398 Chapter 14Partial differential equations
that is, the degenerate eigenfunctions are interchanged (whenp 1 ≠ 1 q).*The symmetries
of some of the wave functions are illustrated in Figure 14.2.
The dashed lines are nodal lines, where the function is zero. We see that, for example,
ψ
1,2andψ
2,1are identical except for orientation.
0 Exercises 8–10
14.5 The particle in a circular box
The motion of a particle in a circular box of radius ais described by the Schrödinger
equation
(14.29)
as in Section 14.4, but with potential-energy function
(14.30)
where ris the distance from the origin at the centre of the box. The equation is
not separable in cartesian coordinates because of the functional form of Vat the
boundary of the box, but it becomes so when the equation is expressed in the plane
polar coordinates rand θ. In these coordinates, the two-dimensional Laplacian
operator is (see equation (9.38) and Example 9.18)
(14.31)
For the particle within the box, withV 1 = 10 , equation (14.29) is then
− (14.32)
∂
∂
∂
∂
∂
∂
=
2222222
11
m
r
rr
r
E
ψψ ψ
ψ
θ
∇=
∂
∂
∂
∂
∂
∂
22222211
r
rr
r θ
Vxy
rxya
()
,
,=
0 =+<
22for (inside the box)
∞welsewhere
−∇ ,+ , ,= ,
222 m
ψψψ()()() ()xy Vxy xy E xy
*An ‘accidental’ degeneracy may also occur that is not an obvious consequence of symmetry. For example, the
states (7, 1) and (1, 7) are degenerate with the state (5, 5).
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...+
+
+
+
+
+
++–
................................................................................................................................................................................................................................................................................................................................................................................................................................................................p=1
q=1
p=1
q=2
p=2
q=1
p=2
q=2
p=1
q=3
p=3
q=1
Figure 14.2