Notes_ Quantum Theory of Light and Matter Waves

Hwa Chong Institution – H3 Physics 2009

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Quantum Theory of Light and Matter Waves

Learning Outcomes

Quantum Theory of Light

Candidates should be able to:

i) show an understanding of what is meant by ideal blackbody radiation.

ii) apply Wien’s displacement law in related situations or to solve problems.

iii) discuss qualitatively the failure of classical theory to explain blackbody radiation at high

frequencies.

iv) describe qualitatively Planck’s hypothesis of blackbody radiation.

v) derive, using relativistic mechanics, the equation p = h/λ for a photon and use the

expression in related situations or to solve problems.

vi) show an understanding of the Compton shift effect and how it supports the concept of the

photon.

vii) derive, using relativistic mechanics, the Compton shift equation and use the equation in

related situations or to solve problems.

viii) show an understanding of discrete electron energy levels in isolated atoms (Bohr theory)

and use the terms absorption, spontaneous emission and stimulated emission to describe

photon processes involving electron transitions between energy levels, including the typical

lifetime of such transitions.

ix) recall and use the equation hf = E1 – E2 to solve problems.

x) apply the relation Nx = Noexp(–(Ex – Eo)/kT) to explain and solve problems on the

population distribution of atoms with energy Ex .

xi) describe the process of population inversion and explain why this cannot be achieved with

just two energy levels.

xii) use population inversion and stimulated emission to explain the action of a laser, using the

He-Ne laser as a specific example. (Details on the structure and operation of the laser are

not required.)

Matter Waves

i) show a qualitative understanding of the concept of wave-particle duality and the principle of

complementarity.

ii) show an understanding that a particle can be described using a wave function Ψ and give a

simple mathematical form of the free particle wave function.

iii) discuss qualitatively the probabilistic interpretation of the wave function and state that the

square of the wave function amplitude IΨ I2

gives the probability density.

iv) show an understanding of the normalisation of a wave function.

v) show an understanding that the resolving power of optical instruments is determined by the

wavelength of the radiation and use this to explain how electron microscopes can achieve

higher resolution than normal optical microscopes.

vi) describe the use of X-ray diffraction to probe crystal structures.

vii) recall and apply Bragg’s equation nλ = 2d sinθ to solve problems.

viii) extend the use of Bragg’s equation to electron diffraction in probing the surfaces of solids.

(Only non-relativistic cases are considered.)


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A Brief History

Throughout scientific history, light has been viewed alternately as either waves or particles.

Dutch scientist Christiaan Huygens perceived light as waves propagating in a "medium" known as

the ether. While it is able to explain the wave behavior of light (interference and diffraction), it

eventually failed as ether is not detected. Then Isaac Newton came up with the corpuscular

theory of light which imagined light as consisting of small particles, while it can explain reflection

of light, it failed to account for interference and diffraction. Furthermore, English scientist

Thomas Young conducted a series of experiments, including the famous Young's double-slits

experiment that supported the wave nature of light. And when James Clark Maxwell formulated

the electromagnetic theory, establishing that light is not a disturbance of ether, but rather

fluctuations in the electromagnetic field, the scientific community had more or less embraced

the wave theory of light. However, by then there are new observations which the

electromagnetic theory was unable to explain, two phenomena that have puzzled scientists at

the end of the 19th century: the radiation spectrum of a blackbody and the photoelectric effect.

Phillip Lenard observed that electrons emitted from a metal surface when light is incident on it,

depended on the frequency of light, not the intensity of the light as predicted by classical physics.

Indeed, no electrons were emitted from the surface if the frequency of the incident radiation was

less than some threshold value that depended on the metal.

The shape of this blackbody spectrum had already been determined experimentally by the end

of the nineteenth century. However, a satisfactory theory of blackbody radiation should

provide a precise mathematical expression for the blackbody spectrum.

1900 – the history of quantum mechanics false starts with Max Planck

From 1895 onwards Planck tried to find a way to derive the blackbody radiation law after

partially successful attempts by Wien and Lord Kelvin.

Planck’s model went through successive refinements in his attempt to obtain a perfect match

between theory and experiment. He eventually succeeded, but only at the cost of incorporating

‘energy elements’ in his model. In this model, the total energy of all the oscillators in a blackbody

is divided into a finite (but very large) number of equal (but tiny) parts, determined by a constant

of nature, which he labelled h. This became known as Planck’s constant. In a letter to R. W. Wood,

Planck called his limited postulate “an act of desperation.”

1905 – The photoelectric effect

Lenard’s experiment was explained by Einstein in 1905 when he modeled light as photons with

energy hfE = . It was Einstein who made the bold assumption that light energy could also be

delivered as ‘packets’ we now call ‘photons’ and gave physical meaning to Planck’s constant.


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1913 – Bohr explains why atoms don’t implode

Ernest Rutherford had earlier proposed an atomic model in which he envisaged the atom as

having a central positive nucleus surrounded by negative orbiting electrons. Unfortunately,

Rutherford's model faced a very fundamental problem. Maxwell's electromagnetic theory

predicted that a charge undergoing acceleration will radiate EM waves, losing energy in the

process. This means that the orbiting electrons, which undergo centripetal acceleration, will lose

energy through EM radiation and rapidly spiral into the nucleus.

The Rutherford-Bohr model of the atom resolved the issue by making some new postulates.

Bohr speculated that each allowed orbit only had ‘room’ for a certain number of electrons so

that electrons further out from the nucleus cannot jump inwards if the inner orbits are already

full. The electrons closest to the nucleus were simply forbidden to jump right into the centre of

the atom. Although it is not yet the full quantum model, Bohr’s model was credible because it

predicted the positions of spectral lines for hydrogen – confirmed by observation.

1923 – Compton, the light quantum has not just energy, but momentum as well

In 1923 Arthur Compton (1892-1962) was scattering x-rays off graphite. He found that some of

the scattered radiation has smaller frequencies than the incident radiation which is dependent

on the angle of scattering. He could only explain his observations if he treated x-ray photons as

particles obeying the conservation of momentum and energy in their collisions with stationary

electrons. So his experiment and explanation confirmed that x-ray photons carry not only

‘energy’ but also ‘momentum’ of chf .

1924 - Louis de Broglie, whose PhD earned him a Nobel prize

de Broglie combined two equations for the photon ( hfE = from the photoelectric effect and

pcE = from relativity) and expressed the result in terms of frequency hpcf = . He could

substitute this into λ= fc , re-arranged in terms of phpchcfc ===λ .

What de Broglie’s equation tells us is that everything has dual wave-particle character but

because Planck’s constant is so small, the ‘waviness’ of an everyday object is so utterly tiny that

it can never be detected. Niels Bohr summarized the situation in his principle of

complementarity,

The wave and particle models are complementary; if a measurement proves the wave character

of radiation or matter, then it is impossible to prove the particle character in the same

measurement, and vice versa.


