14.4 The particle in a rectangular box 395
and the partial differential equation (14.4) in two variables has been reduced to two
ordinary differential equations, both of which we can solve. The equations, (14.9) in
the variable xand (14.10) in the variable y, are separable first-order equations of the
kind discussed in Section 11.3, and they have general solutions
X(x) 1 = 1 Ae
Cx, Y(y) 1 = 1 Be
−Cy(14.11)
A solution of the equation (14.4) in two variables is then the product
f(x, 1 y) 1 = 1 X(x) 1 × 1 Y(y) 1 = 1 Ae
Cx1 × 1 Be
−Cy1 = 1 De
C(x−y)(14.12)
0 Exercises 4–7
Particular solutions, including the possible values of the separation constant,
can often be obtained by the application of initial and boundary conditions, as
exemplified by the important problems discussed in Sections 14.4 to 14.6. In other
cases, as discussed in Section 14.7 for the vibrating string, it may be necessary to
consider more general solutions that are linear combinations of products.
14.4 The particle in a rectangular box
The Schrödinger equation for a particle of mass mmoving in the xy-plane is
(14.13)
where
(14.14)
is the two-dimensional Laplacian operator (see Section 9.6). For the present system,
the potential energy function is (Figure 14.1)
Vxy (14.15)
xa yb
(),=
<< <<
00 and 0for
∞ elsewhere
∇=
∂
∂
∂
∂
22222xy
−∇ ,+ , ,= ,
222 m
ψψψ()()() ()xy Vxy xy E xy
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.V=∞ V=∞
V=∞
V=∞
V= 0
0 a
b
x
y
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...Figure 14.1