Chapter 3: Wave Properties of Particles

• 1905 discovery of the particle properties of waves: A revolutionary concept to explain data.

• 1924 (Louis de Broglie’s PhD thesis) speculation that particles might show wave behavior:

An equally revolutionary concept without a strong experimental mandate.

◦ moving objects have wave as well as particle characteristics

• 1927 The existence of de Broglie waves was experimentally demonstrated.

• The duality principle provided the starting point for Schrödinger’s successful development

of quantum mechanics.

3.1 DE BROGLIE WAVES

• A moving body behaves in certain ways as though it has a wave nature.

• A photon of light of frequency ν has the momentum

• De Broglie suggested that Eq. (3.1) is a completely general one that applies to material

particles as well as to photons.

• Part of de Broglie’s inspiration came from Bohr’s theory of the hydrogen

atom, in which the electron is supposed to follow only certain orbits

around the nucleus.

• Two years later Erwin Schrödinger used the concept of de Broglie waves

to develop a general theory that he and others applied to explain a wide

variety of atomic phenomena.

• The existence of de Broglie waves was confirmed in diffraction

experiments with electron beams in 1927.

• The momentum of a particle of mass m and velocity v is p=γmv, and its

de Broglie wavelength is accordingly

γ is the relativistic factor

Louis de

Broglie

1892-1987

Nobel Prize in

Physics in 1929

• In certain situations, a moving body resembles a wave and in others it resembles a particle.

◦ Which set of properties is most apparent depends on how its de Broglie wavelength

compares with its dimensions and the dimensions of whatever it interacts with.

Example 3.1&3.2

Find the de Broglie wavelengths of (a) a 46-g golf ball with a velocity of 30 m/s, and (b) an electron

with a velocity of 10

m/s.

Find the kinetic energy of a proton whose de Broglie wavelength is 1.000 fm=1.000x10

-15

m, which

is roughly the proton diameter.

3.2 WAVES OF WHAT?

• Waves of probability.

◦ In water waves, the quantity that varies periodically is the height of the water surface.

◦ In sound waves, it is pressure.

◦ In light waves, electric and magnetic fields vary.

◦ What is it that varies in the case of matter waves?

• The quantity whose variations make up matter waves is called the wave

function, symbol Ψ.

• Born came the basic concept that the wave function Ψ of a particle is

probability of finding it.

• The wave function Ψ itself, however, has no direct physical significance.

By itself cannot be an observable quantity.

• |Ψ|

must represent probability density for electrons (or other particles).

• For this purpose, atomic scattering (collisions of atoms with various

particles) processes suggested.

Max Born

1882–1970

Nobel Prize in

Physics in 1954

• |Ψ|

; the square of the absolute value of the wave function, which is known as probability

density.

• The probability of experimentally finding the body described by the wave function at the

point x, y, z, at the time t is proportional to the value of |Ψ|

there at t.

◦ A large value of |Ψ|

means the strong possibility of the body’s presence,

◦ while a small value of |Ψ|

means the slight possibility of its presence.

• There is a big difference between the probability of an event and the event itself.

◦ Although we can speak of the wave function Ψ that describes a particle as being spread

out in space, this does not mean that the particle itself is thus spread out.

◦ When an experiment is performed to detect electrons, for instance, a whole electron is

either found at a certain time and place or it is not; there is no such thing as a 20 percent

of an electron.

◦ However, it is entirely possible for there to be a 20 percent chance that the electron be

found at that time and place, and it is this likelihood that is specified by |Ψ|

• While the wavelength of the de Broglie waves associated with a moving body is given by

the simple formula =h/γmv, to find their amplitude Ψ as a function of position and time is

often difficult.

3.3 DESCRIBING WAVE

• A general formula for waves.

• How fast do de Broglie waves travel? Since we associate a de Broglie wave with a moving

body, we expect that this wave has the same velocity as that of the body??

• Call the de Broglie wave velocity as v

=ν

◦ Because the particle velocity v must be less than the velocity of light c, the de Broglie

waves always travel faster than light!

