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Aquinas College Physics

Module 4.1: Waves & Quantum

Module 4.1

Quantum Physics 4.1.G Photons, atomic spectra & the photoelectric effect 4.1.H Wave-particle duality, de Broglie & quantum phasors

Topic Notes Name:__________ -1-


Aquinas College Physics

Module 4.1: Waves & Quantum

Important resources for this module: All prezi presentations, booklets, homeworks and practical sheets are all available on the departmental website: www.aquinasphysics.com/41-waves-quantum-behaviour1.html

www.alevelphysicsonline.com/quantum Excellent video tutorials made by an A level physics teacher for A level physics students. If you need to go over any concepts again, this is the first place that you should look. Free access to the course textbook (via the departmental website). Follow the instructions on the website for how to log in.

www.aquinasphysics.com/kerboodle.html

Challenging questions from GCSE level to Undergraduate physics problems. If you are hoping for a B, A or A* you must be visiting this site and regularly practicing the problems. They also run excellent workshops. Look out for these!!

isaacphysics.org/

Multiple-choice practice revision questions on your phone. Revise on the bus on the way in to college!!

www.gojimo.com/ -2-


Aquinas College Physics

Module 4.1: Waves & Quantum

Table of Contents Learning Objectives ................................................................................................................................................... - 4 Before we begin‌ ............................................................................................................................................... - 5 1.

Structure of the atom, photons & electron energy levels................................................................................ - 6 Some questions on photon energies ..................................................................................................................... - 8 Space for your own notes ..................................................................................................................................... - 9 -

2.

The electronvolt ................................................................................................................................................ - 10 Some questions on electronvolts ........................................................................................................................ - 10 -

3.

Emission & absorption spectra ....................................................................................................................... - 12 Emission Spectra .................................................................................................................................................... - 14 Absorption spectra ................................................................................................................................................. - 15 Space for your own notes ................................................................................................................................... - 15 -

4.

The photoelectric effect ................................................................................................................................... - 16 The gold-leaf electroscope: observations & explanations ..................................................................................... - 16 The photoelectric equation ..................................................................................................................................... - 19 Some questions on the photoelectric equation: .................................................................................................. - 20 Some Questions on the graph: ........................................................................................................................... - 21 -

5.

Using LEDs to experimentally find Planck’s constant ................................................................................. - 22 -

6.

Electron diffraction, wave-particle duality & de Broglie ............................................................................. - 24 Electron diffraction tubes ....................................................................................................................................... - 24 Space for your own notes ................................................................................................................................... - 25 The de Broglie wavelength .................................................................................................................................... - 26 Some questions on the de Broglie wavelength .................................................................................................. - 26 -

7.

Wave-particle duality, many paths & quantum phasors.............................................................................. - 28 Firing electrons/photons one at a time through a double slit ................................................................................. - 28 Many paths interpretation of interference & quantum phasors .............................................................................. - 29 Questions on quantum phasors .......................................................................................................................... - 29 Space for your own notes ................................................................................................................................... - 33 Using phasors to explain reflection ........................................................................................................................ - 34 -

8.

IsaacPhysics mastery questions ...................................................................................................................... - 36 Questions on Quantum Calculations ...................................................................................................................... - 36 Questions on the photoelectric effect ..................................................................................................................... - 38 -

Space for your own notes......................................................................................................................................... - 40 -

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Aquinas College Physics

Module 4.1: Waves & Quantum

Learning Objectives 4.1 (a) Describe and explain (vi)

evidence that photons exchange energy in quanta E = hf (for example, light emitting diodes, photoelectric effect and line spectra)

(vii)

Quantum behaviour: quanta have a certain probability of arrival; the probability is obtained by combining amplitude and phase for all possible paths

(viii) evidence from diffraction that electrons show quantum behaviour (b) Make appropriate use of the terms (i)

intensity, electronvolt, work function, threshold frequency, phase, phasor, amplitude, probability, superposition, path difference

(c) make calculations and estimates involving (iv) the energy carried by photons across the spectrum, E = hf (a) demonstrate and apply knowledge and understanding of the following practical activities. (v) the wavelength of a particle of momentum p, đ?œ† = â„Žâ „đ?‘? (vi) determining the Planck constant using different coloured LEDs

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Aquinas College Physics

Module 4.1: Waves & Quantum

4.1 Quantum Physics Prezi 4.1.G Photons, atomic spectra & the photoelectric effect goo.gl/HWtcfT

These notes coordinate with Prezis 4.1.G & 4.1.H on the departmental website & pages 144-160(ish) in the course textbook.

