waves + quantum Flashcards
when do waves superpose?
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superposition happens when two or more waves pass through each other
what happens during superpositioning? (simple)
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⋅ superposition happens when two or more waves pass through each other
⋅ the displacements of the waves combine at the instance where the waves cross
⋅ then the waves continue on separately
what is the principle of superposition?
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the principle of superposition states that “when two or more waves cross, the resultant displacement [of the new wave] equals the vector sum of the individual displacements of the waves”
what are the two types of interference?
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interference can be constructive or destructive
what are examples of constructive interference?
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examples of constructive interference are:
⋅ superposing of crest + crest = makes bigger crest
⋅ trough + trough = bigger trough
what are examples of TOTAL destructive interference?
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examples of total destructive interference are:
⋅ crest + equal size trough = nothing
⋅ (and vice versa)
what is needed for total destructive interference?
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⋅ the displacements that will cancel out each other completely (crest + trough) must be equal-sized
⋅ if the displacements are not equal-sized, the destructive interference will not be total, instead it will result in a smaller wave
what are examples of destructive interference?
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examples of destructive interference:
⋅ crest + not equal size trough = smaller wave
⋅ (and vice versa)
what is needed for the interference of waves to be noticeable?
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for the interference of waves to be noticeable, the two amplitudes combining should be nearly equal
what can you use phasors for?
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you can use phasors to show superposition
how do the features of a phasor relate to the features of a wave?
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⋅ the frequency of the phasors are the same as the frequency of the wave
⋅ the length of a phasor is the same length as the amplitude of the wave
how do you superimpose waves using phasors?
what does it mean if [two points on a] wave[s] are in phase?
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two points on a wave are in phase if they are both at the same point in the wave cycle
how many radians (or degrees) are in 1 complete wave cycle, and what does this mean?
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⋅ 1 complete wave cycle = 2π radians (or 360 degrees)
⋅ so the angle the phasor will move through is 2π radians (or 360 degrees)
when are points on a wave in phase?
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⋅ points on a wave are in phase if the phase difference between them is 0 or a multiple of 2π radians (or 360 degrees) (eg, 0, 2π, 4π, 6π)
what directions will the phasors of points that are in phase point?
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phasors of points in phase will point in the same direction
when are points in antiphase?
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points are in antiphase if the phase difference between them is an odd-number multiple of π radians (or 180 degrees) (eg, π, 3π, 5π)
what directions will the phasors of points that are in antiphase point?
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the phasors of points in antiphase will point in opposite directions
what does it mean if two points on a wave are in antiphase?
two points on a wave are in antiphase if they are both at complete opposite points in the wave cycle
what is phase difference
can points on waves only be in phase or antiphase?
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⋅ NO
⋅ points don’t have to just be in phase or antiphase, they can have a phase difference of any angle
are waves usually in phase, and what does it mean if they are?
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⋅ in most situations, there will nearly always be a phase difference between two waves
⋅ if two waves ARE in phase, it’s usually bc the waves came from the same oscillator
what are the conditions to get clear interference patterns?
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to get a visible/clear interference pattern:
⋅ the two (or more) sources must be coherent
⋅ the displacements of the waves from the sources must be equal-sized
what happens to the interference pattern when the two (or more) sources are not coherent?
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⋅ if the two sources (or more) are not coherent, an interference will still happen, but it will not be clear
what does it mean if (two or more) sources are coherent?
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sources are coherent if they emit waves with the same wavelength, the same frequency, and a fixed phase difference between the waves
what will the fixed phase difference between waves usually be in exam questions and what does this mean?
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the fixed phase difference is usually zero, so the sources will then also be in phase
what does the type of interference that occurs depend on?
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⋅ interference will be constructive or destructive depending on the path difference (how much further one wave has travelled than the other to get to the point) of the waves combining
⋅ this is assuming the sources are coherent and in phase
what is path difference?
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⋅ path difference is the amount by which the path travelled by one wave is longer than the wave travelled by the other wave
OR
⋅ path difference is how much further one wave has travelled than another to get to a particular point
when is there constructive interference? (path difference)
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⋅ there is constructive interference at any point an equal distance from both sources (that are coherent and in phase), or where the path difference is a whole number of wavelengths
when is there constructive interference? (path difference, equation form)
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⋅ there is constructive interference where the path difference = nλ
⋅ where n = integer
when is there destructive interference? (path difference)
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⋅ there is total destructive interference at any point where the path difference is an odd number of half wavelengths
when is there destructive interference? (path difference, equation form)
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⋅ there is total destructive interference where the path difference = (2n+1)/2 λ
OR
⋅ where (n+1/2)λ
can you observe interference with sound waves?
