and sin
θ
=
2 cos
(θ/
2
)
sin
(θ/
2
)
2
k
2
(κ
−
)
sin
ka
+
2
k
κ
cos
ka
=
0
or
1
2
cos
ka
+
(κ/
k
−
k
/κ)
sin
ka
=
0
(1.54)
This less familiar formof the conditions for allowed states in a finite square
well potential will be very useful when investigating the allowed energy
levels in a 'diatomic', or double quantumwell in Chapter 2, and also when
using the Kronig-Penney model for periodic solids in Chapter 3.
References
There are many introductory (and advanced) texts on quantummechanics. Amore
detailed discussion of the topics considered here can be found for instance in:
Beiser, A. (2002)
Concepts of Modern Physics
, McGraw-Hill Inc., New York.
Eisberg, R. andR. Resnick (1985)
QuantumPhysics of Atoms, Molecules, Solids, Nuclei,
and Particles
, Second Edition, Wiley, New York.
Davies, P. C. W. and D. S. Betts (1994)
Quantum Mechanics,
Second Edition, Nelson
Thornes, Cheltenham.