Prezi 4.1.H Wave-particle duality, de Broglie & Quantum phasors goo.gl/YZ3FFo

In this topic we will be dipping our toes into the weird and wacky world of quantum physics – a section of physics that attempts to describe the behaviour of the very very small – down at the scale of individual atoms and subatomic particles (or are they sub-atomic waves). We will follow this up next year in the topic of particle physics. Many of the physics rules that hold down at this microscale seem completely nonsensical with our macro-scale, Newtonian physics heads on. We have to put to one side many of the things that we have previously taken for granted and accept many seemingly contradictory observations: that particles such as atoms and electrons can be seen to behave as if they are waves; and that waves such as light can also be thought of as behaving as individual particles. We will start by considering light, and evidence that it behaves as a stream of individual particles – or photons. This is in direct contradiction to previous evidence that we have seen that light is in fact a wave. Having considered this we will turn our attention to electrons, and investigate the evidence that these can be considered as waves. Finally, we will consider a way of estimating the probability that a particular particle will arrive at a point on a screen, and so go a short way to reconciling the apparent contradiction evident when photons or electrons are fired one at a time through a set of double slits – that they build up one at a time as individual discrete arrivals (particle behaviour) to form a distinctive interference pattern on the screen (wave behaviour).

Before we begin… Briefly describe and explain the experimental evidence that we have seen that light behaves as a wave.

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Aquinas College Physics

Module 4.1: Waves & Quantum

1. Structure of the atom, photons & electron energy levels A useful video on the structure of the atom, in particular the different energy levels that an electron can take can be found at goo.gl/nSRrmB (see QR code left). Another video introducing the idea of photons can be found at goo.gl/dxVbvV (see QR code right).

Complete the sentences below giving a brief overview of the structure of an atom: 

An atom consists of a central ………………………………. (made of neutrons and protons)

……………………… “orbit” this central nucleus – different orbits relate to different …………………….. The electrons can only have certain …………………………. Values for their energy, called energy levels.

Only a fixed number of ……………………. are allowed in each energy level.

Electrons normally fill the ………………………. energy levels.

When the electrons are in the lowest possible energy levels the ……………………….. is in its “GROUND STATE”.

If the atom gains energy the outer electron moves to a higher energy level, called an “……………………………. STATE”.

One of the key words that comes up in your explanation on the is the term DISCRETE. Electrons are only able to take one of a number of different discrete energy levels. Give a definition of the term discrete below.

Having established that electrons only inhabit discrete energy levels, a natural next step would be to consider what causes electrons to transition between different energy levels. Electrons do this via interactions with photons. Give an explanation of what a photon is in the space below.

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Aquinas College Physics

Module 4.1: Waves & Quantum

Explain the following interactions that occur when electrons transition between energy leve (1) When an electron transitions to a higher energy level.

(2) When an electron transitions to a lower energy level.

The energy of the photon absorbed or emitted depends on the difference between the two different energy levels that the electron transitions between. The energy of the photon depends on the frequency of the light. State the relationship between photon energy E and frequency f of light (electromagnetic) radiation in the box right:

The constant h that links particle energy E and frequency f is known as Planck’s Constant. State the value and units for Planck’s Constant in the box right:

The equation given above can be combined with the wave equation linking the frequency f to the wavelength  and speed of light c. In doing this we can state the photon energy E in terms of h, c and  as given in the box right. You do not get given this equation in the formula booklet, but it will be a very useful equation to remember!!

E 

h



(value)

(units)

E -7-


Aquinas College Physics

Module 4.1: Waves & Quantum

Some questions on photon energies (1) Calculate the energy of a photon of: (a) light of frequency 6.0×1014 Hz

E = ………………………… J (b) gamma rays of frequency 5.0×1019 Hz

E = ………………………… J (c) radio waves of frequency 1215 kHz

E = ………………………… J (2) Green light has a wavelength of 560 nm. Calculate the frequency of this light and the energy of a photon of this light.

E = ………………………… J (3) Estimate how many photons are emitted by a 100 Watt light bulb each hour.

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Aquinas College Physics

Module 4.1: Waves & Quantum

Space for your own notes …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. ……………………………………………………………………………………………………………………………..

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Aquinas College Physics

Module 4.1: Waves & Quantum

2. The electronvolt Useful videos giving further details of the electronvolt can be found at goo.gl/9rxnuM (see QR code left) and goo.gl/bQ4Hwr (see QR code right).

You will have noticed from your answers to questions above that the energies carried by individual photons is exceptionally small. Just as it would be inappropriate to measure astronomical distances in nanometres (we use units of parsecs or light years instead), similarly it is useful to consider alternative units for these small measures of energy down at the quantum and particle level. In doing this, we use a unit of energy known as the electronvolt. Give a definition for an electronvolt.

