Physical Chemistry Third Edition

716 16 The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics

This relation must hold even if the state corresponds to a definite energy eigenvalue,

for whichσE0. When a system makes a transition into such a state, the energy of

the system does not immediately attain the new energy eigenvalue, but has an inherent

uncertainty that rapidly decreases as time passes. This uncertainty in the energy is

important only if the system is observed very soon after making a transition, but it can

be observed experimentally. It imposes a broadening on spectral lines that is larger

if the system spends a shorter time in a given state. This phenomenon is known as

uncertainty broadening.

EXAMPLE16.25

The lowest energy eigenvalue for a hydrogen atom is−13.6 eV. Calculate the uncertainty in

the energy if a hydrogen atom is known to have been in the state corresponding to this energy

for 1.0 nanosecond.

Solution

∆E≥ ̄

h

2

1

∆t



6. 6261 × 10 −^34 Js

4 π

1

1. 0 × 10 −^9 s

 5. 3 × 10 −^26 J

(5. 3 × 10 −^26 J)

(

1eV

1. 6022 × 10 −^19 J

)

 3. 3 × 10 −^7 eV

This uncertainty is smaller than the energy eigenvalue by a factor of 2. 4 × 10 −^8 , and would

be even smaller if the time period were longer.

Exercise 16.14

If the energy of a system is to be measured to an uncertainty of 1. 0 × 10 −^21 J, find the minimum

time during which the system must be in the state at the measured energy.

PROBLEMS

Section 16.5: The Uncertainty Principle of Heisenberg

16.33a.A free electron (not confined in any kind of box) is
known at some specific time to be passing through a
region 10.0 Å in length and is known to be moving
toward the positive end of thexaxis. Find the
minimum uncertainty inpxand the corresponding
uncertainty invx, thexcomponent of its velocity.
b.If the expectation value of the kinetic energy
corresponding only to the motion of the electron in the
xdirection is equal to 5.00 eV, estimate the smallest
upper bound ofvxand the largest lower bound ofvx.

16.34A free electron is known to be passing through a
three-dimensional cubical region that is 10.0 Å on a side
at a certain time.

a.Estimate the uncertainty in each of the three

components of its momentum.

b.Estimate the uncertainty in the energy of the electron

if its energy is near 1.00 eV.

16.35 a.Calculate the uncertainty productσxσpxfor a particle

in a box of lengthafor then1 state.

b.Without doing any calculations, state whether the

uncertainty product for then2 state would be

smaller than, equal to, or larger than the product for

then1 state.

c.Calculate the uncertainty productσxσpxfor a particle

in a box of lengthafor then2 state. Compare it