16.3 The Operator Corresponding to a Given Variable 693
- A hermitian operator has a set of eigenfunctions and eigenvalues.
- The eigenvalues of a hermitian operator are real.
- Two eigenfunctions of a hermitian operator with different eigenvalues are orthogonal
to each other. - Two commuting hermitian operators can have a set of common eigenfunctions.
- The set of eigenfunctions of a hermitian operator form a complete set for expansion
of functions obeying the same boundary conditions.
EXAMPLE16.6
a.Show that the operatord/dxis linear.
b.Show that it is not hermitian.
Solution
a. d
dx(f+g)
df
dx+
dg
dx
d(cf)
dxc
df
dxb.We integrate by parts:
∫∞−∞f(x)∗dg
dx
dxf(x)∗g(x)∣∣
∣
∣∞
−∞−∫∞−∞g(x)df∗
dx
dxThe functionsfandgmust vanish at the limits of integration, so we have
∫∞−∞f(x)∗
dg
dxdx−∫∞−∞df∗
dxg(x)dxwhich is the negative of what we would require for a hermitian operator.The proofs for Properties 4 and 5 are in Appendix B. Property 5 involves
orthogonality. Two functionsfandgareorthogonalto each other if∫
f∗gdq∫
g∗fdq 0(
definition of
orthogonality)
(16.3-31)
where the integrals are taken over all values of the coordinates. The two integrals in
Eq. (16.3-31) are the complex conjugates of each other so that if one vanishes, so does
the other.EXAMPLE16.7
Show that the first two energy eigenfunctions of the particle in a one-dimensional box of
lengthaare orthogonal to each other.
Solution
The energy eigenfunctions are found in Eq. (15.3-10). The formula contains a constantC,
which we assume to be real. Quantum mechanical integrals are taken over all values of the