Physical Chemistry Third Edition

16.5 The Uncertainty Principle of Heisenberg 715

The speed of propagation of the de Broglie wave is not necessarily the same as the expectation

value of the momentum divided by the mass of the particle. This fact is somewhat surprising,

but reminds us of the fact that the de Broglie wave is not a directly measurable quantity.

EXAMPLE16.24

The energy of a free particle moving in thexdirection is

E ̄

h^2 κ^2

2 m

so the time-dependent wave function can be written as

ΨDexp

[

i

(

κx− ̄

h^2 κ^2 t

2 mh ̄

)]

Dexp

[

iκ

(

x− ̄

hκt

2 m

)]

Find an expression for the speed of propagation of the de Broglie wave and compare it with

the expectation value of the speed of the particle (the momentum divided by the mass).

Solution

The speed is obtained by determining the speed at which a node of the de Broglie wave

moves. By inspection of the arguments of the complex exponentials, the speed of the

de Broglie wave is

speed ̄

hκ

2 m

We also have

〈px〉hκ ̄

so that

〈vx〉

̄hκ

m

which differs by a factor of 2 from the speed of the de Broglie wave.

The Time–Energy Uncertainty Relation

Like position and momentum, energy and time obey an uncertainty relation:

∆E∆t≥

h

4 π

(16.5-10)

The time–energy uncertainty relation is even more mysterious than that of position and

momentum, but it has been verified by many experiments. Time is not a mechanical

variable that can be expressed in terms of coordinates and momenta and does not

correspond to any quantum mechanical operator, so we cannot calculate a standard

deviation of a time. The standard interpretation of the time–energy uncertainty relation

is that if∆tis the time during which the system is known to be in a given state (the

lifetime of the state) then there is a minimum uncertainty∆Ein the energy of the

system that is given by

∆E≥

h

4 π∆t

(16.5-11)