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Section One : Blackbody Radiation

1.1 Blackbody Radiation

All bodies absorb and radiate thermal energy. Isolated atoms (gases) produce discrete spectra

(emission or absorption) which arise from electronic transitions between discrete energy levels.

Dense bodies such as liquids and solids radiate continuous spectra (of any form and shape), in

which all frequencies are present, due to mutual interactions between the atoms in close

proximity with one another (Band Theory).

Different bodies have different rates of emission and absorption. A perfect blackbody can be

visualized as an ideal body that absorbs all electromagnetic (EM) radiation landing upon it,

regardless of frequency. More importantly, a blackbody is also the best emitter of radiation, and

the rate of emission (total energy per unit time per unit area) is a function of the absolute

temperature alone, as expressed in the Stefan-Boltzmann’s Law:

4TσI =

where 4-28K m W 10675

-.σ ×= is called the Stefan-Boltzmann constant.

A perfect blackbody can be modeled by a hollow body with only a small hole that allows entry

into the cavity inside. All EM radiation that enters the hole will be trapped inside the cavity and

will be absorbed. At the same time, the blackbody will also emit EM radiation of all possible

frequencies, which is characteristic of the radiating system only and not dependent upon the

type of radiation which is incident upon it. This is why blackbody radiation is also known as

cavity radiation. This model is especially useful in the attempt to formulate a theory for

blackbody radiation. The radiated energy can be considered to be produced by standing waves

or resonant modes of the cavity (see Fig. 1.2).

The spectrums of the radiation emitted from a blackbody at a few different temperatures are

shown in Fig. 1.1. Radiation of all wavelengths is present in the spectrums, but certain

wavelengths tend to dominate, denoted by the “peak” of the curve. The position of the “peak”

depends solely on the absolute temperature T of the body. The peak wavelength λmax can be

found using the Wien’s Displacement Law:

Km 10 x 2.898 -3max =Tλ

As T increases, the spectrum shifts towards shorter wavelengths.

We can therefore deduce the temperature of a body hot enough to be luminous by observing

the dominant colour of the radiation. For the body in this case, at 5000 K, the peak happens to

be in the visible range, around the yellow-orange region. Our more yellowish sun has a (surface)

temperature of about 6000 K, whereas the cool blue of the distant star Vega indicates a much


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hotter 10000 K. In 1965, Dicke, Penzias & Wilson used the blackbody spectrum of the cosmic

background radiation (CBR) to work out its temperature.

Fig. 1.1 Fig. 1.2

Exercise: The peak of a cosmic blackbody radiation is measured to be at max

λ ≈ 1 mm. What is

the temperature of the cosmic blackbody radiation? (Ans : 2.9K)

Exercise: At what wavelength does the human body emit its maximum temperature radiation?

List assumptions you make in arriving at an answer.

(Ans : 935nm)

1.2 Rayleigh-Jeans Law

British physicists John Rayleigh and James Jeans were able to derive a formula based on purely

classical considerations to explain the shape of the spectrum (Rayleigh-Jeans Law). However, the

formula has very significant disagreement with the observed distribution at high frequencies.

He considered light enclosed in a rectangular box with perfectly reflecting sides. Such a box has a

series of possible normal modes for em waves. It seems reasonable to assume that the

distribution of energy among various modes is given by the equipartition principle, (average

energy per mode, )kTE = . However, the number of resonant modes per frequency interval

increases as the frequency increases ( 2)( ffn ∝ ). Hence, classical physics predicts that all the

energy in the blackbody spectrum should be at high frequencies – and this became known as the

‘ultraviolet catastrophe’ (Fig. 1.2).

4

)(2

λ

π kcT I(f) =

Planck solved this problem by introducing the

concept that energy be quantised.

Fig. 1.2


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1.3 Planck Hypothesis

Max Planck was able to derive a formula (Planck radiation formula) that fits in with the observed

spectrum if he made the assumption that the energy levels of the blackbody are not continuous.

Planck assumed that only adding or removing a discrete amount or quantum of energy could

change the energy. He supposed that the magnitude of the energy quantum must be

proportional to the frequency of the wave ( hfE =∆ ).

The quantum argument assumes that the energy of a standing wave cannot increase

continuously under thermal agitation, but must climb a kind of ladder on which the distance

between the rungs depends on the wavelength. At long wavelengths, the rungs are close

together, many quanta will be excited and as we have seen, the classical case of continuous

energy is approached. At the other extreme, when the wavelength is very short, the rungs on the

ladder are further apart and the probability of even one quantum being excited is very low. In

this situation, the thermodynamic average energy of a standing wave is correspondingly very low.

The ultraviolet catastrophe would be avoided, if instead of the average energy per mode being

kT for all modes as required by classical physics, this average energy actually decreased rapidly

with increasing frequency:

1−

=kT

hf

e

hfE

The Planck radiation can then be expressed in terms of spectral energy density (energy per unit

volume per unit frequency) as:

where h is Planck constant = 341063.6

−× Js, c is the speed of light, k =1.38x10-23 J/K is

Boltzmann constant, T is the absolute temperature, λ is the wavelength, and f is the frequency.

Incidentally, in keeping with the correspondence principle, Stefan’s law and Wein’s displacement

law can be derived from the Planck formula.

( )

1

12,

5

2

−

=

kT

hc

e

hcTI

λλ

πλ


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Exercise

(a) What is the surface temperature of the Sun given that the peak of its blackbody spectrum is

maxλ = 5.1 x 10

-7 m? (Ans: 5682K)

(b) Given that it takes sunlight about 8 minutes to reach us, use Stefan’s law 4

T TR σ= to work

out the solar constant (the solar radiation per unit area intercepted by the Earth’s disc). The Sun

have a mean radius of 6.95 x 108

m.

(Ans: 1400Wm-2

)

(c) The average rate of solar radiation incident per unit area on the earth is or 338 Wm-2

. Attached to

the roof of a house are three solar panels, each 1 m x 2 m. Assume the equivalent of 4 hrs of

normally incident sunlight each day, and that all the incident light is absorbed and converted to

heat. How many litres of water can be heated from 20°C to 80°C each day?

(Ans: 0.116l)

Exercise: Describe qualitatively how Stefan-Boltzmann law and Wien displacement

law can be derived from Planck radiation law.

(Ans: differentiate Planck Law wrt λ to get Wien’s Law)

(Ans: integrate Planck Law wrt λ to get Stefan-Boltzmann law)

Exercise: Show that Planck’s result for average energy per mode

1−

=

kT

hf

e

hfE reduces to the

classical result kTE = at low frequency. (Ans: for small x, e x ∼ 1 – x)


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Section Two : Atomic Model

2.1 Bohr Model of the Atom

In 1913, Niels Bohr improved upon the Rutherford planetary model of an atom by making a

number of assumptions in order to combine the new quantum ideas of Planck and Einstein with

the traditional description of a particle in uniform circular motion.