Give the conversion factor between electronvolts and joules is given in the box right.

1 eV

J



Some questions on electronvolts (1) How many electronvolts are there in 1 Joule?

1 joule = ………………………. eV (2) What is the energy of a photon of red light with a frequency of 4.6×1014 Hz in both joules and electronvolts?

E = ……………………………. J ; E = ………………….. eV (3) What is the energy (in eV) of a photon of violet light with a wavelength of 420 nm?

E = ……………………………….. eV - 10 -


Aquinas College Physics

Module 4.1: Waves & Quantum

(4) What is the wavelength of the photon emitted as an electron transitions between the energy levels shown in the diagram right.

 = …………………………… nm (5) Complete the flow chart below showing how to complete calculations between photon wavelength and photon energy E in electronvolts.

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Aquinas College Physics

Module 4.1: Waves & Quantum

3. Emission & absorption spectra Useful videos on the formation of emission and absorption spectra can be found at goo.gl/kLpnvm (see QR code left) and at goo.gl/ZmV6CT (see QR code right)

The transitions that electrons make between different energy levels – and the specific energies, and therefore wavelength, of photon emitted – explain the particular colours emitted during flame tests in chemistry lessons, or explains why tubes filled with different gases light different colours (compare neon gas tubes with the sulphur used in orange street lights). It also allows us to analyse the elemental composition of distant stars and galaxies, and indeed allows us to establish the initial wavelength of light emitted from distant galaxies when measuring how much the wavelength of this light has been increased due to the expansion of the universe when making observations of cosmological redshift. In lesson we will use a diffraction grating or a spectroscope to observe the diagnostic patterns emitted when tubes of gas have a large potential difference placed across them, exciting the individual gas atoms and causing electrons to transition to a higher energy level and fall back again, emitting a photon of specific wavelength / energy in the process. In the space below, give details explaining your observations when viewing a gas discharge tube. (1) Explain what you know about the wavelength make-up of white light

(2) What happens to white light when it is shone through a diffraction grating and why? You may want to include a diagram in your explanation.

(3) Describe your observations when viewing the gas discharge tube through a spectroscope / diffraction grating

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Aquinas College Physics

Module 4.1: Waves & Quantum

(4) Explain what this suggests about the light emitted from the discharge tube.

Perhaps the simplest way to quantitatively describe and explain emission and absorption spectra is to consider the simplest possible element: hydrogen. Hydrogen consists of a single proton orbited by a single electron. The electron can inhabit each of the energy levels shown in the diagram right. Only the electron transitions between the n = 2 energy level and the energy levels above involve the emission / absorption of photons of energy relating to visible wavelengths of light.

(1) Explain what type of radiation is emitted/absorbed when: (a) electrons transition between the n=1 and n=3 energy levels?

(b) electrons transition between the n=3 and n=4 energy levels?

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Aquinas College Physics

Module 4.1: Waves & Quantum

(2) What is the frequency and wavelength of the photon emitted when the electron transitions from: (a) n=3 to n=1?

frequency f = ………………………… Hz ; wavelength  = ………………… nm (b) n=4 to n=2?

frequency f = ………………………… Hz ; wavelength  = ………………… nm

Complete the explanations below giving details on the formation of emission and absorption spectra:

Emission Spectra

When an element is given ……………………… (for example by heating or by an applied potential difference) the electrons are raised to an ……………………… state.

………………………… of light are emitted when the electrons ……………… back to lower energy levels.

Because only a few …………………………. energy transitions are possible, photons are ……………………….. at only certain discrete wavelengths.

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Aquinas College Physics

Module 4.1: Waves & Quantum

Absorption spectra

When ………………………… light is shone through a gas or vapour, the atoms in the gas can …………………… some of the photons.

Because the …………………………. surrounding the gas atoms only have ……………………….. energy jumps they can make, only …………………………. of certain energy are absorbed, and only certain ……………………………. of the spectrum appear dark.

Space for your own notes …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. ……………………………………………………………………………………………………………………………..

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Aquinas College Physics

Module 4.1: Waves & Quantum

4. The photoelectric effect More information on the photoelectric effect can be found from alevelphysicsonline (goo.gl/HC7o9t; see QR code left). Another good video is from Science Shorts (goo.gl/aT2omJ; see QR code centre) and a third can be found at goo.gl/UX9TVF (QR code right), giving a good explanation of how the kinetic energy of the released electrons can be measured by finding the stopping potential.

