section vi 26 quantum physics

7/28/2019 Section VI 26 Quantum Physics

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Quantum Physics 26. Quantum physics

Content

26.1 Energy of a photon 26.2 Photoelectric emission of electrons

26.3 Wave-particle duality

26.4 Energy levels in atoms

26.5 Line spectra

Learning outcomes

Candidates should be able to:

(a) show an appreciation of the particulate nature of electromagnetic

radiation

(b) recall and use E = hf

(c) show an understanding that the photoelectric effect provides

evidence for a particulate nature of electromagnetic radiationwhile phenomena such as interference and diffraction provide

evidence for a wave nature

(d) recall the significance of threshold frequency

(e) explain photoelectric phenomena in terms of photon energy and

work function energy1


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(f) explain why the maximum photoelectric energy is independentof intensity, whereas the photoelectric current is proportional to

intensity

(g) recall, use and explain the significance ofhf= + mv2max(h) describe and interpret qualitatively the evidence provided by

electron diffraction for the wave nature of particles (i) recall and use the relation for the de Broglie wavelength = h/p

(j) show an understanding of the existence of discrete electron

energy levels in isolated atoms (e.g. atomic hydrogen) and

deduce how this leads to spectral lines

(k) distinguish between emission and absorption line spectra

(l) recall and solve problems using the relation hf= E1E2.

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Bedtime stories

The particle theory(copuscular theory) predicted that light should speed up

when passing from air to water. The wave theory predicted the opposite. In

1862, the speed of light in water was measured and found to be less than in air

So far we have learnt about properties of waves and that light is a wave motion

and introduced concepts of interference and diffraction

In the last 100 years, new phenomena has revolutionized physics

A number of experiments in 1900 showed that light behaves more like astream of particles than as a wave

Then later, it was shown experimentally that electrons, which had alwaysbeen thought of as particles, could also behave as waves

This wave-particle duality is one of the most important concepts of modernphysics

The phenomena that light had a particle nature is the explanation of Albert

Einstein in 1905 for which he received the 1921 Nobel Prize

Then, we shall find that diffraction which is a characteristic of waves isexhibited by electrons which have always been considered to be particles!

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Wave-particle duality of light

In 1924 Einstein wrote:- There are therefore now two

theories of light, both indispensable, and without any

logical connection.

Evidence for wave-nature of light

Diffraction and interference

Evidence for particle-nature of light

Photoelectric effect Compton effect

Light exhibits diffraction and interference phenomena that are only

explicable in terms of wave propertiesLight is always detected as packets (photons); if we look, we never

observe half a photon


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Symmetry in physics


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Demonstration of photoelectric emission

using gold-leaf electroscope

When a clean zinc plate is placed on a gold-leaf electroscope and the

electroscope then charged negatively, the gold leaf deflects proving

that the zinc plate is charged

When visible light of any color is shone on to the plate, the leaf doesnot move even if the intensity (brightness) of the light is increased

However, when ultra-violet radiation is shone on the plate, the leafimmediately falls, showing that it is losing negative charge

This means that electrons are being emitted from the zinc plate

These electrons are called photoelectrons and the process is calledphotoelectric emission

If the intensity of the uv radiation is now increased, the leaf fallsmore quickly, showing that the rate of emission of photoelectronshas increased

The difference between uv radiation and visible light is that uv

radiation has a shorter wavelength and hence a higher frequency than

visible light 8


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Further investigations with apparatus like

these lead to further conclusions If photoelectric emission takes place, it does so instantaneously. There

is no delaybetween illumination and emission (10-9 s)

Photoemission takes place only if the frequency of the incidentradiation is above a certain minimum value called the thresholdfrequency f0

Different metalshave different threshold frequencies

Whether or not emission takes place depends only on whether thefrequency of the radiation used is above the threshold for that surface.It does not depend on the intensity of the radiation. Even very dim

light gives some electrons with high KE For a given frequency (above the threshold frequency), the rate of

emission of photoelectrons is proportional to the intensity of theradiation

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PHOTOELECTRIC EFFECT

I

Vacuum

chamber

Metal

plate

Collecting

plate

Ammeter

Potentiostat

Light, frequency


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Another expt to investigate photoelectric

emission

Pg 239 fig 9.2 Physics by Chris Mee

A is the metal surface and B is the collector across which there is a variablepotential difference

Without any p.d. applied, when uv radiation of a fixed frequency above thethreshold, is shone on to Athe metal surface, it emits photoelectrons

These electrons travel from A to B causing current to flow which is detected bythe micro-ammeter

If now a p.d. is applied between A and B, with B negative wrt A, any electrongoing from A to B will gain potential energy as it moves against the electricfield. The gain in potential energy is at the expense of the kinetic energy of theelectron

i.e. loss in k.e. = gain in p.e. =charge of electron x potential difference (E =eV)

If the voltage is now gradually increased, the current registered decreases andfinally falls to zero. This minimum value of the p.d necessary to stop electronflow completely is the stopping voltageand it measures the max k.e. with whichthe electrons are emitted

The fact that there is current flow detected at voltages less than the stoppingvoltage indicates that there is a range of kinetic energies for these electrons

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cont..