1. An electron in an atom moves in a circular orbit about the nucleus under the influence of

the Coulomb attraction between the electron and the nucleus, obeying the laws of

classical mechanics.

2. Instead of the infinity of orbits which would be possible in classical mechanics, it is only

possible for an electron to move in an orbit for which its orbital angular momentum

L = mvr = n , for the nth orbital level.

3. Despite the fact that it is constantly accelerating, an electron moving in such an allowed

orbit does not radiate electromagnetic energy. Thus, its total energy E remains constant.

4. Electromagnetic radiation is emitted if an electron, initially moving in an orbit of total

energy , discontinuously changes its motion so that it moves in an orbit of total

energy . The frequency of the emitted radiation is equal to the quantity

divided by h.

The Bohr model is certainly not a full quantum mechanical description of the atom. The full

quantum mechanical model, or electron cloud model, describes the electron as a

three-dimensional shape ("cloud") where there is at least a minimal probabilty of finding the

electron.

2.2 Electronic Transitions

To incorporate Einstein's photon concept, Bohr theorized that a photon is emitted only when an

electron jumps orbits from a larger one with a higher energy to a smaller

one with a lower energy (Fig. 3.1).

When an electron in an initial orbit with a larger energy Ei changes to a

final orbit with a smaller energy Ef, the emitted photon has an energy of Ei

- Ef, consistent with the law of conservation of energy.

where f is the frequency of the emitted light. This process is known as

de-excitation. This photon is emitted spontaneously when the atom de-excites and the electron

returns to its previous energy level. The lifetimes of excited states tend to be shorter for high

frequency transitions.

Fig. 2.1

hf = Ei - Ef


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Electrons can also jump from a smaller orbit to a larger orbit by picking up energy through

collision of atoms (which happens more often during heating), by acquiring energy when a high

voltage is applied, or by absorbing a photon of energy equal to the energy difference between

the two levels. This is known as excitation.

However, as well as exciting electrons, resonant photons can stimulate an excited atom to emit

photons of the same energy. This is something that Einstein recognised in 1916 and is called

stimulated emission. Stimulated absorption and emission is analogous to resonance in

mechanical systems. Spontaneous emission has no classical mechanical analogue. The

stimulated light is coherent (i.e. in phase) with the incident beam and hence amplifies it (i.e. the

intensity of the light after stimulated emission has occurred is greater than the intensity before).

For a two-level system with energies E1 and E0, there are three processes which can occur (Fig.

2.2).

• In spontaneous emission, an atom or a molecule in excited state E1 drops to the ground state,

emitting a photon in the process.

• In stimulated absorption, an atom or molecule in ground state E0 absorbs energy from a

photon with frequency given by hf = E1 − E0 and gets excited.

• In stimulated emission, an atom or molecule in excited state is perturbed by an incoming

photon with frequency f given by hf = E1−E0 so that it releases a second photon with the same

frequency, in phase with it.

Stimulated emission and spontaneous emission are competing processes. So how do we know

which process will, in fact, occur when an excited atom is irradiated with resonant photons? It

depends on which process is fastest. Stimulated emission occurs more rapidly when the

stimulating photon beam is of high intensity. However, spontaneous emission is fast when the

lifetime of the excited state is short (i.e. those corresponding to high energy transitions).

Therefore, spontaneous emission competes more strongly with stimulated emission in

transitions that emit blue or ultraviolet light than for transitions that emit red or infrared

radiation.

Fig. 2.2


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2.3 Boltzmann Factor

In a gas, the atoms can be in various excited states. The Maxwell-Boltzmann distribution function,

f(E), determines the probability of finding an atom in the energy level E, and when multiplied by

the total number of atoms in the gas, gives the number of atoms in a particular excited state.

KTEex

ex Aen/−=

Where the Boltzmann constant, k = 1.38 x 10-23 J/K, T is the absolute temperature of the gas, A is

a constant, Eex is the energy of the excited state in Joules.

Rather than try and find the number of atoms in a particular excited state, it is often more useful

to consider the ratio of atom population in two states to check for the possibility of stimulated

emission in lasers. This ratio is then given by,

kT

EE

y

x

yx

en

n)( −−

=

For stimulated emission to occur, we require the ratio of atoms in an upper energy level to that

of a lower energy level to be sufficiently large. If not, a passing photon of appropriate energy will

be more likely to be absorbed then cause stimulated emission to occur.

Question

Estimate the relative populations in thermal equilibrium of two energy levels such that a

transition from the higher level to the lower level gives visible radiation of 550 nm at room

temperature of 300 K. (Ans: 1.16 x 10-38

)

2.4 Experimental support for Bohr’s model

Bohr’s model of electronic transitions between energy levels, thereby emitting photons of

specific amount of energies is verified experimentally. Experimental measurements of emitted

light from hydrogen gas have wavelengths given by

Lyman Series

−=

22

1

1

11

nR

λ , n = 2, 3, 4, …

Balmer Series

−=

22

1

2

11

nR

λ, n = 3, 4, 5, …

Paschen Series

−=

22

1

3

11

nR

λ, n = 4, 5, 6, … R = 1.097 x 107 m-1


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According to Bohr’s model, these photons are emitted from electronic transitions from energy

level n to level 1 for Lyman series, level n to level 2 for Balmer series, level n to level 3 for

Paschen series etc. That means the energy of each level is given by the general formula

2n

hcREn = .

Comparing with the theoretical expression for energy of the nth level according to Bohr’s model,

22

4

28

1

hn

meE

o

nε

−= ,

we have an expression for the Rydberg constant, ch

meR

o

32

4

8ε= , which agrees with

experimental value of R up to 4 significant figures.


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Section Three : Compton Effect

3.1 Associated Momentum of a Photon

Einstein built on the work of Planck and proposed that all EM radiation exists and travels as

discrete quanta, known as photons. Each photon is associated with radiation of a single

frequency f and its energy is directly proportional to f:

λ

hchfE ==

where h is the Planck’s constant. This is known as the quantum theory of light.

While a photon has zero mass, it has momentum due to its motion at the speed of light. From

relativity we know that

( ))0(since

0

22

0

222

==⇒

+=

mpcE

cmpcE

Equating λ

hcE = (wave model) with pcE = (particle model) ,

λ

hp =⇒

Hence the momentum p of a photon is inversely proportional to the wavelength λ of the

associated light wave.


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3.2 Compton Scattering

Besides the photoelectric effect, another phenomenon which constitutes very strong evidence in

support of the quantum theory of light is the Compton Effect.

Arthur Compton observed the scattering of X-rays from electrons in a carbon target and found

that scattered X-rays have a longer wavelength than that incident upon the target. In his

experiment, Compton targeted a block of carbon with a beam of X-ray of a well-defined

wavelength (from an X-ray tube). A detector measured the intensity and wavelength of scattered

X-rays at various scattering angles. The results for four different scattering angles are shown in

Fig. 3,1. At each scattering angle (other than 0°), there are two distinct intensity peaks

corresponding to two different wavelengths. The wavelength of incident X-rays always appears

but a second peak corresponding to a longer wavelength also appears. The difference in the two

wavelengths is known as the Compton shift, which varies with the scattering angle.

