Physical Chemistry Third Edition

16.6 Postulate 5. Measurements and the Determination of the State of a System 717

with the value for then1 state, and explain the

difference.

d.Calculate the uncertainty productσxσpxfor a particle

in a box of lengthafor then3 state. Compare it

with the value for then2 state, and explain the

difference.

e.Obtain a formula forσxσpxfor a particle in a box of

lengthaas a function ofn. Find the limit asn→∞.

16.36a.Calculate the uncertainty productσxσpxfor a
harmonic oscillator in thev0 state.
b.Without doing any calculations, state whether the
uncertainty product for thev1 state would be
smaller than, equal to, or larger than the product for
thev0 state.

c.Calculate the uncertainty productσxσpxfor a harmonic

oscillator for thev1 state. Compare it with the

value for thev0 state, and explain the difference.

d.Calculate the uncertainty productσxσpxfor a harmonic

oscillator for thev2 state. Compare it with the

value for thev1 state, and explain the difference.

16.37a.Find the commutator [̂x,L̂z].

b.Using Eq. (16.5-5), find an uncertainty relation for̂x

andL̂zfor a particular state, the lowest-energy state of

a particle in a cubical box of sizeabyabya.

16.38Construct a graph representing the time-average
probability density for finding a harmonic oscillator at a

given position according to classical mechanics. Start

your graph atx− 0. 99 xtand end it at 0. 99 xt, wherext

represents the magnitude of the coordinate at the turning

point. Comment on the relative value at the turning

points.

16.39Consider a particle in a one-dimensional box of lengtha

with a wave function att0 given by

Ψ(x,0)

1

√

2

(ψ 1 +ψ 2 )

whereψ 1 andψ 2 are the first two energy eigenfunctions.

a.Construct a graph of the wave function att0.

b.Write expressions for the real and imaginary parts of

Ψ(q,t) at timet 2 ma^2 /h.

c.Write an expression forΨ(x,t) at timet 4 ma^2 /h.

Construct a graph of the magnitude of the wave

function at this time.

d.Write an expression forΨ(x,t) at timet 2 ma^2 /h.

Draw separate graphs of the real and imaginary parts

of the wave function at this time.

e.Does this wave function represent a standing wave?

Does it satisfy the time-independent Schrödinger

equation?

f.Calculate〈E〉andσEat timet0.

g.Calculate〈E〉andσEat timet 2 ma^2 /h.

h.Calculate〈E〉andσEat timet 4 ma^2 /h.

16.6 Postulate 5. Measurements and

the Determination of the State of a System

The fifth and final postulate gives information about the state of a system following a

measurement.

Postulate 5. Immediately after an error-free measurement of the mechanical variableA

in which the outcome was the eigenvalueaj, the state of the system corresponds to a wave

function that is an eigenfunction ofÂwith eigenvalue equal toaj.

This postulate says very little about the state of the system prior to a single

measurement of the variableA, because the act of measurement can change the state

of the system. To understand physically how a measurement can change the state of

a system consider the determination of the position of a particle by the scattering of

electromagnetic radiation. When an airplane reflects a radar wave, the effect on the air-

plane is negligible because of the large mass of the airplane. When an object of small

mass such as an electron scatters ultraviolet light or X-rays, the effect is not negligible.