Revision Guide Chapter 7 Quantum Behaviour
Contents CONTENTS... 2 REVISION CHECKLIST... 3 REVISION NOTES... 4 QUANTUM BEHAVIOUR... 4 Random arrival of photons... 4 Photoelectric effect... 5 PHASE AN PHASORS... 6 PHASOR ARROWS AN MANY PATHS... 7 Reflection... 8 Refraction... 10 2 slit Interference... 11 Single slit interference... 12 FERMAT S LEAST TIME PRINCIPLE... 13 QUANTUM BEHAVIOUR SUMMARY... 14 LINE SPECTRA... 14 Emission of photons from atoms... 14 iffraction gratings... 14 Energy levels... 15 ELECTRON IFFRACTION... 15 e Broglie wavelength... 16 2
Revision Checklist I can show my understanding of effects, ideas and relationships by describing and explaining: how phasor arrows come to line up for paths near the path that takes the least time how phasor arrows 'lining up' and 'curling up' account for straight-line propagation, reflection, refraction, focusing, diffraction and interference (superposition) of light that the probability of arrival of a quantum is determined by graphical addition of arrows representing the phase and amplitude associated with each possible path evidence for random arrival of photons evidence for the relationship E = hf evidence from electron diffraction that electrons show quantum behaviour I can use the following words and phrases accurately when describing effects and observations: frequency, energy, amplitude, phase, superposition, intensity, probability path difference, interference, diffraction I can interpret: diagrams illustrating how paths contribute to an amplitude I can calculate: the energy of a photon using the relationship E = hf the de Broglie wavelength of an electron using the relationship = h/mv I can show my ability to make better measurements by: measuring the Planck constant h I can show an appreciation of the growth and use of scientific knowledge by: commenting on the nature of quantum behaviour 3
Revision Notes Quantum Behaviour In Chapter 6 of this course, light was regarded as a wave and as such as being a continuous flow of energy. However this model is not always adequate on its own to explain the observed behaviour of light and because of this quantum physics has developed. Quantisation is about physical quantities that may take discrete values only. The photon is the quantum or smallest lump of electromagnetic radiation. Examples showing evidence for quantum behaviour are: Random arrival of photons The random nature of the arrival of photons is most easily seen using high energy gamma ray photons, which can be heard arriving randomly in a Geiger counter. The pictures below illustrate the random arrival of photons. They are constructed as if made by collecting more and more photons to build up the picture. Where the picture is bright the probability of arrival of a photon is high. Where it is dark, the probability is low. You can see how the random arrival, governed by these probabilities, builds up the final picture. The random arrival of photons on a sensitive photographic film as it is gradually exposed for longer and longer. When the picture is fully built up the random arrival of photons is not evident but when only a few photons have arrived it is very clear. See the photograph of a woman s face below. The random arrival of (gamma)-rays at a Geiger-Muller tube. Quantum behaviour is more obvious when there are few photons and the energy of each is relatively high. 4
Photoelectric effect When light is shone on a metal surface, electrons can be emitted. Photoelectricity photons metal electron absorbs photon and leaves the metal For each metal there is a minimum frequency of light that will cause the emission of electrons. Below this frequency none are emitted whatever the intensity. Above it the rate of emission of electrons increases with increasing intensity. The maximum energy of each photoelectron depends on the frequency (colour) of the light. It is not possible to explain all of these observations by considering the light as a continuous flow of wave energy where the energy depends on the amplitude. However the photoelectric effect makes sense if light is considered as a stream of photons with each photon emitting one electron. It takes a certain amount of energy to emit an electron so photons with less energy than this will never emit one. The more energetic the photon the more energy the electron will have. The photon theory of electromagnetic radiation was proposed by Einstein to explain the photoelectric effect. Einstein s equation is just an application of the law of conservation of energy. Photon energy = energy needed to release electron + (maximum) k.e. of electron hf = + ½mv 2 (max) The energy needed to emit an electron from the metal is called the work function. Emission can be stopped if the metal is made sufficiently positive so that ev s =. Experiments such as the photoelectric effect show that high frequency radiation e.g. ultra-violet has more energy than visible light. The energy E of each photon is proportional to its frequency f E = hf (on formula sheet) The constant h is the Planck constant = 6.6 10-34 Js in SI units. In most applications the quantum behaviour is not obvious as a very large number of low energy photons give the impression of a continuous flow. For example in the case of the visible light emitted from an ordinary light bulb the rate of emission of photons is many millions each second. 5
