1
PHYS809Class10Notes
Separation of variables in Cartesian geometry
Separation of variables in Cartesian geometry, with
(
)
(
)
(
)
( )
,
,
,
x y z
X
x Y
y Z
z
φ
=
leads to the set of
equations
2
2
2
2
2
2
,
,
,
d X
X
dx
d Y
Y
dy
d Z
Z
dz
α
β
γ
=
=
=
(1.1)
where
α
,
β
and
γ
are constants that satisfy
0.
α
β
γ
+
+
=
(1.2)
We see that the constants cannot all have the same sign. The sign of the constants is to some extent
dictated by the particular problem under consideration. The problem of a rectangular box with potential
specified on the sides of the box can be solved by using superposition of solutions for the box in which
the potential is nonzero only on one side of the box. For a box with sides on which
x
= 0
,
x
=
a
,
y
= 0
,
y
=
b
,
z
= 0
and
z
=
c
, suppose the potential is nonzero only on the
z
=
c
side. The boundary conditions on
the adjacent sides require that
α
and
β
are both negative which makes
γ
positive. Furthermore, these
boundary conditions require discrete values for these constants such that
sin
,
sin
,
x
X
m
a
y
Y
n
b
π
π
=
=
(1.3)
where
m
and
n
are integers. We then find that
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 MacDonald
 Vector Space, Rectangle

Click to edit the document details