
396 SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE
7. f(x)y
n−1
∂w
∂x
+
hh
g(x)y
n
−
b
y
ii
∂w
∂y
=
∂
2
w
∂y
2
.
The transformation (A and B are any numbers)
t =
Z
h
n+1
(x)
f(x)
dx + A, ξ =
2
n + 1
yh(x)
n+1
2
,
where h(x) = B exp
−
Z
g(x)
f(x)
dx
, leads to the equation
∂w
∂t
=
∂
2
w
∂ξ
2
+
1 −2β
ξ
∂w
∂ξ
, β =
1 −b
n + 1
,
which is considered in Section 3.2.5 (see also equations in Sections 3.2.1 and 3.2.3).
3.9.2 One-Dimensional Schr
¨
odinger Equation
i~
∂w
∂t
= −
~
2
2m
∂
2
w
∂x
2
+ U (x)w
◮ Eigenvalue problem. Cauchy problem.
Schr¨odinger’s equation is the basic equation of quantum mechanics; w is the wave function,
i
2
= −1, ~ is Planck’s constant, m is the mass of the particle, and U(x) is the potentia ...