Physical Chemistry Third Edition

696 16 The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics

16.9 Find 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.sin(z)

b.cosh(z)

c.z^3

16.10a. Show that the multiplication operatorxis linear and
hermitian.
b. Show that the operatori(d/d x) is linear and hermitian.
c.Show that any hermitian operator is linear.

16.11Show that the first two energy eigenfunctions of the
harmonic oscillator are orthogonal to each other.

16.12Show that the operator for multiplication by a function,
h(x), is linear and hermitian.

16.13Using integration by parts, find the result of performing
the integral
∫∞

−∞

(

d

dx

δ(x−a)

)

f(x)dx

whereδ(x) is the Dirac delta function.

16.14Find the commutator [p^2 x,x^2 ].

16.15a.Find the eigenfunctions and eigenvalues of

̂px−ih ̄(∂/∂x).

b.Are the energy eigenfunctions for a particle in a

one-dimensional box eigenfunctions of this operator?

If so, find the eigenvalues.

c.Are the energy eigenfunctions for a free particle

eigenfunctions of this operator? If so, find the

eigenvalues.

16.16a.Show that [̂Lx,L̂y]ih ̄̂Lz.

b.Argue from the result of part a that [̂Lz,L̂x,]i ̄ĥLy

and that [̂Ly,̂Lz]ih ̄L̂x.

16.17Find the commutator [̂Lx,p̂x].

16.18a.Find the quantum mechanical operator for the variable

Or·pxpx+ypy+zpz

b.Find the commutators [Ô,x] and [Ô,̂px] whereÔis the

operator in part a.

16.19Find an eigenfunction of the following operator and find

its corresponding eigenvalue:

d^2

dx^2

+ 2

d

dx

+ 1

16.4 Postulate 4 and Expectation Values

The first postulate of quantum mechanics asserts that the wave function of a system

contains all available information about the values of mechanical variables for the

state corresponding to the wave function. The third postulate provides a mathematical

operator for each mechanical variable. The fourth postulate provides the mathematical

procedure for obtaining the available information:

Postulate 4.

(a)If a mechanical variableAis measured without experimental error, the only possible

outcomes of the measurement are the eigenvalues of the operatorÂthat corresponds toA.

(b)Theexpectation valuefor the error-free measurement of a mechanical variableAif given

by the formula

〈A〉

∫

Ψ∗ÂΨdq

∫

Ψ∗Ψdq

(16.4-1)

whereÂis the operator corresponding to the variableA, and whereψψ(q, t)is the wave

function corresponding to the state of the system at the time of the measurement.