Phase and phasors 'Phase' refers to stages in a repeating change, as in 'phases of the Moon'. The phase difference between two objects vibrating at the same frequency is the fraction of a cycle that passes between one object being at maximum displacement in a certain direction and the other object being at maximum displacement in the same direction. Phase difference is expressed as a fraction of one cycle, or of 2 radians, or of 360. Phasors are used to represent amplitude and phase in a wave. A phasor is a rotating arrow used to represent a sinusoidally changing quantity. Suppose the amplitude s of a wave at a certain position is s = a sin(2 ft), where a is the amplitude of the wave and f is the frequency of the wave. The amplitude can be represented as the projection onto a straight line of a vector of length a rotating at constant frequency f, as shown in the diagram. The vector passes through the +x-axis in an anticlockwise direction at time t = 0 so its projection onto the y-axis at time t later is a sin(2 ft) since it turns through an angle 2 ft in this time. Phasors can be used to find the resultant amplitude when two or more waves superpose. The phasors for the waves at the same instant are added together 'tip to tail' to give a resultant phasor which has a length that represents the resultant amplitude. If all the phasors add together to give zero resultant, the resultant amplitude is zero at that point. Generating a sine wave B y A +a A B 0 0 T 2 T C C phasor rotating at constant frequency f Ša time t s = a sin (2 ft) 6
Phasor Arrows and Many Paths As a photon travels through space it can be represented by a rotating phasor arrow. The number of complete turns per second is equal to the frequency. You can work out how many times a photon s phasor arrow spins as it moves along a path of known length. Take a photon of frequency 5 10 14 Hz moving along a path of length 0.5 m. Using speed = distance/time Time taken to cover the path = distance/speed = 0.5 m (3 10 8 ms -1 ) = 1.6666667 10-9 s. Number of rotations = (5 10 14 Hz) (1.6666667 10-9 s) = 8.3333333 10 5 This is 833,333 1 / 3 rotations so if the phasor arrow starts off pointing vertically upwards and rotates anticlockwise it will have turned 1 / 3 of a rotation past the vertical at the end of its path. 120 Initial phasor arrow Arrow at end of path Where many of them arrive at a point they add tip to tail like vectors to give the resultant amplitude. The amplitude is squared to give the probability of a photon arriving at the detector and thus the intensity of the light. Intensity probability of arrival of photons (resultant phasor amplitude) 2 P A 2 One of the key ideas in this chapter is that photons travelling from a source to a receiver do not just travel along the expected path e.g. along a straight line between two points but that they try all possible paths. If we then add their photon arrows we find that those along the expected paths make a big contribution to the resultant amplitude but those along other paths tend to make only a small contribution. This idea can be applied to a number of familiar physical situations involving light such as reflection, refraction, diffraction and interference. The phasors arrows give low resultant amplitudes at dark regions and high resultant amplitudes at bright ones. 7
Reflection This diagram shows possible photon paths from the source S to the detector. S Mirror The next diagram shows how long photons take to follow paths via various points on the mirror. The minimum trip time occurs for the shortest path, which is the one for which the angle of incidence equals the angle of reflection. Trip time t for path t t x large t x large t x t x small x t x Position along mirror x The next diagram shows the directions of the phasor arrows at the end of each path. Note that near the minimum trip time path, movement along the mirror surface makes little difference to the trip time and hence the phasor direction is similar. At the edges of the mirror movement from one position to another makes a much bigger difference to the trip time so the phasor arrows show much more change in direction. The final diagram shows the effect of adding the arrows tip-to-tail. For paths near the centre the phasors line up to produce a large resultant amplitude. For paths near the edges the phasor arrows curl up and make little contribution to the amplitude. 8