◦ In order to understand this unexpected result, we must look into the distinction between

phase velocity (wave velocity) and group velocity.

• How waves are described mathematically?

• Fig. 3.1. If we choose t=0 when the displacement y of the string at x=0 is a maximum, its

displacement at any future time t at the same place is given by the formula

◦ where A is the amplitude of the vibrations (that is, their maximum displacement on

either side of the x axis) and ν their frequency.

◦ Equation (3.4) tells us what the displacement of a single point on the string is as a

function of time t.

Figure 3.1 (a) The appearance of a wave in a stretched

string at a certain time. (b) How the displacement of a

point on the string varies with time.

Figure 3.2 Wave propagation

• A complete description of wave motion in a stretched string, however, should tell us what y

is at any point on the string at any time.

• To obtain such a formula, let us imagine that we shake the string at x=0 when t=0, so that a

wave starts to travel down the string in the +x direction (Fig. 3.2).

Since the wave speed v

is given by v

=ν, we have

• Most widely used description of a wave, however, is still another form of Eq. (3.5). The

quantities angular frequency  and wave number k are defined by the formulas.

◦ The unit of  is the radian per second and that of k is the radian per meter.

◦ Angular frequency gets its name from uniform circular motion, where a particle that

moves around a circle ν times per second sweeps out 2ν rad/s.

◦ The wave number is equal to the number of radians corresponding to a wave train 1 m

long, since there are 2 rad in one complete wave.

3.4 PHASE AND GROUP VELOCITIES

• A group of waves need not have the same velocity as the waves themselves.

• The amplitude of the de Broglie waves that correspond to a moving body reflects the

probability that it will be found at a particular place at a particular time.

• de Broglie waves cannot be represented simply by a formula resembling Eq. (3.9),

which describes an indefinite series of waves all with the same amplitude A.

• Instead, we expect the wave representation of a moving body

to correspond to a wave packet, or wave group, like that

shown in Fig. 3.3, whose waves have amplitudes upon which

the likelihood of detecting the body depends.

Figure 3.3 A wave group.

• A familiar example of how wave groups

come into being is the case of beats.

• If the original sounds have frequencies of,

say, 440 and 442 Hz, we will hear a

fluctuating sound of frequency 441 Hz with

two loudness peaks, called beats, per second.

• The production of beats is illustrated in Fig.

3.4.

Figure 3.4 Beats are produced by the superposition of two waves with different frequencies

• A wave group: a superposition of individual waves of different wavelengths whose

interference with one another results in the variation in amplitude that defines the group

shape.

◦ If the velocities of the waves are the same, the velocity with which the wave group

travels is the common phase velocity.

◦ However, if the phase velocity varies with wavelength, the different individual waves do

not proceed together. This situation is called dispersion.

◦ As a result, the wave group has a velocity different from the phase velocities of the

waves that make it up. This is the case with de Broglie waves.

• The de Broglie wave group associated with a moving body travels with the same velocity as

the body.

The phase velocity v

of de Broglie waves is, as we found earlier,

• This exceeds both the velocity of the body v and the velocity of light c, since v<c. However,

has no physical significance because the motion of the wave group, not the motion of the

individual waves that make up the group, corresponds to the motion of the body, and v

< c

as it should be.

• The fact that v

>c for de Broglie waves therefore does not violate special relativity.

Example 3.3

An electron has a de Broglie wavelength of 2.00 pm =2.00x10

-12

m. Find its kinetic energy and the

phase and group velocities of its de Broglie waves.

Electron Microscopes

• The wave nature of moving electrons is the basis of the electron

microscope, the first of which was built in 1932.

• The resolving power of any optical instrument, which is limited by

diffraction, is proportional to the wavelength of whatever is used to

illuminate the specimen.

• In the case of a good microscope that uses visible light, the

maximum useful magnification is about 500; higher magnifications

give larger images but do not reveal any more detail.

