Chapter 3

Wave Properties of Particles

3 Wave Properties of Particles

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3 Wave Properties of Particles: Discovery

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• 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: Matter Waves

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• A moving body behaves in certain ways as it

has a wave nature.

• A photon of light of frequency ν has the

momentum

(Photon wavelength)

• 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.

• De Broglie suggested that Photon wavelength equation is a completely

general one that applies to material particles as well as to photons.

Louis de Broglie

1892-1987

Nobel Prize in

Physics in 1929

3. 1 De Broglie Waves: de Broglie wavelength

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• 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:

(de Broglie wavelength )

• 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.

3. 1 De Broglie Waves

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Example 3.1

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.

3. 1 De Broglie Waves

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Example 3.2

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? Wave function

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• 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

3. 2 Waves of What? Probability Density

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• |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.

3. 2 Waves of What?

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• 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: Phase & Group velocities

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• 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

• De Broglie phase velocity

(de Broglie phase velecity )

• 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?

3.3 Describing Wave: Wave propagation

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• 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.

(3.4)

Figure 3.2 Wave propagation

3.3 Describing Wave: Wave formula

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• 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).

(3.5 Wave formula)

Since the wave speed v

is given by v

=ν, we have

(Wave formula)

• 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.

(Angular frequncy)

(Wave number)

• The unit of  is the radian per second and unit 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.

(3.9 Wave formula)

3. 4 Phase And Group Velocities: Wave packet

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• 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.

3. 4 Phase And Group Velocities: Beats

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• 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.

3. 4 Phase And Group Velocities: A wave group

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• 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.

(De Broglie group velocity )

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,

(De Broglie phase velocity )

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

However, v

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.

3. 4 Phase And Group Velocities

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Example 3.3

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

-12

Find its kinetic energy and the phase and group velocities of its de

Broglie waves.

3 Electron Microscopes

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• 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

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.

3 Electron Microscopes

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• 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

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• 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 & Lester Germer and G. P. Thomson

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

3.5 Particle Diffraction: The Davisson-Germer experiment

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• 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!

3.5 Particle Diffraction: Polar graphs

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• 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.

3.5 Particle Diffraction

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• Let us see whether we can verify that de Broglie

waves are responsible for the findings of Davisson

and Germer. In a particular case,

Figure 3.8 The diffraction of the

de Broglie waves by the target is

responsible for the results of

Davisson and Germer.

• 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°.

• 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 here d=0.091 nm, θ=65° and n=1.

• the de Broglie wavelength of the diffracted electrons is λ=0.165 nm.

Now we use de Broglie’s formula to find the expected

wavelength of the electrons.

3.5 Particle Diffraction

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• The electron kinetic energy of 54 eV is small compared with its rest

energy mc

of 0.51 MeV, so we can let γ=1. Since KE=1/2mv

the

electron momentum mv is

The electron wavelength is therefore

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

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• 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

(Fig. 3.10).

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.

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. (Boundary Condition)

• 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.

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

(De Broglie wavelengths of trapped particle)

3.6 Particle In A Box

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• The kinetic energy of a particle of momentum mv is

(3.18 Particle in a box)

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, 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 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.

3.6 Particle In A Box

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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.

The minimum energy the marble can have is

5.5x10

-64

J, corresponding to n=1. A marble

with this kinetic energy has a speed of only

3.3x10

-31

m/s and therefore cannot be

experimentally 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 n=10

3.7 Uncertainity Principle I: Wave approach

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• 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.

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.

• 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.

3.7 Uncertainity Principle I : Wave approach

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• But where is the particle located? 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.

Werner Karl

Heisenberg

1901-1976

Nobel Prize in

Physics in 1932

• 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

(Uncertainity principle)

3.7 Uncertainity Principle I: Wave approach

11 March 2018 30MSE 228 Engineering Quantum Mechanics © Dr.Cem Özdoğan

• 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.

(Uncertainity principle)

3.7 Uncertainity Principle I: Wave approach

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.

3 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 x

, 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.

(Standart deviation)

The Gaussian function f (x) that describes the

curve is given by

where f(x) is the probability that the value x be found in

a particular measurement.

(Gaussian function)

3 Gaussian Function

• The probability that a measurement lie inside a certain range of x

values, say between x

and x

, 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 (2) of the mean

value.

3.8 Uncertanity Principle II: Particle approach

• 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.

Figure 3.17 An electron cannot be observed

without changing its momentum.

• 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

(3.23)

3.8 Uncertanity Principle II: Particle approach

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

uncertainty in the electron’s momentum.

• A reasonable estimate of the minimum uncertainty in the measurement

might be one photon wavelength, so that

(3.24)

• 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

• Uncertanity 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.

3.9 Applying Uncertanity Principle

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.

3.9 Applying Uncertanity Principle

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.

3.9 Applying Uncertanity Principle

Example 3.9

An “excited” atom gives up its excess energy by emitting a photon of

characteristic frequency. 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.

3.9 Applying Uncertanity Principle: Energy and Tme

• 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

(Uncertainties in energy and time)

• A more precise calculation based on the nature of wave groups

changes this result to

3 Solved Problems

1. Show that de Broglie wavelength of electron with having kinetic

energy value of 100 keV is 0.0037.

3 Solved Problems

2. An electron moves in the x direction with a speed of 3.6x10

m/s.

We can measure its speed to a precision of 1%. With what precision

can we simultaneously measure its x coordinate?

3 Solved Problems

3. What kinetic energy (in electron volts) should neutrons have if they

are to be diffracted from crystals with interatomic distance of 1.00

Å?

3 Solved Problems

4. Show that the spread of velocities caused by the uncertainty

principle does not have measurable consequences for macroscopic

objects (objects that are large compared with atoms) by considering

a 100-g racquetball confined to a room 15 m on a side. Assume the

ball is moving at 2.0 m/s along the x axis

3 Solved Problems

5. Estimate the kinetic energy of an electron confined within a nucleus

of size 1.0x10

m by using the uncertainty principle.

3 Solved Problems

6. Although an excited atom can radiate at any time from t=0 to t=,

the average time after excitation at which a group of atoms radiates

is called the lifetime, , of a particular excited state. (a) If =1.0x10

s (a typical value), use the uncertainty principle to compute the line

width ν of light emitted by the decay of this excited state. (b) If the

wavelength of the spectral line involved in this process is 500 nm,

find the fractional broadening ν/ν.

3 Solved Problems

7. (a) A charged pi meson has a rest energy of 140MeV and a lifetime

of 26 ns. Find the energy uncertainty of the pi meson, expressed in

MeV and also as a fraction of its rest energy. (b) Repeat for the

uncharged pi meson, with a rest energy of 135MeV and a lifetime of

8.3x10

-17

s. (c) Repeat for the rho meson, with a rest energy of

765MeV and a lifetime of 4.4x10

-24

3 Solved Problems

8. Find the quantized energy levels of an electron constrained to move

in a one-dimesional atom of size 0.1 nm.

Chapter 3 Wave Properties of Particles

Additional Materials