9/11/2018 Wave Packets
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Wave Packets
The above discussion suggests that the wavefunction of a massive particle of momentum
and energy , moving in the positive -direction, can be written
(82)
where and . Here, and are linked via the dispersion
relation (79). Expression (82) represents a plane wave whose maxima and minima
propagate in the positive -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 wavefunction is that
is proportional to the probability density
of finding the particle at position at
time . However, the modulus squared of the wavefunction (82) is , which depends
on neither nor . In other words, this wavefunction represents a particle which is
equally likely to be found anywhere on the -axis at all times. Hence, the fact that the
maxima and minima of the wavefunction propagate 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 wavefunction 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 wavenumbers: , i.e.
(83)
Here, represents the complex amplitude of plane waves of wavenumber 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,
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in Sect. 3.15, this is indeed the case. Incidentally, a plane wave which varies as
and has a negative
(but positive ) propagates in the negative -
direction at the phase velocity . Hence, the superposition (83) includes both
forward and backward propagating waves.
Now, there is a useful mathematical theorem, known as Fourier's theorem, which states
that if
(84)
then
(85)
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
wavefunction at a given time.
For instance, suppose that at the wavefunction of our particle takes the form
(86)
Thus, the initial probability density of the particle is written
(87)
This particular probability distribution is called a Gaussian
distribution, and is plotted in
Fig. 7. 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, (86) is the wavefunction of a particle which is initially localized around
in some region whose width is of order . This type of wavefunction is known
as a wave packet.
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Figure 7: A Gaussian probability distribution
in -space.
Now, according to Eq. (83),
(88)
Hence, we can employ Fourier's theorem to invert this expression to give
(89)
Making use of Eq. (86), we obtain
(90)
Changing the variable of integration to , this reduces to
(91)
where . The above equation can be rearranged to give
(92)
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where . The integral now just reduces to a number, as can easily be seen by
making the change of variable . Hence, we obtain
(93)
where
(94)
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 wavenumber
yielding the value . (Recall that , so a measurement of the particle's
wavenumber, , is equivalent to a measurement of the particle's momentum, ).
According to Eq. (93),
(95)
Note that this probability distribution is a Gaussian
in -space. See Eq. (87) and Fig. 7.
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 just seen that a Gaussian probability distribution of characteristic width in
-space [see Eq. (87)] transforms to a Gaussian probability distribution of characteristic
width in -space [see Eq. (95)], where
(96)
This illustrates an important property of wave packets. Namely, if we wish to construct a
packet which is very localized in -space ( , if i.e.
is small) then we need to combine
plane waves with a very wide range of different -values ( , will be large). i.e.
Conversely, if we only combine plane waves whose wavenumbers differ by a small
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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
Richard Fitzpatrick 2010-07-20
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