• Fast electrons, however, have wavelengths very much shorter than

those of visible light and are easily controlled by electric and

magnetic fields because of their charge.

Figure 3.5 Because the wavelengths of the fast electrons in an electron microscope are shorter than those of the light

waves in an optical microscope, the electron microscope can produce sharp images at higher magnifications.

• In an electron microscope, current-carrying coils produce magnetic fields that act as lenses

to focus an electron beam on a specimen and then produce an enlarged image on a

fluorescent screen or photographic plate (Fig. 3.5).

◦ The technology of magnetic “lenses” does not permit the full theoretical resolution of

electron waves to be realized in practice.

◦ For instance, 100-keV electrons have wavelengths of 0.0037 nm, but the actual

resolution they can provide in an electron microscope may be only about 0.1 nm.

• However, this is still a great improvement on the ~200-nm resolution of an optical

microscope, and magnifications of over 1,000,000 have been achieved with electron

microscopes.

3.5 PARTICLE DIFFRACTION

• An experiment that confirms the existence of de Broglie waves.

• A wave effect with no analog in the behavior of Newtonian particles is diffraction.

• In 1927 Clinton Davisson and Lester Germer in the United States and G. P. Thomson in

England independently confirmed de Broglie’s hypothesis by demonstrating that electron

beams are diffracted when they are scattered by the regular atomic arrays of crystals. (All

three received Nobel Prizes for their work)

• Classical physics predicts that the scattered electrons will emerge in all directions with only

a moderate dependence of

◦ their intensity on scattering angle and

◦ even less on the energy of the primary electrons.

• Using a block of nickel as the target, Davisson and Germer verified

these predictions.

• In the middle of their work an accident occurred that allowed air to

enter their apparatus and oxidize the metal surface. To reduce the

oxide to pure nickel, the target was baked in a hot oven.

• After this treatment, the target was returned to the apparatus and the

measurements resumed.

Figure 3.6 The Davisson-Germer experiment.

• Now the results were very different. Instead of a continuous variation of scattered electron

intensity with angle, distinct maxima and minima were observed whose positions depended

upon the electron energy!

• Typical polar graphs of electron intensity after the accident are shown in Fig. 3.7.

Figure 3.7 Results of the Davisson-Germer

experiment, showing how the number of scattered

electrons varied with the angle between the

incoming beam and the crystal surface. The Bragg

planes of atoms in the crystal were not parallel to

the crystal surface, so the angles of incidence and

scattering relative to one family of these planes

were both 65° (see Fig. 3.8).

• The method of plotting is such that the intensity at any angle is proportional to the distance

of the curve at that angle from the point of scattering.

• If the intensity were the same at all scattering angles, the curves would be circles (scattered

electrons emerge in all directions) centered on the point of scattering.

• De Broglie’s hypothesis suggested that electron waves were being diffracted by the target,

much as x-rays are diffracted by planes of atoms in a crystal.

• Let us see whether we can verify that de Broglie waves are responsible for the findings of

Davisson and Germer. In a particular case,

◦ a beam of 54-eV electrons was directed perpendicularly at the

nickel target and

◦ a sharp maximum in the electron distribution occurred at an angle

of 50° with the original beam.

◦ The angles of incidence and scattering relative to the family of

Bragg planes shown in Fig. 3.8 are both 65°.

Figure 3.8 The diffraction of the de Broglie waves by the

target is responsible for the results of Davisson and Germer.

• The spacing of the planes in this family, which can be measured by x-ray diffraction, is

0.091 nm. The Bragg equation for maxima in the diffraction pattern is

• which agrees well

with the observed

wavelength of 0.165

nm. The Davisson-

Germer experiment

thus directly verifies

de Broglie’s

hypothesis of the

wave nature of

moving bodies.

3.6 PARTICLE IN A BOX

• Why the energy of a trapped particle is quantized.

• The wave nature of a moving particle leads to some remarkable consequences when the

particle is restricted to a certain region of space instead of being able to move freely.