Emission and absorption spectra provide us with our first compelling evidence that light behaves as distinct quanta or discrete packets of energy (photons). A second piece of evidence comes from observation on what happens when light is shone on to the surface of metals – the photoelectric effect. These observations were first explained by Albert Einstein in 1905 – winning him the Nobel Prize in physics in 1921. The photoelectric effect can be demonstrated using a piece of equipment called a gold-leaf electroscope.

The gold-leaf electroscope: observations & explanations (1) In the space below, sketch a labelled diagram of a gold leaf electroscope that has been negatively charged.

(2) Describe / explain the following main observations from the experiment: (a) Explain why the gold leaf stands away from the solid metal stem of the electroscope.

(b) Describe what happens when light of low frequency shines on the metal plate.

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Aquinas College Physics

Module 4.1: Waves & Quantum

(c) Describe what happens when light of a high frequency shines on the metal plate.

There are two important variables that we can change about the incoming light: the intensity and the frequency. (3) Give a definition for the term INTENSITY of light.

With slightly more sophisticated equipment, it is possible to measure the kinetic energy with which the electrons leave the surface of the metal plate. (4) If light were behaving as a wave‌ (a) What wave property does the intensity of light radiation relate to?

(b) What you would expect to observe if light of low frequency was shone for a long time at the surface of the metal? Explain why.

(5) With light behaving as a stream of photons‌ (a) Describe what is the effect is (in terms of the photons) of increasing the intensity of light?

(b) Describe what is the effect is (in terms of the photons) of increasing the frequency of light?

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Aquinas College Physics

Module 4.1: Waves & Quantum

(c) Complete the table below explaining the changes to the photoelectrons emitted from the surface of the metal Change to the light shining onto the metal

Number of photoelectrons released each second

Energy with which photoelectrons leave each second

Change expected

Change expected

Explanation

Explanation

Change expected

Change expected

Explanation

Explanation

Increasing the intensity of the light

Increasing the frequency of the light

(6) Complete the following: “…………… photon interacts with ………………. electron” (7) The energy delivered to the electrons by the photons of light fills two roles. What are they? 1. The energy delivered by the photon…

2. Any energy left over…

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Aquinas College Physics

Module 4.1: Waves & Quantum

The photoelectric equation Your answer to Q(7) on the facing page, can also be stated as Einstein’s photoelectric equation linking the photon energy E to the work function  and the maximum kinetic energy of the electrons, KEmax. Give this equation in the box right:

An alternative way of writing this is by giving the photon energy in terms of the frequency f and Planck’s constant h and the maximum kinetic energy of the electrons in terms of the mass m and the maximum velocity vmax with which the photoelectrons escape.

E

hf

The idea of the work function of a metal is important here. Give a definition of the term work function  of a metal. Underline any key terms.

Explain why the terms KEmax and vmax are used in the equations above. Why do many photoelectrons in practice have speeds / energies less than this?

Explain what is meant by the term THRESHOLD FREQUENCY in the context of the photoelectric effect.

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Aquinas College Physics

Module 4.1: Waves & Quantum

Some questions on the photoelectric equation: (1) Calculate the minimum frequency of light that will release electrons from the surface of zinc (work function of zinc = 4.3 eV)

max. frequency = ………………………….. Hz (2) Ultraviolet light of frequency 1.7×1015 Hz is shone onto the surface of a zinc plate. What is maximum speed that electrons are released? (electron mass: 9.1×10-31 kg)

max. speed = ………………………….. m s-1 (3) Radiation of wavelength 320 nm strikes a metal surface, releasing photoelectrons with a maximum kinetic energy of 1.4×10-19 J. Calculate the work function and threshold frequency for the metal.

work function = ……………………………. J ; threshold frequency = ……………………. Hz (4) The work function of magnesium is 3.7 eV. If radiation of wavelength 170 nm is incident on the surface of the metal, calculate the maximum kinetic energy with which photoelectrons are released.

max.kinetic energy = ………………………………. J

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Aquinas College Physics

Module 4.1: Waves & Quantum

As mentioned previously, it is possible to quantify the maximum kinetic energy of emitted photoelectrons by measuring the stopping potential of the photoelectrons when connected up with a photocell (see video at goo.gl/ghV7Li ; QR code right). Having done this experiment, the results may be plotted on the axes shown below.

Some questions on the graph: (1) Plot the line for a metal with a moderately low work function, for example Caesium (ď Ś = 2.1 eV) (2) Explain the significance of the following features of the graph: (a) The intercept with the y-axis

(b) The intercept with the x-axis

(c) The gradient of the graph

(3) With a different colour (or dashed line) plot the line for a metal with a slightly higher work function (for example zinc with a work function ď Ś = 4.3 eV) (4) These lines should have the same gradient. Explain why.

