12.6 The particle in a one-dimensional box 355
where the differential operator
(12.55)
is called the Hamiltonian operator, or simply the Hamiltonian, of the system. The effect
of operating with Hon ψis to generate a multiple of ψ. The (time-independent)
Schrödinger equation can always be written in the form (12.54) as an eigenvalue equa-
tion, with the system specified by the Hamiltonian operator and appropriate boundary
conditions. The permitted wave functionsψ 1 = 1 ψ
nare called the eigenfunctionsof
the Hamiltonian, and the corresponding energiesE 1 = 1 E
nare the eigenvalues ofH.
An important property of eigenfunctions is that of orthogonality. In the present
case, consider the integral
(12.56)
whereψ
mandψ
nare two differenteigenfunctions (12.53). Then
(12.57)
= 1 0 when m 1 ≠ 1 n
Proof. The integral is evaluated by making use of the addition properties of
trigonometric functions, as discussed in Section 6.2. Thus,
so that
Now,
sincesin 101 = 10 and the sine of a multiple, m 1 ± 1 n, of πis also zero.
=
±
±−
=
l
mn
mn
()
sin( ) sin
π
π 00
Z
001
llmnx
l
dx
mn
mn x
l
cos
()
()
sin
± ()
=
±
π ±
π
π
I
mnx
l
dx
mnx
l
ll=
−
−
1 +
2
1
2
00ZZcos
()
cos
ππ()
dx
sin sin cos
()
cos
mx ()
l
nx
l
mnx
l
ππ π mn
=
−
−
1 +
2
ππx
l
I
l
mx
l
nx
l
dx
l=
2
0Zsin sin
ππ
Ixxdx
lmn=Z
0ψψ() ()
H=−
2222 m
d
dx