Compton scattering cannot be explained by classical electromagnetic theory. According to

classical theory, two effects should be observed,

i) radiation pressure should cause the electrons to accelerate in the direction of

propagation of the waves, and

ii) the oscillating electric field of the incident radiation should set the electrons into

oscillation at an apparent frequency that is different due to the Doppler Effect. Each

electron first absorbs radiation as a moving particle and then re-radiates as a moving

particle, thereby exhibiting two Doppler shifts in the frequency of radiation.

Because different electrons will move at different speeds after the interaction, depending on the

amount of energy absorbed from the electromagnetic waves, the scattered wave frequency at a

given angle to the incoming radiation should show a distribution of Doppler shifted values.

However, neither of these two effects are observed in the experiment.

If X-rays are interpreted as particles (photons) instead, the collision between an X-ray photon and

an electron (in a carbon atom) can be analysed using standard relativistic mechanics. Indeed this

was what Compton postulated and he managed to obtain a relationship between the Compton

shift and the scattering angle. For his effort, Compton was awarded the Nobel Prize for Physics in

1927.

3.3 Compton Shift Equation

To derive the Compton shift equation, Compton had to make two assumptions:

• The electron is considered free. The energy of a Molybdenum Kα X-ray photon is about

17.5 KeV, much higher than the binding energies of the outer electrons, which are

normally in the order of eV.

• The electron is taken to be stationary initially.


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From Fig. 3.2,

momentum of the photon before collision ipr

=

momentum of the photon after collision fpr

=

momentum of the electron after collision epr

=

energy of photon before collision i

hc

λ=

energy of photon after collision f

hc

λ=

energy of electron before collision 2cme= (rest mass energy, electron not moving)

energy of electron after collision 4222 cmcp ee +=

From the principles of conservation of energy and momentum,

(2)---------

(1)---------

efi

ee

f

e

i

ppp

cmcphc

cmhc

rrr+=

++=+ 42222

λλ

Rearrange equation (1), divide by c and square the result,

(3)---------

−+−+=

+++−−=+

+−=+

+−=+

fi

e

fiif

e

e

i

e

if

e

fif

ee

e

if

ee

e

if

ee

chmhhh

p

cmchmhchmhh

cmp

cmhh

cmp

cmhchc

cmcp

λλλλλλ

λλλλλλ

λλ

λλ

1122

222

2

2

2

2

2

2

22

2

22

2

2

222

222

24222

But from equation (2),

Fig. 3.2


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( ) ( )

(4)---------

θλλλλ

θ

coshhh

cospppp

pppp

ppppppp

fifi

fifi

fifi

fifieee

2

2

2

2

2

22

22

2

2

2

2

−+=

−+=

⋅−+=

−⋅−=⋅=rr

rrrrrr

Equating equation (3) and (4),

)cos(cm

h

)cos(cm

h

)cos(h

cm

coshhh

chmhhh

e

if

efi

fi

fifi

e

fififi

e

fiif

θλ∆λλ

θλλ

λλ

θλλλλ

θλλλλλλλλλλ

−==−

−=

−

−=

−

−+=

−+−+

1

111

111

211

22

2

2

2

2

22

2

2

2

2

Equation Shift Compton

The Compton shift depends upon the angle of scattering and the mass of the scatterer, but is

independent of the wavelength of the incident X-rays. The shift ranges from a minimum value of

0 ("glancing" collision; θ = 0°) to a maximum value of cm

h

e

2("head-on" collision; θ = 180°)

(compare with Fig. 2.1). The factor cm

h

e

is known as the Compton wavelength and has a value

of 2.426 × 10-12 m.

The peak near the initial wavelength is considered to be scattering off inner electrons in the

carbon atoms which are more tightly bound to the carbon nucleus. This causes the entire atom

to recoil from the X-ray photon, and the larger effective scattering mass proportionally reduces

the wavelength shift of the scattered photons. For a carbon atom, the nuclear mass is

approximately 22,000 times larger than that of an electron. In this case, the scattering equation

yields a wavelength shift almost 22,000 times smaller than that for an unbound electron, so the

wavelengths of the photons suffer minimal changes after being scattered.


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Section Four : Application of the Quantum Theory of Light - Lasers

Laser is an acronym for light amplification by the stimulated emission of radiation. Lasers have

found many applications because of their particular properties. A laser beam is intense,

monochromatic, narrow and highly directional, with little lateral dispersion. Its properties

depend on many atoms being stimulated to emit radiation which are of equal frequency and

phase, and travel in the same direction. The radiation can be made to gain energy exponentially

through amplification in an optical cavity (resonant cavity). The optical cavity also accounts for

the directionality of the laser beam.

4.1 Population Inversion and Metastable States

Let begin by examining the requirements for lasing.

Stimulated emission produces identical photons that are coherent and directional, but for

stimulated emission to take place a "passer-by" photon whose energy is just equal to the

de-excitation energy must approach the excited atom before it de-excites via spontaneous

emission. Typically, a photon emitted by the spontaneous emission serves as the seed to trigger

a collection of stimulated emissions. Still, if the lifetime of the excited state is too short, then

there will not be enough excited atoms around to undergo stimulated emission. So, the first

criterion for lasing to occur is that the upper lasing state must have a relatively long lifetime,

otherwise known as a metastable state. The mean lifetime of an excited atom is about 10-8 s.

Decays from so-called metastable states may be much slower, some as long as 10-3 s.

In addition to the requirement of a long lifetime, the likelihood of absorption of the "passer-by"

photons needs to be minimized. This likelihood is directly related to the ratio of the atoms in

their ground state (or a lower energy state) versus those in the excited state. The smaller the

ratio, the more likely the "passer-by" photon will cause a stimulated emission rather than get

absorbed. So, to satisfy this requirement, a population inversion is needed: create more atoms

in the excited state than those in the ground state: 12 nn > .


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Consider a system with two energy levels. At thermal equilibrium, the lower energy state is more

populated than the higher energy state (the Boltzmann factor): 12 nn < . As the temperature

increases, the population in the high-energy state (n2) increases, but will never exceeds n1; its

only at infinite temperature do n2 and n1 become equal. In other words, a population inversion

(12 nn > ) can never exist for a two level system at thermal equilibrium. To achieve population

inversion therefore requires pushing the system into a non-equilibrium state.

Achieving population inversion in a two level system by pushing it into a non-equilibrium state is

not very practical. Such a task would require a very strong pumping transition that would send

any decaying atom back into its excited state; very energy costly and inefficient. To achieve

non-equilibrium conditions, an indirect method of populating the excited state must be used. In

order to maintain a population inversion, a third or even a fourth energy level is required. The

Helium-Neon laser is an example of a four level laser.

