# Unit 2: Section 1 Waves and Quantum Behaviour

Chapter 6 and 7 Advancing physics

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• Created by: R_Hall
• Created on: 01-05-13 20:31

## Superposition and Coherence

• When two or more waves cross, the resultant displacement equals the vector sum of the individual displacements
• Crest + crest or trough + trough is constructive interference, a bigger crest/trough is made
• Crest+ trough is destructive interference, there is no resultant
• Rotating arrows represent the phase of each point on a wave. Called phasors and rotate anti-clockwise in a full circle as a wave completes a full cycle
• Two waves are in phase- both at the same point in the wave cycle. One complete wave cycle is 360° (or 2π radians)
• If there is a phase difference of 0 or a multiple of 360°, the waves are in phase (phasors in same direction). Points of a phase difference of odd number multiples of 180° (π radians) are in antiphase (exactly out of phase) Phasors in opposite direction
• In order to get a clear interference patter, two or more wave sources must be coherent- same wavelength and frequency, fixed phase difference
• Whether you get constructive/destructive interference depends on path difference- the amount by which the path of one wave travels is longer that that of another wave
• At any point at a equal distance from both sources- constructive. When path difference is a whole number of wavelengths- constructive
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## Standing Waves

• A standing  wave is the superposition of two progressive waves with the same wavelength, moving in opposite directions.
• No energy is transmitted by a standing wave
• Standing waves in strings form oscillating 'loops' seperated by nodes
• Each particle vibrates at right angles to the string. Nodes are where the amplitude is 0. Antinodes are where the amplitude is maximum
• At resonant frequencies, an exact number of 1/2 wavelengths fits onto the string
• Longitudinal standing waves form in a wind instrument. The node will form at the closed end. Lowest resonant frequency when length of pipe= 1/4 wavelength
• Antinodes form at open ends. If both open, lowest resonant frequency when l= 1/2 wavelength
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## Diffraction

• Diffraction- waves spread out as they come through a narrow gap. Gap lots bigger than wavelength= diffraction unnoticeable. Gap same size as wavelength= most diffraction
• If gap smaller than wavelength, waves mostly reflected back
• When a wave meets an obstacle, diffraction around edges and a 'shadow' behind object. Wider obstacle-longer shadow
• With light waves, you get a pattern of light and dark fringes. If wavelength= size of aperture -> diffraction pattern. Bright central fringe with alternating dark/light fringes on either day. Narrower the slit, wider the diffraction pattern
• Brightest point is where the light passes in a straight line from slit to screen. All waves are in phase
• All other bright points, constant phase difference, so the phasors all point in a slightly different direction- smaller resultant
• Dark fringes are where the phase difference between waves mean phasors add up to form a circle, 0 resultant
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## Two-Source Interference

• For two-source interference to occur, you need coherent sources (same wavelength and frequency). Easy with sound and water, but hard with light-
• A single laser light is shone through two slits, as it is coherent and monochromatic
• Slits have to be same size as wavelength of laser light so it is diffracted. Light from the slits acts as two coherent point sources
• Pattern of dark and light fringes depending whether constructive or destructive interference.
• Thomas Young came up with an equation to work out wavelength of light from this experiment- Young's double-slit formula
• The fringes are so small that it's hard to get an accurate value of X. Measure across several fringes and divide by number of fringe widths between them
• Newton suggested light was made up of particles called 'corpuscles'. Explained reflection and refraction, but only waves (as put forward by Huygens) can show diffraction and interference
• Young's double slits confirmed Huygen's wave theory- light can diffract (through narrow slits) and interfere (form interference pattern)
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## Diffraction Gratings

• When you diffract through more slits, the patterns get sharper. Bright bands are brighter and dark bands are darker. Sharper fringes make for more accurate measurements
• For monochromatic light, there is a line of maximum brightness at the centre- zero order line. The lines either side are the 1st order, and the next pair are the 2nd order
• The larger the wavelength, the more the pattern will spread out.
• The coarser the grating, the less the pattern will spread out
• White light is a mixture of colours. If diffracted, the patterns due to different wavelengths within the white light spread out by different amounts
• Each order becomes a spectrum. 0 order stays white as all the wavelengths just pass through
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## Light and Photons

• Light travels very fast, makes speed of light hard to measure.
• The photoelectric effect can only be explained if light acts as a particle- a photon
• If light of high frequency is shone onto the surface of a metal, it will emit electrons. Free electrons on the surface absorb energy and vibrate, and eventually the bonds holding it to the metal break and the electron (photoelectron) is released
• No electrons are emitted below a certain frequency- threshold frequency. Electrons have different KE's, value of maximum KE increases with frequency of radiation, not intensity. Number of electrons emitted per s proportional to intensity
• Einstein suggested than EM waves can only be released in discrete packets- quanta, and that they can only exist in discrete packets called photons
• Higher the frequency, more energy carried
• He believed photons act as particles, will either transfer all or none of its energy
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## Energy Levels and Photon Emission

• Electrons in an atom can only exist in certain defined energy levels. Each level has a number, n=1 is ground level. Electrons can move down a level by emitting a photon
• Since transitions are between definite energy levels, the energy of each photon emitted can only take a certain value. Energy carried by each photon= difference in energies between two levels
• If a gas is heated, electrons move to a higher energy levels before falling back down, emitting energy as photons
• If the light from a hot gas is split (with prism or diffraction grating) get a line spectrum (bright lines on black background). Each line corresponds to the wavelength of emitted light
• The spectrum of white light is continuous, there are no gaps in the spectrum. Hot things emit a continuous spectrum in the visible and infrared
• Line absorption spectrum- light with a continuous spectrum passed though cool gas. A low temps, most electrons will be in ground state. Photons of correct wavelength are absorbed by electron to promote to high energy levels
• See a spectrum with black line corresponding to absorbed wavelengths
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## The 'Sum Over Paths' Theory

• Feynman believed that a photon will explore all possible paths from source to detector in one go. This can be mapped using phasors
• In quantum mechanics, phasors are used to see how probable it is for a quantum to arrive in a particular place.
• As a phasor travels, it will rotate anti-clockwise until it reaches the detector. Using the energy and Planck's constant, frequency can be calculated
• As there are an infinite number of phasors, only calculate the straightest/quickest possible paths
• You can find the probability that a quantum will arrive at a point by squaring the resultant phasor amplitude
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## Using 'Sum Over Paths'

• When you reflect a photon off a mirror, it will explore every possible path. By finding the phasors for each path, you can find the final resultant
• The phasore nearest the quickest path have phasors that almost line up, giving the most amplitude of the resultant, so the most probability that the photon will reach the detector
• The phasors for longer, slower paths curl up to cancel themselves out- adding almost nothing to the resultant
• The final phasor of the quickest path will contribute the most to the resultant amplitude and the probability of a quantum arriving at a point
• This explains why light travels in a straight line- shortest and quickest path
• When light travels in water, it slows down, but frequency is constant. The photons still have the same energy, the phasor will have the same amplitude and frequency despite material
• To focus photons, you need to make sure all straight line paths from source to focus point take the same amount of time
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## Quantum Behaviour of Electrons

• The de Broglie equation relates a wave property to a moving particle property
• Diffraction patterns are observed when accelerated electrons in a vacuum tube interact with the spaces in a graphite crystal. As an electron hits a fluorescent screen, it causes a photon to be released. Using phasors, the higher the probability, the brighter the point.
• This confirms electrons show quantum behaviour
• Increase the electron speed and the diffraction pattern circles squash together towards the middle
• You only get a diffraction if a particle interacts with an object about the same size as its de Broglie wavelength
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