• The simplest case is that of a particle that bounces back and forth between the walls of a

box, as in Fig. 3.9.

• From a wave point of view, a particle trapped in a box is like a

standing wave in a string stretched between the box’s walls.

• In both cases the wave variable (transverse displacement for the string,

wave function for the moving particle) must be 0 at the walls, since

the waves stop there.

Figure 3.9 A particle confined to a box of width L. The particle is

assumed to move back and forth along a straight line between the walls of the box.

• The possible de Broglie wavelengths of the particle in the box

therefore are determined by the width L of the box, as in Fig. 3.10.

• The kinetic energy of a particle of momentum mv is

Figure 3.10 Wave functions of a particle trapped in a box L wide.

• The particle has no potential energy in this model, the only energies it can have are

• Each permitted energy is called an energy level, and the integer n that specifies an energy

level E

is called its quantum number.

• We can draw three general conclusions from Eq. (3.18). These conclusions apply to any

particle confined to a certain region of space (even if the region does not have a well-defined

boundary), for instance an atomic electron held captive by the attraction of the positively

charged nucleus.

1 A trapped particle cannot have an arbitrary energy, as a free particle can. The fact of its

confinement leads to restrictions on its wave function that allow the particle to have only

certain specific energies and no others.

2 A trapped particle cannot have zero energy. Since the de Broglie wavelength of the particle

is =h/mν, a speed of v=0 means an infinite wavelength. But there is no way to reconcile an

infinite wavelength with a trapped particle, so such a particle must have at least some kinetic

energy.

3 Because Planck’s constant is so small-only 6.63x10

-34

J.s-quantization of energy is

considerable only when m and L are also small. This is why we are not aware of energy

quantization in our own experience.

Example 3.4&3.5

An electron is in a box 0.10 nm across, which is the order of magnitude of atomic dimensions.

Find its permitted energies.

A 10-g marble is in a box 10 cm across. Find its permitted energies.

3.7 UNCERTAINITY PRINCIPLE I

• We cannot know the future because we cannot know the present

• Wave group of Fig. 3.3: The particle that corresponds to this wave group may be located

anywhere within the group at a given time. The probability density |Ψ|

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 |Ψ|

is not actually 0.

• The narrower its wave group, the more precisely a particle’s

position can be specified (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 =h/γmv, the particle’s momentum γmv

is not a precise quantity.

• 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.

• 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.

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 momentum) cannot be established because 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.

• Thus, we have the uncertainty principle.

• It is impossible to know both the exact position and exact momentum of

an object at the same time.

• This principle, which was discovered by Werner Heisenberg in 1927, is

one of the most significant of physical laws.

• The de Broglie wavelength of a particle of momentum p

is =h/p and the corresponding wave number is

• In terms of wave number, the particle’s momentum

is therefore

•

• Hence an uncertainty k in the wave number of the de Broglie waves

associated with the particle results in an uncertainty p in the particle’s

momentum according to the formula

Werner Karl

Heisenberg

1901-1976

Nobel Prize in

Physics in 1932

• This equation states that the product of the uncertainty x in the position of an object at

some instant and the uncertainty p in its momentum component in the x direction at the

same instant is equal to or greater than h/4.

◦ If we arrange matters so that x is small, corresponding to a narrow wave group, then p

will be large.

◦ If we reduce p in some way, a broad wave group is inevitable and x will be large.

• Since we cannot know exactly both where a particle is right now and what its momentum is,

we cannot say anything definite about where it will be in the future or how fast it will be

moving then.

• We cannot know the future for sure because we cannot know the present for sure.

• But we can still say that the particle is more likely to be in one place than another and that its

momentum is more likely to have a certain value than another.

Example 3.6

A measurement establishes the position of a proton with an accuracy of ±1.00x10

-11

m. Find the

uncertainty in the proton’s position 1.00 s later. Assume v<<c.

Gaussian Function

• When a set of measurements is made of some quantity

x in which the experimental errors are random, the

result is often a Gaussian distribution whose form is

the bell-shaped curve shown in Fig. 3.15.