When the experiment is repeated with radiation of greater intensity but thesame frequency, the max current increases but the value of the stoppingpotential is unchanged. The graph ofmax k.e. vs frequency of radiation givesa straight line

Conclusions to be drawn:

The photoelectrons have a range of k.e. from zero up to some max value

If the frequency of the incident radiation is increased, the max k.e. of the

photoelectrons also increases

For constant frequency, the max k.e. is unaffected by the intensity of

radiation

If the graph of max k.e. vs frequency of radiation is extrapolated to the

point where the max k.e. of the electrons is zero, the min frequency

required to cause emission i.e.the threshold frequencycan be found The above 2 experiments on photoemission showed conclusively that

radiation of frequency below the threshold, no matter how intense or forhow long, does not produce photoelectrons

This cast a major doubt as to whether light is a continuous wave (which was

until then fully accepted due to interference and diffraction effects)

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PHOTOELECTRIC EFFECT

When UV light is shone on a metal plate in a vacuum, it emits

charged particles (Hertz 1887), which were later shown to be

electrons by J.J. Thomson (1899).

Electric field E of light exerts force F=-

eE on electrons. As intensity of light

increases, force increases, so KE ofejected electrons should increase.

Electrons should be emitted whatever

the frequency of the light, so long as E

is sufficiently large

For very low intensities, expect a time lag

between light exposure and emission,

while electrons absorb enough energy to

escape from material

Classical expectations

Hertz J .J . Thomson

I

Vacuum

chamber

Metal

plate

Collecting

plate

Ammeter

Potentiostat

Light, frequency
http://images.google.co.uk/imgres?imgurl=br.geocities.com/saladefisica3/fotos/hertz.gif&imgrefurl=http://br.geocities.com/saladefisica3/&h=302&w=223&prev=/images%3Fq%3Dhertz%26svnum%3D10%26hl%3Den%26lr%3D%26ie%3DUTF-8%26oe%3DUTF-8%26sa%3DG


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PHOTOELECTRIC EFFECTEinstein

Millikan

Verified in detail

through subsequentexperiments by

Millikan

Maximum KE of ejected electrons is

independent of intensity, but

dependent on

For


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Wave theory shaken and quantum theory is

born! In classical wave theory :

When an em wave of amplitude and frequency interacts with anelectron, the electron will absorb energy from it and when itabsorbs enough energy it will escape free from the metal

Thus even a low frequency wave whose energy can be boosted byincreasing the amplitude (i.e. increasing the intensity or brightness)can be used to cause photoemission

Alternatively, we can use less bright light but shine it on the metalfor a longer time until it has enough energy to cause emission

Classical wave theory of em radiation predictions:

Whether an electron is emitted or not should depend on the powerof the incident wave i.e. on its intensity. A very intense wave of

any frequency should cause photoemission The max k.e. of the photoelectrons should be greater if the

radiation intensity is greater

There is no reason why photoemission should be instantaneous

Hence a new concept of the quantum theoryarose

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Albert Einsteins theory of photoelectric

emission

In 1901, Max Planck put forward the theory that energy cannot bedivided into smaller and smaller amounts and had suggested that theenergy carried by em radiation might exist as discrete packets calledquantaand gave his expression E = hf

where fis the frequency of the radiation and h is a constant called thePlanks constant (h = 6.63 x 10-34 J s)

In 1905 Albert Einstein developed the theory of quantized energy to

explain all the observations associated with photoelectric emission.

Einstein proposed that light radiation consists of a stream of energy

packets called photons. This is a special name given to a quantum of

energy when the energy is in the form of em radiation When a photon interacts with an electron, it transfers all its energy to

the electron

It is only possible for a single photon to interact with a single electron;

the photon cannot share its energy between several electrons

This transfer of energy is instantaneous 16


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Photon theory of photoelectric emission

If the frequency of the incident radiation is less than the threshold

frequency for the metal, the energy carried by each photon isinsufficient for an electron to escape the surface of the metal

If the photon energy is insufficient for an electron to escape, it is

converted to thermal energy in the metal

The minimum amount of energy required for an electron to escape

from the surface is called the work function energy,

If the frequency of the incident radiation is equal to the thresholdfrequency, the energy carried by each photon is just sufficient forelectrons at the surface to escape

If the frequency of the incident radiation is greater than thethreshold frequency, surface electrons will escape and have surplusenergy in the form of kinetic energy. These electrons will have themax k.e.

