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## Wave-packets

The above discussion suggests that the wave-function of a massive particle of momentum and energy can be written
 (64)

where and . Here, and are linked via the dispersion relation (61). Expression (64) represents a plane-wave which propagates in the -direction with the phase-velocity . As we have seen, this phase-velocity is only half of the classical velocity of a massive particle.

From before, the most reasonable physical interpretation of the wave-function is that is proportional to the probability density of finding the particle at position at time . However, the modulus squared of the wave-function (64) is , which depends on neither nor . In other words, this wave-function represents a particle which is equally likely to be found anywhere on the -axis at all times. Hence, the fact that the plane-wave wave-function (64) propagates at a phase-velocity which does not correspond to the classical particle velocity does not have any real physical consequences.

So, how can we write the wave-function of a particle which is localized in : i.e., a particle which is more likely to be found at some positions on the -axis than at others? It turns out that we can achieve this goal by forming a linear combination of plane-waves of different wave-numbers: i.e.,

 (65)

Here, represents the complex amplitude of plane-waves of wave-number in this combination. In writing the above expression, we are relying on the assumption that particle waves are superposable: i.e., it is possible to add two valid wave solutions to form a third valid wave solution. The ultimate justification for this assumption is that particle waves satisfy a differential wave equation which is linear in . As we shall see, in Sect. 3.12, this is indeed the case.

Now, there is a useful mathematical theorem, known as Fourier's theorem, which states that if

 (66)

then
 (67)

Here, is known as the Fourier transform of the function . We can use Fourier's theorem to find the -space function which generates any given -space wave-function at a given time.

For instance, suppose that at the wave-function of our particle takes the form

 (68)

Thus, the initial probability density of the particle is written
 (69)

This particular probability distribution is called a Gaussian distribution, and is plotted in Fig. 4. It can be seen that a measurement of the particle's position is most likely to yield the value , and very unlikely to yield a value which differs from by more than . Thus, (68) is the wave-function of a particle which is initially localized around in some region whose width is of order . This type of wave-function is known as a wave-packet.

Now, according to Eq. (65),

 (70)

Hence, we can employ Fourier's theorem to invert this expression to give
 (71)

Making use of Eq. (68), we obtain
 (72)

Changing the variable of integration to , this reduces to
 (73)

where . The above equation can be rearranged to give
 (74)

where . The integral now just reduces to a number, as can easily be seen by making the change of variable . Hence, we obtain
 (75)

where
 (76)

Now, if is proportional to the probability density of a measurement of the particle's position yielding the value , then it stands to reason that is proportional to the probability density of a measurement of the particle's wave-number yielding the value . (Recall that , so a measurement of the particle's wave-number, , is equivalent to a measurement of the particle's momentum, ). According to Eq. (75),

 (77)

Note that this probability distribution is a Gaussian in -space--see Eq. (69) and Fig. 4. Hence, a measurement of is most likely to yield the value , and very unlikely to yield a value which differs from by more than . Incidentally, a Gaussian is the only mathematical function in -space which has the same form as its Fourier transform in -space.

We have seen that a Gaussian probability distribution of characteristic width in -space [see Eq. (69)] transforms to a Gaussian probability distribution of characteristic width in -space [see Eq. (77)], where

 (78)

This illustrates an important property of wave-packets. If we wish to construct a packet which is very localized in -space (i.e., if is small) then we need to combine plane-waves with a very wide range of different -values (i.e., will be large). Conversely, if we only combine plane-waves whose wave-numbers differ by a small amount (i.e., if is small) then the resulting wave-packet will be very extended in -space (i.e., will be large).

Next: Evolution of wave-packets Up: Wave-particle duality Previous: Quantum particles   Contents
Richard Fitzpatrick 2006-12-12