starting with a plane mirror S not much chance of getting photons here set up detector where we would like to get a focus start bending the mirror to get the arrows to line up S keep bending until the arrows line up? up a little here down a little here up a little more here 9
Refraction Refraction Š explorations through a surface S Place the source, detector and surface. Light appears to travel more slowly below the surface, so we reduce the speed of the exploring phasor. The frequency is unchanged. S S Choose a photon frequency and define a characteristic set of paths going via the surface. Explore each path by moving a phasor along the path. Start with a fresh phasor each time and record the final arrow. Record these arrows in order. The trip time is calculated in two parts: above and below the surface. The phasor spins at the same frequency. The time taken determines the angle through which it has turned. Obtain and square the amplitude to find the chance that a photon ends up at this detector. Refraction occurs Š quantum mechanics says that there is a large chance that the photon be found at the detector. Most of the final amplitude comes from paths just to the right of the straight line path; paths close to the path of least time. near least time path far from least time path Explore more paths to get more arrows, a clearer picture and greater accuracy. The pattern is clear. Most of the amplitude comes from the paths close to the path that takes least time, only a little from those far out. 10
2 slit Interference If light from a narrow source is passed through a pair of closely spaced slits onto a screen, a pattern of interference fringes is seen on the screen. Photons have two paths to the screen, and must be thought of as trying both. There is a phasor quantity (amplitude and phase) associated with each path. Since the paths are nearly equal in length the magnitude of the amplitudes for each path is similar, but the phases differ. The phasor for a path rotates at the frequency of the light. The phase difference between two paths is proportional to the path difference. At points on the screen where the phasors have a phase difference of half a turn, that is 180, dark fringes are observed because the phasors added 'tip to tail' give zero resultant. Where the phasors are in phase (zero or an integer number of turns difference) there are bright fringes. The intensity on the screen is proportional to the square of the resultant phasor. Interference dark if phasors give zero resultant path difference L slits 11
Single slit interference In the propagation of photons from source to detector across an empty space, the probability of arrival of photons anywhere but close to the straight line from source to detector is very low. This is because, not in spite of, the many other possible paths. The quantum amplitudes for all these paths add to nearly zero everywhere except close to the straight line direction. As soon as the space through which the light must go is restricted, by putting a narrow slit in the way, the probability for photons to go far from the direction of straight line propagation increases. This is because the cancelling effect of other paths has been removed. The net effect is that the narrower one attempts to make the light beam, the wider it spreads. Trying to pin down photons Very wide slit x The photon has lots of space to explore between x and y: as a result its likely arrival places are not much spread out. S Only near the straight through path do the phasor arrows make a large resultant. y barrier to restrict paths explored scan detector to predict brightness on a screen chance the photon ends up at each place Wide slit x As the photon passes xy it has only a few paths to explore. Path differences are small. S Phasor arrows add to a large resultant at a wide spread of places. y barrier to restrict paths explored scan detector to predict brightness on a screen chance the photon ends up at each place 12
Fermat s least time principle Fermat had the idea that light always takes the quickest path the path of least time. You see below a number of paths close to the straight line path from source to detector. A graph of the time for each path has a minimum at the straight line path. Path of least time Negligible change in time near the minimum 0 distance of mid-point from that of straight line path phasors from paths near minimum are in phase and combine to give large resultant Near the minimum the graph is almost flat. This is a general property of any minimum (or maximum). That is, near the minimum the times are all almost the same. The amount by which a photon phasor turns along a path is proportional to the time taken along the path. Thus, for paths near the minimum all the phasors have turned by more or less the same amount. They are therefore all nearly in phase with one another. They line up, giving a large resultant amplitude. This is the reason why Fermat s idea works. Only for paths very close to the path of least time is there a large probability for photons to arrive. The photons try all paths, but all except the paths close to the least-time path contribute very little to the probability to arrive. The idea explains photon propagation in a straight line, reflection and refraction. 13