4.2 Action of a Laser – He-Ne Laser

This population inversion can be achieved by pumping and can be described using the

helium-neon (He-Ne) laser as an example. A tube contains a mixture of helium and neon gas. An

electrical discharge excites some helium atoms that cannot normally return to their ground state

(selection rules forbid it). However, by coincidence, there are excited neon levels at almost the

same energy as the excited helium level, and collisions between helium and neon atoms can

allow the excited helium atom to transfer its energy to the neon atom. (Collisional transitions are

not subject to the same selection rules that govern radiative transitions.) Thus the excited

helium atom returns to its ground state and the neon atom goes into one of its excited states.

This process can ‘pump’ a large number of neon atoms into the upper of the two excited levels

shown in Fig. 4.1, so that a population inversion exists between the excited neon level and the

almost empty lower level.

The laser process in a He-Ne laser starts with collision of electrons from the electrical discharge

with the helium atoms in the gas. This excites helium from the ground state to the 23S1 and 21S0

long-lived, metastable excited states (using Paschen notation). When the excited helium atoms

collide with the ground-state neon atoms, there is a transfer of energy to the neon atoms,


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exciting them into the 2s and 3s states. The transfer is possible due to a coincidence of energy

levels between the helium and neon atoms.

This process is given by the reaction equation:

He* + Ne → He + Ne* + ΔE where (∗) represents an excited state, and ΔE is the small energy

difference between the energy states of the two atoms, of the order of 0.05 eV.

The number of neon atoms entering the excited states builds up as further collisions between

helium and neon atoms occur, causing a population inversion between the neon 3s and 2s, and

3p and 2p states. Spontaneous emission between the 3s and 2p states results in emission of

632.8 nm wavelength light, the typical operating wavelength of a He-Ne laser.

After this, fast radiative decay occurs from the 2p to the 1s energy levels, which then decay to

the ground state via collisions of the neon atoms with the container walls. Because of this last

required step, the bore size of the laser cannot be made very large and the He-Ne laser is limited

in size and power.

With the correct selection of cavity mirrors, other wavelengths of laser emission of the He-Ne

laser are possible. The 3s→3p and 2s → 2p transitions give infrared operation at 3.39 µm and

1.15 µm wavelengths, and a variety of 2s → 1s transitions are possible in the green (543.5 nm),

the yellow (594 nm) and the orange (612 nm).

4.3 Properties of Laser

The laser beam is intense because the light beam between the mirrors passes many times

through the amplifying medium.

High-power lasers are used for cutting, and for welding sheets of metal together. Lasers are

being used to generate sufficient energy to initiate nuclear fusion. Lasers fire simultaneously and

their beams intersect to deliver 200 kJ of energy to a fuel pellet in less than a nanosecond (10-9 s).

Fig. 4.1


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The aim is to make the implosion so rapid that individual nuclei in the pellet are driven into

contact with one another with enough force to overcome their mutual Coulomb repulsion.

The laser beam is monochromatic and is described by an almost pure sine wave. This is because

the beam, moving to and fro between the mirrors, forms a standing wave. Thus an integral

number of half-wavelengths must fit exactly between the mirrors. No other wavelengths can be

amplified. This makes laser light much more monochromatic than light emitted by spontaneous

emission.

The fact that laser light is highly monochromatic ensures that any interference patterns

produced will be very clear and easy to interpret. This has an application in compact disc (CD)

players, where an interference pattern is produced between a reference laser beam and a laser

beam that is reflected from steps cut in the playing surface of the disc. Another famous

application of interference of laser light is the production of holograms or ‘three-dimensional

photographs’.

Laser light forms a narrow, well-directed beam, which spreads at large distances by diffraction

only. The spread by diffraction is through an angle of only about dλ where d is the beam

diameter. This angle is normally only a few milli-radians. The high directionality is explained by

the fact that all the photons produced by stimulated emission travel in the same direction.

The fact that laser light forms a narrow beam can be used in eye surgery. It can also be used to

measure distances to extreme accuracy. For example, the Apollo 14 astronauts left behind a

special reflector on the Moon’s surface. By firing a laser through an Earth-bound telescope at

this reflector, the time of flight of its round trip can be measured, confirming that the Moon is

receding from us at the rate of a few centimetres per year.

Question

A laser emits a beam of light with a circular cross-section of diameter 1 mm (and wavelength λ =

633 rim). How big is the spot 100 m away? (Ans: circle of radius 6.33cm)


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Section Five : Wave nature of Particle

Wave-Particle Duality applies to matter as well as radiation.

- De Broglie

Louis de Broglie postulated in 1924 that, since light waves could

exhibit particle-like behaviour, matter can also exhibit wave

behaviour. He proposed that all moving particles can be

mathematically modeled as a wave with wavelength given by,

p

h=λ

where p is momentum of the particle. i.e. p = mv,

λ is the de Broglie wavelength of the particle,

h is the Planck constant, h = 6.63 x 10- 34 J s

Note that for particles, mEp 2= and for photons, c

Ep = . These two relationships to

energy are not interchangeable because photons are massless and particles do not travel at the

speed of light.

Radical as the idea seems, it directly translates into the quantization of angular momentum that

was assumed in Bohr’s atomic model. If electrons are indeed waves, then they should form

standing waves around their orbit. This imposes the condition that the circumference of the

orbit, must be a multiple of the electron’s wavelength, ie,

mv

hnnr == λπ2

Rearranging yields, π2

hnmvr = .

de Broglie's idea was confirmed

experimentally in 1927 by American physicists

Clinton J. Davisson (1881–1958) and Lester H.

Germer (1896–1971). When they bounce a

stream of electrons off a nickel crystal, they

obtained strong maxima in the intensity of

reflected electron beam occurring at specific

angles, instead of a smooth variation of

intensity with angle as expected for particles.

This pattern is characteristic of diffraction

pattern from waves reflecting off a diffraction

grating.


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Exercise :

Calculate the de Broglie wavelength for

(a) a proton of kinetic energy 70 MeV kinetic energy

(Ans : 3.4 x 10-15

m)

(b) a 100 g bullet moving at 900 m s-1

(Ans : 7.4 x 10-36

m)

Section Six : Bragg Reflection

Röntgen discovered X-rays in 1895. Early experiments suggested that they were electromagnetic

waves with wavelengths of ~ 10-10 m. It was also strongly suspected that atoms in crystalline

solids were arranged in a regular lattice, with the spacing between planes of atoms about the

same as the wavelength of x-rays. Max von Laue (1879-1960) put these ideas together in 1913 by

suggesting that a crystal might serve as a three-dimensional diffraction grating for x-rays.

The interference pattern obtained by x-rays (wave) is the same as that by using electrons

(particle) thus showing that electrons can exhibit wavelight behaviour.

