Matter waves
Notes on Quantum Mechanics
http://NotesOnQuantumMechanics.com/MatterWaves.pdf
Last updated Thursday, September 5, 2002 12:54:11
Copyright © 2002 Dan Dill (dan@bu.edu)
Department of Chemistry, Boston University, Boston MA 02215
à Matter waves: wavefunctions
Bohr's analysis of one-electron atoms teaches that an electron in such an atom evidently can
have only certain energies, and so can exchange only certain energy differences with light.
We have seen that de Broglie's hypothesis,
lelectron = h ê pelectron ,
accounts for these results, if we use it to associate with the electron a wavelength and
impose the requirement that an integer number of wavelengths fit on the circumference of
the (classical) electron orbit about the nucleus.
It is remarkable that de Broglie was able to obtain Bohr's result so simply, but the de Broglie
relation introduces two profound new questions. First, just what could it be that is waving,;
that is, what could be meant by an "electron wave?" Second, anyhow, how can a particle be
represented as a wave; that is, we have the idea that the electron is a "particle," so how can it
behave as a "wave?"
à Born interpretation: probability amplitudes
Let's consider the first question first. One idea is that, perhaps, the electron itself is
following a wiggly pattern around the nucleus, with the length of the wiggles corresponding
to the wavelength. This picture could not be correct, however, because the wiggles do not
correspond to any wave disturbance and so could not constructively or destructively
interfere with one another, according to whether an integer number of wavelengths fits on
the circumference. That is, electron waves, whatever they are, are not wiggling electrons.
It was Max Born who provided the answer we use today. Born postulated that electrons are
represented in terms of probability amplitudes, or wavefunctions (usually represented by the
Greek letter y). Wavefunctions are abstract quantities rather than physical objects. They
have physical meaning according to the recipe (or rule) that the probability of an electron
being within „ x of a particular place, say x0 , is
Matter waves
2
» y* Hx0 L yHx0 L »2 „ x,
or, in three dimensions, the probability of the electron being within a volume „ r centered at
a particular place, say r0 = Hx0 , y0 , z0 L, is
» y* Hr0 L yHr0 L »2 „ r.
Here » … »2 denotes the squared absolute value; this is necessary since wavefunctions can
è!!!!!!!
have imaginary parts (parts proportional to -1 = Â). The expression » yHxL »2 has the units
of probability per unit length and is called the probability density.
The terminology proability density is used because in three dimensions, the quantity
» y HrL »2 has the dimensions probability per unit volume. Because of this connection
between the wavefunction and probability, wavefunctions are sometime referred to as
probability amplitdues.
The probability of finding the particle within a larger (one-dimensional) region, say between
x = a and x = b, is
‡
b
a
» yHxL »2 „ x.
Since a particle must be found somewhere, the Botn recipe requires that the probability of
finding it anywhere at all be 1,
‡
a
b
» yHxL »2 „ x = 1.
This relation is known as the normalization condition and wavefunctions that satisfy this
condition are said to be normalized.
It is crucial to understand that the Born recipe gives only indirect physical meaning to the
wavefunction, as a means to compute probabilities. In particular, as far as we know today,
the wavefunction itself does not appear to be directly accessible. Indeed, so far as I am
aware we do not have any experimental evidence of the reality of wavefunctions; in this
senese they may simply be calculational tools.
à Working with wavefucntions
Here is a wavefunction of a particle confined in the region 0 § x § 1,
y3 HxL =
è!!!
2 sinH3 p xL.
and here is the corresponding probability density,
Copyright © 2002 Dan Dill (dan@bu.edu). All rights reserved
Matter waves
3
»yHxL»2
2
1.5
1
0.5
0.2
0.4
0.6
0.8
1
x
Probability density of a particle confined to the region 0 § x § 1 .
Here is an expression for the probability of finding the particle somewhere between x = a
and x = b,
‡
b
a
2
è!!!!
I 2 Sin@3 π xDM x
2 J−
a
b
Sin@6 a πD
Sin@6 b πD
+
+
−
N
2
2
12 π
12 π
Show that the probability of finding the particle somewhere between x = 0 and x = 1 is 1.
What is the probability of finding the particle within „ x = 0.01 of x = ÅÅÅÅ61 ? Answer: 0.02
1
1
What is the probability of finding the particle between x = ÅÅÅÅ61 - ÅÅÅÅ
ÅÅ and x = ÅÅÅÅ61 + ÅÅÅÅ
ÅÅ ?
12
12
Answer: 0.27 .
What is the probability of finding the particle within „ x = 0.01 of x = ÅÅÅÅ13 ? Answer: 0.