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Aquinas College Physics

Module 4.1: Waves & Quantum

5. Using LEDs to experimentally find Planck’s constant A useful video explaining this experiment can be found from www.alevelphysicsonline.com at goo.gl/XD6JMn (see QR code right)

Complete the sentences below to explain the basic workings of LEDs.  LEDs consists of two parts: (1) an

n-type

layer

where

the

impurities mean that there are a ………….…….. of electrons. (2) a

p-type

layer

where

the

impurities mean that there are a ………….…….. of electrons.  In order for electrons to flow across the p-n junction, they need to …………… energy.  The energy gained by the electron to flow across the p-n junction can be found by multiplying the ………………………………….. by the ……………………………….  When the electron passes across the p-n junction, this electron drops back to its ground state. In doing so it emits a ……………………………  LEDs with different striking voltages therefore emit light of different ……………………………… The photon energy is therefore equal to the energy drop experienced by the electron as it passes across the p-n junction in the LED. In the box right, give an equation linking the electron energy change (in terms of electron charge e and striking voltage V0) to the photon energy (in terms of wavelength , Planck’s constant h and the speed of light c).



(1) Explain the experimental set-up, giving details of the method you took and readings taken to find the Planck constant h. A circuit diagram would be useful.

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Aquinas College Physics

Module 4.1: Waves & Quantum

(2) Sketch the graph that you expect to see if you plot the striking voltage V0 on the yaxis against the reciprocal of the wavelength of light, 1â „đ?œ†, emitted by the LED.

(3) Explain how you would find a value for the Planck constant h from your results.

1 Îť

(m-1)

In your answer you should refer to the equation you wrote on the facing page.

(4) Describe what steps you took to minimise experimental uncertainty.

(5) Explain how the addition of a sensitive ammeter to the circuit could help improve the reliability of your results.

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Aquinas College Physics

Module 4.1: Waves & Quantum

6. Electron diffraction, wave-particle duality & de Broglie So far we have solely considered how light – which we have previously seen concrete evidence supporting the interpretation that it behaves as a wave – can also behave as a particle. Whether light shows wave or particle characteristics depends solely on the experiment that we chose to investigate it with. So how about items (such as electrons, protons, atoms and molecules) that we have previously considered to behave as particles? Is there any evidence that they show wave behaviour? The short answer is yes. The following sections explore this idea further.

Electron diffraction tubes More information on electron diffraction is presented in the video from Sixty Symbols found at goo.gl/ZGv83z (see QR code left) and from Science Shorts at goo.gl/8ni1DZ (see QR code right).

Proof that small particles such as electrons show wave-like behaviour can be seen with a piece of equipment called an electron diffraction tube. An overview of this piece of equipment can be seen below.

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Aquinas College Physics

Module 4.1: Waves & Quantum

(1) In the spaces provided on the facing page, add details about the two components of the electron diffraction tube labelled (1) and (2)

(2) Explain what happens when electrons arrive at the phosphor-coated screen at the end of the diffraction tube. In your explanation you need to talk about electrons in the phosphor transitioning between energy levels and photon emission.

(3) Complete the diagram by drawing on the observed pattern that is seen on the screen at the end of the electron diffraction tube.

(4) What do you know about the wavelengths of the electrons fired through the target sample and the spacing between the atoms in the sample?

Space for your own notes …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. - 25 -


Aquinas College Physics

Module 4.1: Waves & Quantum

The de Broglie wavelength More detail on the de Broglie wavelength can be found from the fantastic video from Sixty Symbols (goo.gl/UhYpn5 ; see QR code left) and from www.alevelphysicsonline.com (goo.gl/DJEfaZ ; see QR code right)

A reasonable question to ask, if we accept the idea that particles such as electrons can behave as waves, is what are the factors that affect the wavelength  of these “matter waves”? This conundrum was solved by a French scientist (and Charlie-Chaplin lookalike) named Louis de Broglie in 1924. De Broglie found that the wavelength of “matter waves” is related to the momentum p of the particles considered to be behaving as waves.