If a photon interacts with an electron below the surface, some energy is

used to take it to the surface, so that it is emitted with less than the max

k.e. This gives rise to a range of values of k.e. 17


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The photoelectric equation

Albert Einstein used the principle of conservation of energy to derive the

photoelectric equation

Photon energy = work function energy + maximum k.e. of the photoelectron

i.e. hf = + mevmax2

For radiation incident at the threshold frequency,

hf0 = since mevmax2 = 0 Hence the complete photoelectric equation can be written as

hf = hf0 + mevmax2

Materials with a low work function emit electrons in visible light whereas those

with a higher work function require the higher energy photons of ultraviolet

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Work function energies and threshold

frequencies of some materials

Metal /eV f0

/Hz

Sodium 2.4 5.8 x 1014

Calcium 2.9 7.0 x 1014

Zinc 3.6 8.8 x 1014

Silver 4.3 1.0 x 1014

Platinum 5.6 1.4 x 1014

(Remember; 1 eV = 1.6 x 10-19 J)

A quantum is an extremely small amount of energy

E.g. quantum of red light =2 eV

quantum of violet light =4 eV

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Example

The work function energy of platinum is 9.0 x 10-19 J. Planks constant h =

6.63 x 10-34 J s

Calculate

a) the threshold frequency for the emission of photoelectrons from

platinum

b) the maximum k.e. of a photoelectron when radiation of frequency2.0 x 1015 Hz is incident on a platinum surface

Solution

a) using hf0 =, f0 =/h =1.4 x 1015 Hz

b) using hf = hf0 + mevmax2

mevmax2 = hf - hf0 = h(ff0) =4.0 x 10

-19 J

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Compton


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COMPTON SCATTERING

X-ray source

Target

Crystal

(selects

wavelength)

Collimator

(selects angle)

Compton (1923) measured intensity of scattered X-rays from

solid target, as function of wavelength for different angles. He

won the 1927 Nobel prize.

Result: peak in scattered radiation shifts

to longer wavelength than source.

Amount depends on (but not on the

target material). A.H. Compton,Phys. Rev. 22 409 (1923)

Detector

Compton


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COMPTON SCATTERING

Comptons explanation:billiard ball collisions between particles of light

(X-ray photons) and electrons in the material

Classical picture: oscillating electromagnetic field causes oscillations in positions of

charged particles, which re-radiate in all directions atsame frequency and wavelength as

incident radiation.

Change in wavelength of scattered light is completely unexpected classically

ep

pBefore After

Electron

Incoming photon

p

scattered photon

scattered electron

Oscillating electronIncident light wave Emitted light wave


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Electron diffraction

If light waves can behave like particles (photons) perhaps particlescan behave like waves?

If a beam of X-rays of a single wavelength is directed at a thin metalfoil, a diffraction pattern is produced.

This is a similar effect to the diffraction pattern produced when lightpasses through a diffraction grating.

The foil contains many crystals and the gaps between neighboring

planes of atoms in the crystals act as slits, creating a diffractionpattern

If a beam of electrons is directed at a thin metal foil, a similardiffraction pattern is produced

The electrons which we normally consider to be particles, are

exhibiting a property which we would normally associate with waves To observe diffraction, we know that the wavelength of the radiation

should be comparable to the size of the aperture

The fact that diffraction is observed with electrons suggests that theyhave a wavelength of about the same magnitude as the separation of

planes of atoms in crystals of the order of 10-10 m24


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Davisson G.P. Thomson

Davisson, C. J.,

"Are Electrons

Waves?," Franklin

Institute Journal

205, 597 (1928)

The Davisson-Germer experiment:

scattering a beam of electrons froma Ni crystal. Davisson got the 1937

Nobel prize.

At fixed accelerating voltage (fixed

electron energy) find a pattern of sharp

reflected beams from the crystal

At fixed angle, find sharp peaks in intensity

as a function of electron energy

G.P. Thomson performed similar interference

experiments with thin-film samples

i

i

ELECTRON DIFFRACTION

The Davisson-Germer experiment (1927)


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a

i

r

cosi

a

cosr

a

Path difference:

Constructive interference when

Note iandrnot

necessarily equal

ELECTRON DIFFRACTION

(cos cosr i

a

(cos cos )r i

a n


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Double-slit experiment for electrons

When electrons fired toward a slit

for which d < , a diffraction

pattern is seen

If current is lowered so that only1 electron is allowed at a time -

pattern is STILL SEEN!


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Diffraction


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Electron diffraction


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Electron Microscopes


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Wave-particle duality

In 1924 Louis de Broglie suggested that all moving particles have a

wave-like nature.