1
1
What is the probability of finding the particle between x = ÅÅÅÅ13 - ÅÅÅÅ
ÅÅ and x = ÅÅÅÅ13 + ÅÅÅÅ
ÅÅ ?
12
12
Answer: 0.061.
Make a sketch that shows why it is that the probability of finding the particle near x = ÅÅÅÅ61
is different from the probability of finding the particle near x = ÅÅÅÅ13 .
à Representing a "particle" by a wave: wavepackets
Now that we know a little about what a matter wave—a wavefunction—is, let's consider the
second question raised by de Broglie's analysis: How can a particle, which we think of as
something being in a particular region of space ("here, and not there"), be represented by a
wave, which is by definition spread throughout space (else its length—the distance from
peak to peak—has no meaning).
A caution on the use of language:
We will show now one way to reconcile the wave aspect of electrons with the image of an
Copyright © 2002 Dan Dill (dan@bu.edu). All rights reserved
Matter waves
4
electron as being a localized object—a "particle." But at a more fundamental level,
whether quantum objects appear as waves or particles has meaning only in terms of the
kinds of measurements we make on them.That is, the so-called wave-particle duality is an
inescapable aspect of every quantum object.
This duality can trap us in a thicket of fantasy that can be avoided only by being careful to
not to ascribe to quantum objects a preexisting particle or wave character, but instead to
do so only in terms of the behavior exhibited in measurements we make upon them. That
is, if in a measurement a quantum object appears as a particle, that does not mean that the
object behaved as a particle before we measured it. In particular, quantum mechanics tells
us nothing about what quantum objects are before we make measurements on them.
For example, if you find yourself visualizing a photon, say, either as a particle or as a
wave, then you are indulging in fantasy as far as quantum mechanics is concerned. Please
keep this in mind when we speak of something as behaving as a wave or as a particle.
In particular, we are now going to see how to represent quantum objects as being
localized in space. This is quite different than to say (incorrectly) that we are now going to
see how to represent localized quantum objects, for quantum objects have no such
characteristic except in terms of measurements made on them
We might agree that perhaps an electron does behave as a wave inside an atom. But surely it
behaves as a particle as it traces out images on the insides of our television tubes and
computer monitors. (Remember, as noted above, such language is unjustified by quantum
mechanics, but seems reasonable to us because objects in our everyday world can be talked
about as either waves or particles.)
IIn fact it is possible to reconcile a wave picture and a localized particle picture, by using
the property of waves that they oscillate between positive and negative values, for this
means that different possible probability amplitudes can be combined in various ways to
make so-called wavepackets that can have any shape we choose. Localized particles, in
particular, are wavepackets composed of probability amplitudes of many different
wavelengths, so many that they interfere destructively everywhere except the region of
space where the particle is.
A nice illustration of using interference to build a probability amplitude localized in space is
to construct a packet composed of sine waves,
2
x
y j HxL = $%%%%%%
ÅÅÅÅÅÅ sinJ j p ÅÅÅÅÅÅ N
L
L
with j loops over the region 0 § x § L. We can write a general wavepacket as a linear
combination
YHxL = N ‚ g j y j HxL
n
j=1
of component functions, where the weights gi determine the form of a given wavepacket.
We interpret the squared amplitude of the wave as the probability density, the probability
per unit length of the particle being at a give position. The particle must be somewhere, so
Copyright © 2002 Dan Dill (dan@bu.edu). All rights reserved
Matter waves
5
the sum of probabilities over all positions must be equal to 1. The normalization factor N is
chosen so that this total proability does equal 1,
‡
1
0
» YHxL »2 „ x = 1.
What we want to see is the effect of including increasing numbers of components y j , by
plotting wavepackets with unit weights, g j = 1 (for simplicity) for j § nmax , and zero
weights, g j = 0, for j > nmax ,
max
1
Ynmax HxL = $%%%%%%%%
ÅÅÅÅÅÅÅÅÅÅÅÅ%Å%%%Å ‚ y j HxL,
nmax j=1
n
for five different numbers of wavefunctions in the packet, nmax = 1, 2, 5, 10 and 20. Let's
chose length L = 1 and write the wavepacket explicitly as
max
2
Ynmax HxL = $%%%%%%%%
ÅÅÅÅÅÅÅÅÅÅÅÅ%Å%%%Å ‚ sinH j p xL,
nmax j=1
n
For example, here is the wavepacket consisting of the first three functions y j ,
2
$%%%%%%
HSin@π xD + Sin@2 π xD + Sin@3 π xDL
3
Show that this expression is correct.