In the box right, state the equation giving the de Broglie wavelength  in terms of the particle’s momentum p.

p

Some questions on the de Broglie wavelength Answer the questions below to practise your use of the de Broglie equation (mass of an electron me = 9.1×10-31 kg) (1) Calculate the wavelength of an electron moving at 1×106 m s-1

= ……………………….. m 6

(2) Calculate the wavelength of an electron moving at 5×10 m s

-1

= ……………………….. m (3) Calculate the wavelength of a man of mass 70 kg walking at 2 m s

-1

= ……………………….. m (4) Calculate the wavelength of a car of mass 1000 kg moving at 10 m s

-1

= ……………………….. m

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Aquinas College Physics

Module 4.1: Waves & Quantum

**Stretch & challenge question (5) (a) Estimate how fast you would have to walk to diffract as you go through the door of the classroom on your way out the lesson.

v = ……………………….. m s-1 (b) The Universe is 13.8 billion years old. How far would you have moved if you had travelled at the speed calculated in Q5(b) for the entire history of the universe?

d = ………………………….. m

Complete the sentence below to summarise the relationship between a particle’s momentum and its wavelength. As momentum increases, the wavelength…

This means that if we can accelerate particles up to very high speeds (for example using a particle accelerator), this allows us to image objects at increasingly smaller scales. This is how a scanning electron microscope works. For example, the image right shows a close up view of an ant’s head.

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Aquinas College Physics

Module 4.1: Waves & Quantum

7. Wave-particle duality, many paths & quantum phasors We have seen using different experiments that photons, electrons, sub-atomic particles, atoms and even molecules – quantum objects - show both wave and particle characteristics. This idea is looked at in more detail in the videos from Science Shorts (goo.gl/fSd19B; QR code left) and Bozeman Science (goo.gl/sNsGBh; QR code right).

A true understanding of the wave-particle duality shown by quantum objects has flummoxed and baffled scientists since they were first observed. The Nobel-prize winning physicist Richard Feynman famously said “if you think you understand quantum mechanics, you don’t understand quantum mechanics. Perhaps the most evocative description of this conundrum is the analogy given by Austrian physicist Erwin Schrodinger – that quantum objects behaving as both a wave and a particle, until scientific observation forces one type of behaviour over the other, is as ridiculous as a cat in a box being both alive and dead at the same time. More information on Schrodinger’s cat can be found at goo.gl/EEwJDq (see QR code right). Nonetheless, evidence does indeed exist that quantum objects have both wave a particle behviours.

Firing electrons/photons one at a time through a double slit  

When single electrons or photons are fired through a double slit one at a time they arrive at a distant screen as a single speck – showing distinctive particle behaviour. However, when this process is repeated and the pattern allowed to build up, one arrival at a time, an interference pattern begins to emerge (see diagram below).

The only way to interpret this is that wave-type behaviour is being shown by the particles as they travel from the source, through the slits, and to the distant screen. The video from Dr. Quantum explains this in a bit more detail (goo.gl/akqGVb; QR code right).

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Aquinas College Physics

Module 4.1: Waves & Quantum

Many paths interpretation of interference & quantum phasors Whilst perhaps we cannot truly understand this strange behaviour, we are able to describe it mathematically and make predictions on the likelihood of a particular photon / electron arriving at any particular point on a distant screen. Consider the set-up right, with a source of photons at point X which pass through a triple slit to arrive at the screen. Using a technique called the many paths or sum-over-paths theory of quantum physics, if we consider all of the possible paths that a photon can take to the screen, we can combine the probabilities of each of paths using quantum phasors to give the final probability of an arrival at any point on the final screen. Quantum phasors work in much the same way as the phasors we have used previously to describe positions along a wave, and to help us find the resultant amplitude when two component waves superpose. 

The wave speed and length of each path can be used to find the travel time of the quantum object between the source and screen for each of the paths taken



The energy of the quantum object can be used to find its frequency: đ?‘“ =



Knowing the frequency and travel time will allow you to find the final orientation of the phasor for each path



The probability of arrival at each point is found from the resultant amplitude of the phasors for each of the paths, when added tip-to-tail.



The probability P of arrival can be found from the resultant amplitude A of the phasors using the relationship:

đ??¸ â„Ž

(from E = hf)

probability P � To illustrate how this works, answer the questions below.

Questions on quantum phasors (1) A source of radiowaves releases photons with an energy of 6.63Ă—10-26 J. (a) What is the frequency at which the phasors rotate?

frequency = ‌‌‌‌‌‌‌‌. Hz - 29 -


Aquinas College Physics

Module 4.1: Waves & Quantum

(b) There are two likely paths that the radiowave photons may travel to reach a distant screen: (i) One of these paths is 16.5 m long. How much time does it take the photon to arrive at the distant screen?

(ii)

t = ……………….……. s How many times does the quantum phasor used to represent the photon rotate during its journey from source to screen?

no. of rotations= ……………….……. (c) The other likely path a photon can take is 24 m long. (i) What is the travel time for the photon if it takes this path?

t = ……………….……. s (ii)

How many times does the quantum phasor used to represent the photon rotate during this journey from source to screen?

no. of rotations= ……………….……. (d) Draw the phasors for the photon taking each of the two paths considered to arrive at the distant screen.