X – ray Diffraction Image Electron Diffraction Image

The situation can be modeled in two-dimensions using an array of posts in a ripple tank to

represent atoms in a lattice. The total interference pattern is the superposition of all the waves

scattered from these posts. Constructive interference can result in two ways; scattering from

adjacent atoms in a row, and scattering from atoms in adjacent rows. For the first case, we see

maxima in the pattern at angles where the angle of reflection is the same as the angle of

incidence to the plane. For the second case, there is constructive interference only when the

scattered radiation from each row is in phase (i.e. differs by an integer number of wavelengths).

The figure below shows that the path difference for adjacent rows is θsin2d .


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This leads to the Bragg equation θ=λ sin2dn (n = 1, 2, 3 ….)

where d is the spacing between planes of atoms and θ is the scattering angle.

Exercise :

Derive the Bragg Equation.

At angles where this condition is satisfied, a strong maximum in the interference pattern is

observed. These ‘diffraction’ patterns are often referred to as Bragg reflections.

If the crystal lattice spacing d is known, we can determine the x-ray wavelength from the

diffraction pattern, just as we determined wavelengths of visible light using diffraction gratings.

Generally, for a crystal, there can exist several families of crystal planes, each with different

spacing.

Conversely, knowing the wavelength of light used, we can investigate the crystal structure of

solids. Indeed, x-ray diffraction was instrumental in determining the structure of DNA. X-ray

photographs by Franklin were the key. She did not get due credit and sadly she died (in 1958)

before others were awarded the 1962 Nobel prize that she should at least have shared.


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Exercise :

The Bragg angle for a certain reflection from a powder specimen of copper is 47.75º at 20ºC and

46.60º at 1000ºC. Calculate the coefficient of linear expansion of copper, α. (∆L=αL∆T)

(Ans: 1.91 x 10-5 K-1)

Given wave-particle duality, particles (electrons for example) can also be used in crystal

diffraction if they have de Broglie wavelengths comparable to the inter-atomic spacing (i.e.

sufficiently small momenta).

Exercise :

(a) Show that to achieve a de Broglie wavelength of 6.3 x 10-10

m, electrons would need a kinetic

energy of 38eV while neutrons would need 2 x 10-3

eV.

(b) Explain why, in practice, electrons would need higher energies than 38 eV.

Exercise :

A beam of x-rays with wavelength 0.154nm is directed at certain planes of a silicon crystal. As

you increase the angle of incidence from zero, you find the first strong interference maximum

from these planes when the beam makes an angle of 34.5° with the planes.

a) How far apart are the planes?

b) Will you find other interference maxima from these planes at larger angles?

(Ans: 0.136 nm, no other angles for maxima)

Unfortunately, much higher energies have to be used if we wish to observe electron diffraction

because the interaction of electrons with the atoms in a crystal is so strong that electrons with

energy of a few eV would be completely absorbed by a specimen. In order that an electron beam

can penetrate even a very thin specimen, say 100 nm thick, energies of the order of 50-100 keV

are needed.

In general x-ray diffraction is the cheapest and the most convenient method and is by far the

most widely used. X-rays can be detected photographically or with a counter. X-rays are not

absorbed very much by air and so the specimen need not be in an evacuated chamber. However,

they do not interact very strongly with lighter elements. Electrons are scattered strongly in air so

that diffraction experiments must be carried out in a high vacuum. However, electrons can be

focused into narrow beams by electrostatic or magnetic lenses. This enables the diffracted

beams to be used to form a direct magnified image of the structure, as in the electron

microscope. But why is there a need for electron microscopes?


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Section Seven : Application of Matter Waves - Microscopes

To view images at atomic scales, instruments can no longer use visible light to form images.

Instead, instruments use electrons instead of visible light as electrons have much smaller

associated wavelengths, allowing for the greater resolution needed for details at atomic scales.

Even then, wavelengths of the electrons are usually comparable to the object separation and

results in wave interference effects. These effects limit formation of clear images.

Exercise :

a) What is the wavelength of electrons accelerated by a voltage of 10kV? (Ans: 1.23 x 10-11

m)

b) How does this compare to the typical wavelengths of visible light? (Ans:40,000 times smaller)

The resolving power of an optical instrument is

given by the Rayleigh criterion. This states that it

is possible to separate the images of two point

objects if the centre of the central maximum of

one image lies on the first dark fringe of the other.

This is illustrated in figure. The criterion is satisfied

when the central maxima are separated by an

angle given by the relationship:

D

λ=θ

1.22 sin (for circular apertures of diameter D )

In most optical instruments the angles are so small that the criterion is usually written, with θ

in radians, as:

D

λθ

1.22 =

Clearly, the shorter the wavelength of light used, the closer two objects can be and still be

resolved. Using a wavelength of 500 nm, typical of visible light, no optical instrument can resolve

objects smaller than a few hundred nanometres.

The diffraction patterns of two

point sources (solid curves) and

the resultant pattern (dashed

curve) for three angular

separations of the sources. (a)

The sources are separated such

that their patterns are just

resolved. (b) The sources are

closer together and their

patterns are barely resolved. (c)

The sources are so close

together that their patterns are

not resolved.


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Exercise :

Light of wavelength 558nm is used to view an object under an optical microscope. If the aperture

of the objective in the microscope has a diameter of 0.900 cm, what is the limiting angle of

resolution? (Ans: 7.56 x 10-5

rad)

Application: Determining molecular structure of proteins

The main technique used to determine the molecular structure of proteins, DNA, and RNA is

x-ray diffraction using x-rays of wavelength of about 1.0 Å. It allows the experimenter to “see”

individual atoms that are separated by about this distance in molecules. The figure below shows

a classic x-ray diffraction image of DNA made by Rosalind Franklin in 1952, used to determine the

double-helix structure of DNA by F.H.C. Crick and J.D. Watson in 1953.

Application: Electron Diffraction (only nonrelativistic cases)

Electron diffraction is a technique used to study matter by firing electrons (particles) at a

sample and observing the resulting interference pattern.

This technique is most frequently used in solid state physics and chemistry to study the crystal

structure of solids. These experiments are usually performed in a transmission electron

microscope (TEM) or a scanning electron microscope (SEM) as electron backscatter diffraction.

In these instruments, the electrons are accelerated by an electrostatic potential in order to gain

the desired energy and wavelength before they interact with the sample to be studied. The

periodic structure of a crystalline solid acts as a diffraction grating, scattering the electrons in a

predictable manner. Working back from the observed diffraction pattern, it is possible to deduce

the structure of the crystal producing the diffraction pattern.

In order to show diffraction, an electron ‘wave’ must travel through a gap of the same order as

its associated wavelength. The atomic spacing in crystal atoms or powdered carbon (graphite)

provides such gaps.

An x-ray diffraction photograph of DNA.

The cross pattern of spots was a clue that

DNA has a helical structure.

The double-helix structure of DNA.