Here are the probability densities corresponding to all of the packets,
»YHxL»2
20
15
10
5
0.2
0.4
0.6
Wavepacket probability densities, » Ynmax »2 , for nmax = 1, 2, 5, 10 and 20 .
0.8
1
x
We see that as nmax increases the packets become increasingly localized near the left edge of
the region. We can illustrate the effect of nmax on localization more strikingly with a surface
plot.
Copyright © 2002 Dan Dill (dan@bu.edu). All rights reserved
Matter waves
6
»YHxL»2
20
15
20
10
5
0
15
10 nmax
0.2
0.4
x
5
0.6
0.8
1
Wavepacket probability densities, » Ynmax »2 , for nmax = 1 to 20 .
The surface makes it easier to see how adding more different wavelengths leads to greater
localization.
One final comment about using wavepackets to represent quantum objects as being
localized in space. Might we use a reverse analysis to represent delocalized quantum
à Heisenberg indeterminacy principle
What wavepackets show is that to represent a localized particle, many different wavelengths
need to be combined. But, by the de Broglie relation, p = h ê l, this means equivalently that
many different linear momenta, p, need to be combined. That is, the uncertainty in where
the particle is, dx, is inversely proportional to the uncertainty in the momentum, d p = h ê dl ,
of the particle. The precise relation is
h
dx d p ¥ ÅÅÅÅÅÅÅÅÅÅ .
2p
This inverse relationship is known as the Heisenberg indeterminacy principle. It means that
there is a fundamental limitation on how precisely we can localize a particle. If we are very
precise, then there is a very large uncertainty in the momentum of the particle, so that a
short time later we will not know where the particle will be.
Root-mean-squared (rms) deviation
The quantities dx and d p that appear in Heisenberg's indeterminacy relation are root-meansquared (rms) deviations. Here is how to compute an rms deviation: Take the square root of
the mean (or average) of the squared deviations, where a deviation is the difference between
a particular value and the mean of all of the values.
Copyright © 2002 Dan Dill (dan@bu.edu). All rights reserved
Matter waves
7
Here is an example to illustrate how this works. Say thirteen students took an exam, and the
scores were
scores = 878, 51, 43, 56, 63, 49, 61, 91, 33, 87, 44, 67, 59<;
out of a possible 100. The mean is
mean =
Plus @@ scores
Length@scoresD
;
60.2
The deviations from the mean and the squared deviations are
deviations = mean − scores;
squaredDeviations = deviations2 ;
and here are their values together with the corresponding exam scores,
x
78.0
51.0
43.0
56.0
63.0
49.0
61.0
91.0
33.0
87.0
44.0
67.0
59.0
<x>−x
−17.8
9.2
17.2
4.2
−2.8
11.2
−0.8
−30.8
27.2
−26.8
16.2
−6.8
1.2
H<x>−xL2
318.5
83.8
294.3
17.3
8.1
124.4
0.7
951.5
737.3
720.7
260.9
46.9
1.3
The mean of the squared deviations is
meanSquaredDeviation =
Plus @@ deviations2
Length@scoresD
;
274.3
Finally, the square root of this gives the rms deviation.
rmsDeviation =
è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!
meanSquaredDeviation ;
16.6
There is an equivalent way to compute the rms deviation, namely, as
dx =
"################
######
Xx2 \ - Xx\2 ,
where X…\ denotes the mean of the enclosed quantity, that is, as the square root of
difference between the mean of the squared scores and the square of the mean of the scores.
Copyright © 2002 Dan Dill (dan@bu.edu). All rights reserved
Matter waves
8
See if you can show that this equivalence is true.
For the example above, we get
2
2
Plus @@
scores
Plus @@ scores
y
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%
%%%%%%%%
z
j
rmsDeviation = $%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
−i
z%%% ;
j
Length@scoresD k Length@scoresD {
16.6
the same as we get by the earlier method.
A particle's position is measured in four separate experiments, and the values obtained
along the x axis were - 0.4, 1.1, 0.3, and 0.7, all in meters. Calculate the rms deviation,
d x, including units, of the particle position corresponding to this data.
The same particle's velocity is measured in four separate experiments, and the values
obtained along the x axis were 1.3, 0.9, - 1.6 , and 0.1, all in meters per second. The mass
forth particle is 1.00 kg. Calculate the rms deviation of the particle momentum, d p,
including units, corresponding to this data.
Calculate the uncertainty product, d x d p, including units, of the particle.
Copyright © 2002 Dan Dill (dan@bu.edu). All rights reserved