(e) From these two phasors, do you consider there to be a a high or low probability of arrival at this point on the distant screen?

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Aquinas College Physics

Module 4.1: Waves & Quantum

(2) Using the example of the triple slit shown in the diagram right: (a) Annotate the diagram showing all the likely possible paths that a photon may take between source X and point S on the distant screen. (b) Using a different colour, draw on all possible paths between source X and point Q on the distant screen.

(3) (a) In the space below, give an example of how the phasors may arrive at point S, given that it is in the centre of a bright spot in the interference patterns formed at the screen.

(b) If each individual phasor has an amplitude A, what is the resultant amplitude at point S? resultant amplitude = ……………… (c) In the space below, give an example of how the phasors may arrive at point Q, given that it is in the centre of a dark spot in the interference patterns formed at the screen.

(d) If each individual phasor has an amplitude A, what is the resultant amplitude at point S? resultant amplitude = ………………

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Aquinas College Physics

Module 4.1: Waves & Quantum

(4) The table below gives common combinations of phasors (each combination is commonly asked about in exam questions – so it is vital you can combine them to find the correct resultant phasor). A student finds that when the resultant amplitude of all the phasors is equal to A, the intensity of light at that spot is 5 W m-2. The light intensity is proportional to the probability of arrival at that point. Use this information to complete the table below.

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Aquinas College Physics

Module 4.1: Waves & Quantum

Space for your own notes …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. …………………………………………………………………………………………………………………………….. ……………………………………………………………………………………………………………………………..

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Aquinas College Physics

Module 4.1: Waves & Quantum

Using phasors to explain reflection The idea of using quantum phasors to represent the probability that a photon takes a particular path can also be used to explain why we see reflected images where we do:

(1) On the diagram right draw in the most likely path light will take between source and observer, if it reflects off the mirrored surface. Use this to sketch in the position of the image of the source on the mirror.

(2) With a quantum mechanical interpretation, the photon can take a number of different paths. Some of these paths are illustrated in the second diagram right. (a) What do you know about the path difference between each of these paths?

(b) What does this mean about the difference in the travel time for photons taking each of these paths?

(c) What does this mean about the alignment of the phasors for each of the paths?

(d) Draw the resultant phasor diagram in the box provided. (e) What does the amplitude of the resultant phasors for all of these paths tell you about the likelihood that the photons will take this path?

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Aquinas College Physics

Module 4.1: Waves & Quantum

(3) The diagram right shows some other alternative paths that the photons could take. (a) What can you say about the path differences between each possible paths shown in the diagram?

(b) What does this mean about the difference in travel time for photons taking each of these paths?

(c) What does this mean about the alignment of the phasors for each of the paths?

(d) Draw the resultant phasor diagram in the box provided. (e) What does the amplitude of the resultant phasors for all of these paths tell you about the likelihood that the photons will take this path?

(4) In the box provided sketch out the phasors for all possible paths and complete the sentences below: Where

there

is

a

low

probability of the photons arriving by different paths the phasors ……………….. Where

there

is

a

high

probability of the photons arriving by different paths the phasors ………………..

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Aquinas College Physics

Module 4.1: Waves & Quantum

8. IsaacPhysics mastery questions These questions have their background in the ethos that anything is hard when you don’t spend much time on it, and that things become much easier the more your practice. You will know this if you are learning to drive, or when you pick up any other new skill for the first time. I would say that playing the piano is really really hard. But I’ve probably not spent more than a few hours when I was very young trying. Someone who has spent hours practising playing the piano would probably tell you that it is easy as it comes naturally to them – neglecting to mention all the hours of practice that they have put in. PHYSICS IS NO DIFFERENT!! THE MORE YOU PRACTICE THE BETTER YOU GET. Have a go at the mastery questions on the following pages. You can input your answers and check they are right on the isaacphysics.org website (see separate links for each section). You should be aiming to get correct at least the number given at the top of each section before you can consider yourself to have mastered each of these core concepts.

Questions on Quantum Calculations *Starts easy, and gets harder – aim for 16/19 at least to gain a mastery of these concepts. These questions come from the Isaac Physics skills mastery book (buy this and the CGP revision guide through ParentPay for just £10!!). You can enter your answers and complete these questions with the Isaac Physics board at isaacphysics.org/s/ok1nKw

(1) Fill in the blanks in the table below Frequency of light / Hz

Wavelength of light / nm

Photon energy /J

Photon energy / eV

…………………………

…………………………

…………………………

…………………………

…………………………

…………………………

1.5

…………………………

…………………………

…………………………

2.5

…………………………

500

…………………………

…………………………

…………………………

1013

…………………………

…………………………

…………………………

…………………………

…………………………

6.0×1014

2.0×1015

(2) A laser diode requires 3.2 V across it to make it work. This means that its photons will have an energy of 3.2 eV. Calculate the wavelength of light emitted.