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The circles correspond to the angles where constructive interference takes place. They are circles

because the powdered carbon provides every possible orientation of gap. A higher accelerating

potential for the electrons would result in a higher momentum for each electron. According to

the de Broglie relation, the wavelength of the electrons would thus decrease. This would mean

that the size of the gaps is now proportionally bigger than the wavelength so there would be less

diffraction. The circles would move in to smaller angles. The predicted angles of constructive

interference have been accurately verified experimentally.

(1) Transmission electron microscope (TEM)

A practical device that relies on the wave characteristics of electrons is the electron microscope.

The first transmission electron microscope (TEM) with magnetic lenses was constructed by

electrical engineers Max Knoll and Ernst Ruska. It is a device that focuses electron beams with

magnetic lenses and creates a flat-looking two-dimensional shadow pattern on its screen.

(a) Schematic drawing of a

transmission electron

microscope with magnetic

lenses.

(b) Schematic of a

light-projection microscope.


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In the past, light microscopes have been used mostly for imaging due to their relative ease of use.

However, the maximum resolution that one can image is determined by the wavelength of the

photons (recall Rayleigh’s criterion) that are being used to probe the sample. Visible light has

wavelengths of 400-700 nm; larger than many objects of interest. Ultraviolet could be used, but

soon runs into problems of absorption. Even shorter wavelengths such as X-rays exhibit a lack of

interaction, both in focusing and interacting with the sample.

No microscope can resolve details that are significantly smaller than the wavelength of the

radiation used to illuminate the object. The electron microscope has a much greater resolving

power than light microscopes because it can accelerate electrons to very high kinetic energies,

giving them very short wavelengths

=

mE

h

2λ . Typically, the wavelengths of electrons are

about 100 times smaller than those of visible light; falling in the x-ray region of the spectrum.

The high energy electron beam in an electron microscope is controlled by electrostatic or

magnetic deflection, which acts on the electrons to focus the beam to an image. Rather than

examining the image through an eyepiece as in an optical microscope, the viewer looks at an

image formed on a fluorescent screen.

Electron microscope picture (with

magnification of about x 200) showing a

family of dust mites gently grazing in a field.

(a) Diagram of a TEM for viewing thin samples. (b) An electron microscope.


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iAlthough electron microscopes have much higher resolution than optical microscopes, there are

some disadvantages. Electron microscopes are expensive and hard to maintain. As they are

sensitive to vibration and external magnetic fields, suitable facilities are required to house them

so as to achieve high resolutions. The samples have to be viewed in vacuum as the molecules

that make up air would scatter the electrons.

(2) Scanning Electron Microscope (SEM)

A second type of electron microscope with less resolution and magnification than the TEM, but

capable of producing striking three-dimensional images, is the scanning electron microscope

(SEM). Such a device might be operated with 20 keV electrons and have a resolution of about 10

nm and a magnification ranging from 10 to 100,000. As shown in figure, an electron beam is

sharply focused on a specimen by magnetic lenses and then scanned (rastered) across a tiny

region on the surface of the specimen. The high energy primary beam scatters lower energy

secondary electrons out of the object depending on specimen composition and surface

topography. These secondary electrons are detected by a plastic scintillator coupled to a

photomultiplier, amplified, and used to modulate the brightness of a simultaneously rastered

display CRT.

SEM usually image conductive or semi-conductive materials best. In order to image

non-conductive materials, a common preparation technique is to coat the sample with several

nanometers layer of conductive material, such as gold, from a sputtering machine. However,

this process has the potential to disturb delicate samples.

Exercise:

Smaller objects may be distinguished in electron microscopes than in optical microscopes

because

A electrons are smaller than visible quanta.

B the electrons travel much faster than light.

C there is no chromatic aberration with electrons.

D the electron wavelength is much shorter than that of visible light.

E the electrons are not diffracted.

SEM micrograph showing blood

cells in a tiny artery.

The working parts of a SEM.


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Section Eight : So what exactly is a matter wave?

To appreciate what we meant by the wave nature of matter, let us consider the following three

experiments.

Experiment 1: Using bullets

Experiment 2: Using water waves

Experiment 3: Using electrons

The experimental set-up is shown on the left of the figure and the

results of three different experiments on the right. It is shown that

bullets passed through slit 1 as open circles and slit 2 as black

circles. The column labeled P1 shows the distribution of bullets

arriving at the detector boxes when slit 2 is closed and only slit 1

open. The column labeled P2 shows the distribution of bullets

arriving at the detector boxes when slit 1 is closed and only slit 2

open. The result obtained with both slits open is showed in

column P12. The important point to notice is that the total obtained

in each box when both slits are open is just the sum of the

numbers obtained when only one or other of the slits is open.

P12 = P1 + P2

The detectors are a line of small floating buoys whose jiggling up

and down provides a measure of the wave energy. The wave

crests spreading out from each slit are shown and column I1

shows the smoothly varying wave intensity obtained when only

gap 1 is open. Notice that this is very similar to the pattern P1

obtained with bullets. The column labeled I2 shows the wave

intensity pattern when gap 1 is closed and gap 2 open. The result

obtained with both gaps open is showed in column I12. It is

dramatically different from the pattern obtained for bullets with

both slits open. It is not equal to the sum of the patterns I1 and I2

obtained with one of the gaps closed. This rapidly varying

intensity curve is called an interference pattern.

I12 = (h12)2 = (h1 + h2)2 , where I1 = (h1)2 and I2 = (h2)2

Electrons always arrive with a flash at the phosphor detector at

one point, in the same way that bullets always end up in just one

of the detector boxes rather than the energy being spread out, as

in a wave. Electrons that have gone through slit 1 are represented

as open circles, like the bullets. The column labeled P1 shows the

pattern obtained when slit 2 is closed and only slit 1 open. The

column labeled P2 shows the same thing when slit 1 is closed and

only slit 2 open, the electrons indicated by black circles. The

difference lies in the column P12 which shows the pattern obtained

for electrons when both slits are open. It is not the sum of P1 and

P2 and so we cannot say which slit any electron goes through.

2

2112 ψψ +=P where 2

11 ψ=P and 2

22 ψ=P


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For the electrons to have the same distribution pattern as that of a wave would mean that each

electron somehow passes through both slits at the same time just like a water wave. Yet we

know that the electron is a particle and cannot possibly do that.

This can be resolved in the following way. Since we do not know which slit the electron passes

through, it is analogous to the situation to someone who is blindfolded, trying to guess the

location of the electron as it moves towards the slits. The best the person can do is to describe

the probability of the electron at each location as a probability wave which as waves do,

interferes after passing through the slits to generate the observed distribution pattern.

What if we remove our blindfolds then? Will we see the electron transform into a wave as it

passes through the slits?

To test this, the electron experiment is repeated but this time a light is placed at the slit so that if

the electron passes through the slit, we will see a flash as light scatters off the electron and so

we will know for each electron which slit it passes through. When this is done, what we observe

is that the electron remains as a particle as it goes through one of the slits. However, it is the

resulting distribution that changes! What we get, is no longer that of water waves but amazingly

the same distribution as that of the bullet experiment!