• The standard deviation  of the measurements is a

measure of the spread of x values about the mean of

, where  equals the square root of the average of

the squared deviations from x

Figure 3.15 A Gaussian distribution. The probability of finding a value of x is given by the Gaussian function f(x).

The mean value of x is x

, and the total width of the curve at half its maximum value is 2.35, where  is the

standard deviation of the distribution. The total probability of finding a value of x within a standard deviation of x

is equal to the shaded area and is 68.3 percent.

• If N measurements were made,

• The width of a Gaussian curve at

half its maximum value is 2.35.

• The Gaussian function f (x) that describes the above curve is given by

where f(x) is the probability that the value x be found in a particular measurement. Gaussian

functions occur elsewhere in physics and mathematics as well.

• The probability that a measurement lie inside a certain range of x values, say between x

and

, is given by the area of the f (x) curve between these limits. This area is the integral

• An interesting question is what fraction of a series of measurements has values within a

standard deviation of the mean value x

. In this case x

= x

-  and x

= x

+ , and

• Hence 68.3 percent of the measurements fall in this interval, which is shaded in Fig. 3.15. A

similar calculation shows that 95.4 percent of the measurements fall within two standard

deviations of the mean value.

3.8 UNCERTAINITY PRINCIPLE II

• A particle approach gives the same result.

• The uncertainty principle can be arrived at from the point of view of the particle properties

of waves as well as from the point of view of the wave properties of particles.

• To measure the position and momentum of an object at a certain moment, we must touch it

with something that will carry the required information back to us.

• Suppose we look at an electron using light of

wavelength , as in Fig. 3.17.

• Each photon of this light has the momentum h/.

When one of these photons bounces off the electron

(which must happen if we are to “see” the electron),

the electron’s original momentum will be changed.

• The exact amount of the change p cannot be

predicted, but it will be of the same order of

magnitude as the photon momentum h/. Hence

Figure 3.17 An electron cannot be observed without changing its momentum.

• The longer the wavelength of the observing photon, the smaller the uncertainty in the

electron’s momentum.

• Because light is a wave phenomenon as well as a particle phenomenon, we cannot expect to

determine the electron’s location with perfect accuracy regardless of the instrument used.

• A reasonable estimate of the minimum uncertainty in the measurement might be one photon

wavelength, so that

◦ The shorter the wavelength, the smaller the uncertainty in location. However, if we use

light of short wavelength to increase the accuracy of the position measurement, there

will be a corresponding decrease in the accuracy of the momentum measurement

because the higher photon momentum will disturb the electron’s motion to a greater

extent.

◦ Light of long wavelength will give a more accurate momentum but a less accurate

position.

◦ Combining Eqs. (3.23) and (3.24) gives

• This result is consistent with Eq. (3.22).

3.9 APPLYING THE UNCERTAINITY PRINCIPLE

• A useful tool, not just a negative statement.

• Planck’s constant h is so small that the limitations imposed by the uncertainty principle are

significant only in the realm of the atom.

Example 3.7

A typical atomic nucleus is about 5.0X10

-15

m in radius. Use the uncertainty principle to place a

lower limit on the energy an electron must have if it is to be part of a nucleus.

Example 3.8

A hydrogen atom is 5.3x10

-11

m in radius. Use the uncertainty principle to estimate the minimum

energy an electron can have in this atom.

Energy and Time

• Another form of the uncertainty principle concerns energy and time. We might wish to

measure the energy E emitted during the time t in an atomic process.

• If the energy is in the form of em waves, the limited time available restricts the accuracy

with which we can determine the frequency ν of the waves.

• Let us assume that the minimum uncertainty in the number of waves we count in a wave

group is one wave.

• Since the frequency of the waves under study is equal to the number of them we count

divided by the time interval, the uncertainty ν in our frequency measurement is

The corresponding energy uncertainty is

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

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 1.0x10

-8

s. Find the inherent uncertainty in the frequency of the photon.