Quantum behaviour summary Quantum behaviour can be described as follows: 1. Particles are emitted and absorbed at distinct space-time events. 2. Between these events there are in general many space-time paths. 3. The presence of all possible space-time paths influences the probability of the passage of a particle from emission to absorption. 4. Each path has an associated amplitude and phase, represented by a rotating phasor arrow. 5. The phasor arrows for all possible paths combine by adding 'tip to tail', thus taking account of amplitude and phase. 6. The square of the amplitude of the resultant phasor is proportional to the probability of the emission event followed by the absorption event. A photon, although always exchanging energy in discrete quanta, cannot be thought of as travelling as a discrete 'lump' of anything. Photons (or electrons) arriving at well-defined places and times (space-time events) are observable. But their paths between emission and detection are not well-defined. Photons are not localised in time and space between emission and absorption. They must be thought of as trying all possible paths, all at once. Line spectra Emission of photons from atoms When an electron moves from a higher to a lower energy level in an atom, it loses energy which can be released as a photon of electromagnetic energy. Since the energy of a photon = h f, then if an electron transfers from an energy level E 2 to a lower energy level E 1, the energy of the photon released = h f = E 2 E 1. In this way, the existence of sharp energy levels in atoms gives rise to sharp line spectra of the light they emit. iffraction gratings Light can be split up into its component wavelengths by passing it through a diffraction grating to produce a spectrum. When the spectrum of a gas that has been made to glow by passing an electric discharge through it is observed it consists of a series of discrete wavelengths that is characteristic of the atoms present. This gives a series of lines as opposed Spectral to a continuous lines and spectrum. energy levels It is called a line spectrum. energy energy levels of an atom n = 4 n = 3 4 3 3 2 low energy long wavelength n = 2 4 2 2 1 3 1 4 1 n = 1 E = hf photon emitted as electron falls from one level to a lower level high energy short wavelength Spectral lines map energy levels 14 E = hf is the energy difference between two levels
Energy levels This occurs because the electrons in the gas atoms can only exist at certain energies called energy levels. emission spectrumeach line represents the energy of a particular level. Each type of atom has a characteristic set of energy levels. If an electron is knocked up to a higher level (it cannot have an energy between levels) it will fall back and emit a photon whose energy is equal to the difference in energy of the levels the electron falls between. An atom emits a photon as a result of an electron transferring to a lower energy level. If an electron transfers from energy level E 2 to a lower energy level E 1, the emitted photon has energy E = h f = E 2 E 1. The main application of this is in spectroscopy. By looking at the line spectrum emitted by a gas we can tell which atoms or molecules are present in the gas by looking to see which characteristic sets of lines are present. This is especially useful in examining the light emitted by hot gas clouds in space, as we cannot analyse their composition in any other way. Electron diffraction If a beam of electrons is accelerated by a high potential difference V applied between the negative cathode and the positive anode of an evacuated tube then the kinetic energy they gain is equal to the decrease in electrical potential energy ½ mv 2 = ev (where e is the charge on an electron) graphite target phosphor screen electron gun ~ 3-5 kv If the electrons pass through a thin piece of graphite they are diffracted by the layers of carbon atoms in the graphite and a series of concentric rings is formed. 15
Possible paths for electrons being scattered by successive layers of atoms differ in length, and so in the phase of the associated phasor. The phasors for paths going via successive layers of atoms only combine to give a large amplitude in certain directions. e Broglie wavelength If the quantum behaviour of a free electron is thought of as associated with a wave motion, the wavelength of the waves is the de Broglie wavelength Broglie wavelength h (where mv is the momentum) so h mv p Electron diffraction 2 nd order 1 st order film (or screen) electron beam thin crystal zero order 1 st order 2 nd order 16