bei48482_FM

The corresponding energy uncertainty is

E h 

and so

E or E t h

A more precise calculation based on the nature of wave groups changes this result to

E t  (3.26)

Equation (3.26) states that the product of the uncertainty Ein an energy meas-

urement and the uncertainty tin the time at which the measurement is made is equal

to or greater than 2. This result can be derived in other ways as well and is a gen-

eral one not limited to em waves.

Example 3.9

An “excited” atom gives up its excess energy by emitting a photon of characteristic frequency,

as described in Chap. 4. The average period that elapses between the excitation of an atom and

the time it radiates is 1.0  10 ^8 s. Find the inherent uncertainty in the frequency of the

photon.

Solution

The photon energy is uncertain by the amount

E5.3  10 ^27 J

The corresponding uncertainty in the frequency of light is

 8  106 Hz

This is the irreducible limit to the accuracy with which we can determine the frequency of the

radiation emitted by an atom. As a result, the radiation from a group of excited atoms does not

appear with the precise frequency . For a photon whose frequency is, say, 5.0  1014 Hz,

1.6  10 ^8. In practice, other phenomena such as the doppler effect contribute more

than this to the broadening of spectral lines.

E



h

1.054  10 ^34 J s



2(1.0  10 ^8 s)



2 t



2

Uncertainties in

energy and time

h



t

116 Chapter Three

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