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Aquinas College Physics

Module 4.1: Waves & Quantum

(3) When an electron annihilates a positron (its antiparticle), two photons are produced, each with an energy of 511 keV. Calculate the frequency of each photon.

(4) Complete the table below using the de Broglie equation. (melectron = 9.1Ă—10-31 kg ; mneutron = 1.7Ă—10-27 kg) wavelength / nm

particle

momentum / kg m s-1

kinetic energy / eV

3.0

Electron

‌‌‌‌‌‌‌‌‌

‌‌‌‌‌‌‌‌‌

3.0

Neutron

‌‌‌‌‌‌‌‌‌

‌‌‌‌‌‌‌‌‌

‌‌‌‌‌‌‌‌‌

Electron

‌‌‌‌‌‌‌‌‌

105

(5) Calculate the momentum of an electron if its kinetic energy is 10 keV (clue: the kinetic energy of a particle can be found in terms of its momentum using the equation đ??¸đ?‘˜ =

đ?‘?2â „ 2đ?‘š)

(6) An electron’s wavelength is 3.0×10-7 m. What is its momentum?

(7) The tandem electrostatic accelerator can accelerate carbon-12 nuclei to a kinetic energy of 60 MeV. How fast are they going? Assume m = 12 Ă— mneutron.

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Aquinas College Physics

Module 4.1: Waves & Quantum

(8) An electron is travelling at 2.0×106 m s-1. Calculate its momentum and its kinetic energy. Then use the momentum to calculate the wavelength and use the energy to calculate its frequency.

(9) Use c = f to ‘calculate’ the speed of the electron in Q(8) above using the frequency and wavelength. What do you notice?

Questions on the photoelectric effect *Starts easy, and gets harder – aim for 13/16 at least to gain a mastery of these concepts. These questions come from the Isaac Physics skills mastery book (buy this and the CGP revision guide through ParentPay for just £10!!). You can enter your answers and complete these questions with the Isaac Physics board at isaacphysics.org/s/cQwaVV

(1) Complete the table below frequency of light / Hz

wavelength of light / nm

6.0×1014

……………………

1.2×10-19 J

6.0×1014

……………………

2.6 eV

……………………

……………………

350

2.6 eV

……………………

work function

max KE of photoelectrons

stopping potential /V

……………………

350 …………………… …………………… Clue: the maximum kinetic energy of an electron (in eV) is equal to its stopping potential (in V)

1.35

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Aquinas College Physics

Module 4.1: Waves & Quantum

(2) A material has a work function of 1.3 eV. Calculate its threshold frequency

(3) A material will not emit photoelectrons unless it is irradiated by light with a wavelength less than 380 nm. Calculate its work function in electronvolts.

(4) Calculate the maximum speed of the photoelectrons emitted when a material with an 8.4×10-20 J work function is illuminated by light of frequency 7.0×1014 Hz.

(5) What is the minimum speed of the photoelectrons emitted in Q(4) above?

(6) A graph of stopping potential (y) against frequency of light (x) is plotted for zinc and also for aluminium. Without knowing any more information, answer the following: (a) Are the lines straight or not?

(b) Are the y-intercepts positive, negative or zero?

(c) Are the gradients positive, negative or zero?

(d) Are the gradients of the two lines the same or different?

(e) Are the y-intercepts of the two lines the same or different?

(f) What is the significance of the x-intercept?

(g) If you answered ‘same’ to parts (d) or (e), write down the value of the common gradient or intercept.

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Aquinas College Physics

Module 4.1: Waves & Quantum

(7) A material has a work function of 3.4 eV, and is illuminated by 5.0 eV photons. Calculate the stopping potential of its photoelectrons.

Space for your own notes …………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………… ………………………………………………………..…………………………………………………………………. …………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………… ………………………………………………………..…………………………………………………………………. …………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………… ………………………………………………………..…………………………………………………………………. …………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………… ………………………………………………………..………………………………………………………………….

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Module 4.1: Waves & Quantum

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Aquinas College Physics

Module 4.1: Waves & Quantum

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Aquinas College Physics

Module 4.1: Waves & Quantum

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4.1.G&H Quantum Physics  
4.1.G&H Quantum Physics  
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