Physical Chemistry Third Edition

16.3 The Operator Corresponding to a Given Variable 695

The Eigenfunction of a Coordinate Operator

Since the operator for a coordinatexor any other coordinate must have a set of eigen-

functions, the situation is a little bit strange. We must require

xδ(x−a)aδ(x−a) (16.3-36)

whereδ(x−a) is the eigenfunction andais the eigenvalue. This looks problematic

because we have a variable times a function set equal to a constant times the same

function. To make the equation valid, we must define the eigenfunction so that it is

nonzero only whenxa. However, the integral of the eigenfunction must be nonzero,

so we specify that

∫c

b

δ(x−a)dx 1 (16.3-37)

ifb<a<c. This eigenfunction is called theDirac delta function, defined such that

δ(x−a)→∞ifxaandδ(x−a)0ifxa, in such a way that Eq. (16.3-37)

is obeyed. A graph of this function has unit area under a single point. It is highly

discontinuous, and some mathematicians refuse to call it a function.

EXAMPLE16.8

Show that the following integral:

∫∞

−∞

δ(x−a)f(x)dx

is equal tof(a).

Solution

Since the delta function is nonzero only atxa, we can factorf(a) out of the integral

without changing its value:

∫∞

−∞

δ(x−a)f(x)dxf(a)

∫∞

−∞

δ(x−a)dxf(a) (16.3-38)

PROBLEMS

Section 16.3: The Operator Corresponding to a Given
Variable

16.6 Determine whether each of the following operators is

linear and whether it is hermitian.

a.

d^2

dx^2

b.

d^3

dx^3

c.sin(...)

16.7 Determine whether each of the following operators is

linear and whether it is hermitian.

a.ln(...)

b.

c

x

+i

d

dx

, wherecis a constant andi

√

−1.

c.ix

d

dx

, wherei

√

−1.

16.8Find the complex conjugate of each of the following

wherezx+iy, andxandyare real. Do it once by

replacingiby−i, and once by separating the real and

imaginary parts.

a.sinh(z)

b.cos(z)

c.z^2