The minimum energy the marble can have is 5.5 10
64
J, corresponding to n1. A marble
with this kinetic energy has a speed of only 3.3 10
31
m/s and therefore cannot be experi-
mentally distinguished from a stationary marble. A reasonable speed a marble might have is, say,
1
3
m/s—which corresponds to the energy level of quantum number n10
30
! The permissible
energy levels are so very close together, then, that there is no way to determine whether the
marble can take on only those energies predicted by Eq. (3.18) or any energy whatever. Hence
in the domain of everyday experience, quantum effects are imperceptible, which accounts for
the success of Newtonian mechanics in this domain.
3.7 UNCERTAINTY PRINCIPLE 1
We cannot know the future because we cannot know the present
To regard a moving particle as a wave group implies that there are fundamental limits
to the accuracy with which we can measure such “particle” properties as position and
momentum.
To make clear what is involved, let us look at the wave group of Fig. 3.3. The par-
ticle that corresponds to this wave group may be located anywhere within the group
at a given time. Of course, the probability density
2
is a maximum in the middle of
the group, so it is most likely to be found there. Nevertheless, we may still find the
particle anywhere that
2
is not actually 0.
The narrower its wave group, the more precisely a particle’s position can be speci-
fied (Fig. 3.12a). However, the wavelength of the waves in a narrow packet is not well
defined; there are not enough waves to measure accurately. This means that since
hm, the particle’s momentum mis not a precise quantity. If we make a series
of momentum measurements, we will find a broad range of values.
On the other hand, a wide wave group, such as that in Fig. 3.12b, has a clearly
defined wavelength. The momentum that corresponds to this wavelength is therefore
a precise quantity, and a series of measurements will give a narrow range of values. But
where is the particle located? The width of the group is now too great for us to be able
to say exactly where the particle is at a given time.
Thus we have the uncertainty principle:
It is impossible to know both the exact position and exact momentum of an ob-
ject at the same time.
This principle, which was discovered by Werner Heisenberg in 1927, is one of the
most significant of physical laws.
A formal analysis supports the above conclusion and enables us to put it on a quan-
titative basis. The simplest example of the formation of wave groups is that given in
Sec. 3.4, where two wave trains slightly different in angular frequency and wave
number kwere superposed to yield the series of groups shown in Fig. 3.4. A moving
body corresponds to a single wave group, not a series of them, but a single wave group
can also be thought of in terms of the superposition of trains of harmonic waves. How-
ever, an infinite number of wave trains with different frequencies, wave numbers, and
amplitudes is required for an isolated group of arbitrary shape, as in Fig. 3.13.
At a certain time t, the wave group (x) can be represented by the Fourier integral
(x)
0
g(k) cos kx dk (3.19)
108 Chapter Three
Figure 3.12 (a) A narrow de
Broglie wave group. The position
of the particle can be precisely
determined, but the wavelength
(and hence the particle's momen-
tum) cannot be established be-
cause there are not enough waves
to measure accurately. (b) A wide
wave group. Now the wavelength
can be precisely determined but
not the position of the particle.
∆x small
∆p large
(a)
∆x
λ = ?
∆x large
∆p small
(b)
λ
∆x
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