Using ideas based upon the quantum theory and Alert Einsteins theory

of relativity, he suggested that the momentum p of a particle and itsassociated wavelength are related by the equation

= h/p where h is the Planck constant. is known as the deBroglie wavelength

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Example

Calculate the de Broglie wavelength of an electron traveling with a speed

of 1.0 x 107 m/s. (Planck constant h = 6.6 x 10-34 J s; electron mass me =

9.1 x 10-31 kg)

Solution

Using = h/p , p= h/= 7.3 x 10-11 m

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Spectra

White light is not a single colour. If white light from a tungsten

filament is passed through a prism, the light due to refraction is spreadout or is dispersed into its component colors and the band of differentcolors is called a continuous spectrum

In the case of white light it ranges from 400 nm to 700 nm (800 THz to

400 THz) (VIBGYOR)

Since this spectrum has been produced by the emission of light fromthe tungsten filament lamp, it is referred to as an emission spectrum

When a high potential difference is placed across 2 electrodes of a

discharge tube (which is a transparent tube containing a gas at low

pressure) light is emitted

Examination of the light with a diffraction grating shows that theemitted spectrum is no longer continuous but consists of a number ofbright lines. Such a spectrum is known as a line spectrum

The wavelengths corresponding to the lines of the spectrum are

characteristic of the gaswhich is in the discharge tube

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How are line spectra produced?

In order to understand this, we need to understand how electrons in atoms behave

Bohrs allowed-orbit analysis only works for the simplest atom, hydrogen. Thismodel has now been replaced by a mathematical wave mechanics model of the atomin which allowed orbits are replaced by allowed energy levels

Electrons in an atom can only have certain specific energies called the electronenergy levelsof the atom

The energy levels are usually represented as a series of lines against a vertical scale

of energy and the electron can have any of these energy levels but cannot haveenergies in between them

Normally electrons occupy the lowest energy levels available and under these

conditions the atom and its electrons are said to be in the ground state

However, if the electron absorbs energy perhaps by being heated or by collision with

another electron, it may be promoted to a higher energy level The energy absorbed is exactly equal to the difference in energy of the 2 levels and

under these conditions the atom is described as being in an excited state

Light waves come from atoms. A burst of wave energy is given off whenever anelectron loses energy by dropping to a lower orbit. This burst of energy which actsas a particle is what we call a photon

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Electron energy levels in atoms

Pg 247 fig 9.9 Physics by Chris Mee

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Electron energy levels.

An excited atom is unstable and hence after a short time, the excited

electron will return to a lower level In order to achieve this it must lose energy

It does so by emitting a photon of electromagnetic radiation whoseenergy is given by hf =E = E2E1where E2 is the energy of the higher level and E1energy of the lowerlevel and h is the Planck constant

In terms of the speed of light c, = hc/E This movement of an electron between energy levels is called an

electron transition The larger the energy of the transition, the higher the frequency (and

hence shorter the wavelength) of the emitted radiation

This downward transition results in the emission of a photon The atom can be raised to an excited state by the absorption of a

photon, but the photon must have the right energy corresponding tothe difference in energy of the excited state and the initial state

Hence a downward transition corresponds to photon emission and anupward transition to photon absorption

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Spectroscopy

Because all elements have different energy levels, the energydifferences are unique to each element, consequently each element

produces a different and characteristic line spectrum

Spectra can be used to identify the presence of a particular element and

the study of spectra is called spectroscopy and instruments used to

measure wavelengths of spectra are called spectrometers Spectrometers for accurate measurements of wavelength make use of

diffraction gratings to disperse the light

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Continuous spectra

Whilst the light emitted by isolated atoms such as those in low-pressure gases in tubes produce line spectra, the light emitted by atoms

in a solid, liquid or gas at high pressure produces a continuousspectrum

This is due to the proximity of the atoms to each other which causes

interaction between the atoms resulting in a broadening of theelectron energy levelsseen as a continuous spectrum

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Absorption spectra

If white light passes through a low-pressure gas and the spectrum ofthe white light is then analyzed, it is found that light of certainwavelengths is missingand in their place are dark lines

This type of spectrum is called an absorption spectrum

As the white light passes through the gas, some electrons absorb

energyand make transitions to higher energy levels

The wavelengths of the light they absorb correspond exactly to theenergies needed to make the particular upward transition

When these excited electrons return to lower levels, the photons are

emitted in all directions rather than in the original direction of the

white light, hence some wavelengths appear to be missing

The wavelengths missing from an absorption spectrum are exactlythose present in the emission spectrum of the same element

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Example

Calculate the wavelength of the radiation emitted when the electron in ahydrogen atom makes a transition from the energy level at -0.54 x 10-18 J

to the level at -2.18 x 10-18 J (Planck constant h = 6.62 x 10-34 J s, c = 3.00

x 108 m/s)

Solution

Using E = E2E1 = 1.64 x 10-18 J

And using = hc/E =121 nm

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section vi 26 quantum physics

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