We may now conclude the following,

i) If we do not observe which slit the electron passes through, we have to represent

the electron as a probability wave which interferes as waves do when it passes

through the slits, generating the same distribution as water waves.

ii) If we observe which slit the electron passes through, the electron can be precisely

located and as a particle, goes through the slits generating the same distribution as

the bullet experiment.

We may try to explain the different distribution patterns obtained in the following way. In trying

to observe the electron, we shine a light on the electron and in so doing; the light interfered with

the electron and thereby affects its motion.

To affirm this, we now use light of an increasing wavelength (thereby reducing the amount of

interference on the electron). Initially we will still be able to tell which slit each electron goes

through and the resulting distribution resembles that of the bullet experiment. As the

wavelength of the light increases, we will reach a point where at a certain wavelength, the flash

at the slit becomes big and fuzzy such that we are unable to ascertain which slit the electron

goes through and at this point, the distribution pattern becomes that of a water wave! We may

now conclude that it is impossible to arrange the light in such a way that one can tell which hole

the electron went through and at the same time not disturb the pattern. It was suggested by

Heisenberg that the then new laws of nature could only be consistent if there were some basic

limitation on our experimental capabilities not previously recognized.


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Heisenberg’s Uncertainty Principle

We have seen that quantum mechanics does not allow us the

comfort of being able to visualize the motion of a quantum

particle.

Heisenberg’s uncertainty principle can be written down in a

precise mathematical form which relates the uncertainties in

position and momentum measurements as follows:

For position & momentum, π4

hpx

x≥∆∆ or

2

h≥∆∆

xpx

(where )2π

h=h

For energy & time interval, 2

h≥∆∆ tE

Consider the following thought experiment introduced by

Heisenberg.

Suppose you wish to measure the position

and linear momentum of an electron as

accurately as possible by viewing the

electron with a powerful light microscope.

For one to see the electron and determine

its location, at least one photon of light

must bounce off the electron and pass

through the microscope into your eye as

shown.

When it strikes the electron, however, the photon transfers some unknown amount of its

momentum to the electron. Thus, in the process of locating the electron very accurately (that is,

by making x∆ very small), the light that enables you to succeed in your measurement changes

the electron’s momentum to some indeterminable extent (making xp∆ very large).

In making measurements on a quantum system, it is not possible to measure the quantities x

and p as accurately as we would wish.

Why is this so? Well, to determine the position very accurately, it is necessary to use light with a

very short wavelength – since the wavelength of the light determines the minimum distance

within which we can locate the particle. Very short wavelength light has a very high frequency

and according to the formula first guessed by Max Planck, hfE = where the constant of


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proportionality, h , is known as Planck’s constant.

We see that in order to locate the particle very precisely, we must use high frequency light with a

very large f , but such high frequency light will arrive in photons with a very large energy and

give the quantum system a very large ‘kick’. On the other hand, if we want to know the

momentum very accurately, we must give the system a very small kick and according to Planck’s

formula, this means using light of low frequency. Low frequency means long wavelength and this

in turn means a large uncertainty in the measurement of position.

In short, when one is trying to measure a quantum system, the act of measurement has

already disturbed the system.

Exploratory Question :

Since its impossible to measure precisely the position or momentum of a particle, is it valid then

to even discuss the position and momentum of a particle?

We are now almost ready to move into a formal introduction of Quantum Theory. But first, we

need to be introduced to the concept of probability amplitude.

Recall the intensity distribution pattern of

the electrons. After numerous electrons are

fired, the resulting intensity distribution

pattern represents the probability of finding

an electron at each location and can be

described by a probability function given by

2

2112 ψψ +=P where 2

11 ψ=P and 2

22 ψ=P

( )xψ is a complex valued function over space and is known as the probability amplitude

function, also known as the wave function. Note that at this point in time, ( )xψ is simply a

theoretical construct we used to predict the resulting probability distribution of the two slit

experiment.


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Section Nine : The 1 Dimensional Wave Function ( )tx,ψ

The electron plays dice

The wave function (probability amplitude function) ),( txψ of

a particle is the de Broglie wave. It is a complex valued

function. For example, for a particle moving in the x direction

with a precise value of linear momentum and energy, we have

−= ft

xAtx

λπψ 2sin),(

The squared wave function, ψψψ ∗=2

gives the

probability density function, which is the probability of

finding the particle at a particular location, x and time t.

(Note: if iba +=ψ , then iba −=*ψ .)

Mathematically, we state that dxtxdxxP2

),()( ψ=

Where 1),(2

=∫∞

∞−dxtxψ (normalization condition)

Exercise :

i) Write down the expression for the probability of finding the particle in the interval a < x < b as

well as bxa ≤≤ .

ii) What is the range of possible values for the probability density function ψψψ ∗=2 ?

),( txψ can be used to predict measureable averaged values of the particle, such as its energy,

its position, its momentum and other quantities. Thus once the wave function of a particle is

obtained, we essentially know the state of the particle.

Mathematically, we state that dxxx ∫∞

∞−= ψψ *

To obtain the wave function of the particle, we need to solve the Schrödinger equation,

proposed in 1926 by Austrian-German physicist, Erwin Schrödinger.

Exercise :

Imagine a particle with wavefunction kxAsin=ψ confined to an infinite square well with

walls at 0=x and .ax = The normalization condition is ∫ =a

dx0

21ψ . Find the

normalization constant, A .


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Case study : The wave function of a free particle

The plane wave representation for a free particle is

{ })sin()cos(),()(

tkxitkxAAetxtkxi

k ωωω −+−==Ψ −

The wavenumber k and angular frequency ω of free particle matter waves are given by the

de Broglie relations

h

pk = and

h

E=ω

For non-relativistic particles ω is related to k as m

k

2

2h

=ω

which follows from the classical connection mpE 22= between the energy E and

momentum p for a free particle.

Exercise :

i) Show that the probability density )(22

A=Ψ is uniform.

ii) What does this say about the particle’s location at any instant in time?

Summary :

In this topic, we first began by establishing that light can be modeled as a particle as well as a

wave in explaining various characteristics of light behavior. We then move on to extend this

wave-particle duality for particles as well. Having established empirical evidence for particles

that can only be explained by modeling them as waves, we then moved on to consider several

applications of the use of electrons in designing optical instruments that can have a resolution

beyond that of light-based instruments. While we have at this point established the feasibility of

a wave model for particles, we still do not have an interpretation of this matter wave. What is a

matter wave? In a series of two-slit experiments, we furthered our understanding in that the de

Broglie matter waves are in actual fact referring to probability amplitude waves, which is a

theoretical construct used to generate the probability of finding the particle at a certain time

and location. With this understanding of matter waves, the mechanics governing the wave

function is developed, which shall be introduced in greater detail in the next topic, Quantum

Mechanics.

Notes_ Quantum Theory of Light and Matter Waves

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