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yes
how do you observe interference with sound waves? (method)
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1) connect speakers to same oscillator (so they’re coherent + in phase) + place them in line with each other
2) slowly move microphone in straight line parallel to line of speakers
3) using data logger + computer, you can see where sound is loudest + quietest - these are locations of maximum constructive + destructive interference
when do you get a standing wave?
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you get standing waves when a progressive wave is reflected at a boundary onto another wave
what is a standing wave?
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a standing wave is a superposition of two progressive and coherent waves, moving in opposite directions
is energy transmitted by a standing wave?
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no energy is transmitted by a standing wave, unlike a progressive wave
how do you demonstrate a standing wave? (simple)
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1) you can demonstrate a standing wave by setting up a driving oscillator at one end of a stretched string with the other end fixed
2) the wave generated by the oscillator is then reflected back and forth
when do resonant frequencies occur and what do you get at them?
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1) resonant frequencies occur when the oscillator happens to produce an exact number of waves in the time it takes for a wave to get to the end [of the string] and back again
2) the original wave and the reflected waves then reinforce each other
⋅ at other frequencies the resultant pattern is a jumble
⋅ at these ‘resonant frequencies’ you get a standing wave where the resultant pattern doesn’t move (standing waves)
do standing waves occur at resonant frequencies?
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yes
what do standing waves on strings form?
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standing waves on strings form oscillating ‘loops’ (antinodes) separated by nodes
how do you investigate standing waves?
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method:
1) take a piece of string + fix it in place at one end
2) attach the other end to the oscillator
3) adjust the frequency of the oscillator, until a standing wave is formed
4) this is when a wave is reflected back on itself + interferes causing “loops” to form, with antinodes (positions of maximum amplitude on a standing wave) + nodes (positions of zero amplitude)
5) you can then use an oscilloscope to calculate resonant frequency
what happens at a standing wave’s fundamental frequency?
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at the fundamental frequency, a standing wave is vibrating{/oscillating] at its lowest possible resonant frequency
(example of fundamental frequency:)
what harmonic is the fundamental frequency (f0) the same as?
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the fundamental frequency (f0) is the same as the first harmonic
what does the first harmonic/funadmental frequency (f0) look like on a wave [/string] with fixed ends?
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⋅ the standing wave has a node at each end and an antinode in the middle
⋅ the first harmonic = f0
⋅ L (length of string) = 0.5λ
what does the second harmonic/first overtone look like on a wave [/string] with fixed ends?
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⋅ the standing wave has 3 nodes and 2 antinodes
⋅ the second harmonic [or first overtone] = 2f0
⋅ L (length of string) = λ
what does the third harmonic/second overtone look like on a wave [/string] with fixed ends?
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⋅ the standing wave has 4 nodes and 3 antinodes
⋅ the third harmonic [or second overtone] = 3f0
⋅ L (length of string) = 1.5λ
what is an easier way to calculate what harmonic or overtone a wave [/string] with fixed ends is?
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⋅ the nth harmonic = the n number of antinodes
⋅ the nth overtone = n-1 number of antinodes
(except for the fundamental frequency/first harmonic)
what type of waves are the notes played by stringed and wind instruments?
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the notes played by stringed and wind instruments are standing waves
how do the standing waves that form notes form on stringed instruments?
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1) transverse standing waves form on the strings of stringed instruments
2) your finger or bow sets the string vibrating at the point of contact
3) waves are sent out in both directions from the point of contact and reflected back at both ends
are the standing waves formed in wind instruments (or other air columns) longitudinal or transverse?
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the standing waves that form in wind instruments (or other air columns) are longitudinal
what happens if a source is placed at the open end of a column of air? (structure of standing waves)
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1) if a sources is placed at the open end of a column of air, there will be some frequencies for which resonance occurs and a standing wave is set up
2) if the instrument has a closed end, the node will be formed at the closed end
3) if there are open ends, antinodes will form at the open ends of the pipes
with a column of air that has one closed end, when do you get the lowest resonance frequency
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with a column of air that has one closed end, you get the lowest resonant frequency when the length (L) of the pipe is a quarter of the wave’s wavelength
with a column of air that has both ends opens, when do you get the lowest resonance frequency?
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if both ends are open, you get the lowest resonant frequency when the length (L) of the pipe is half the wave’s wavelength
how do you read an oscilloscope?
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⋅ the screen of an oscilloscope is split into squares called divisions
⋅ the vertical axis of an oscilloscope shows voltage, and the gain dial controls the voltage represented by each division
⋅ the horizontal axis of the oscilloscopes shows time, and the timebase dial controls the time represented by each division
how does data reach the oscilloscope?
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⋅ a cathode ray measures the voltage across something (component or whole circuit)
⋅ the cathode ray then displays the waves from the oscillator as a function of voltage over time
(signal generator = oscillator)
how do you calculate the frequency of a wave using an oscilloscope?
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1) to calculate the frequency of a wave, first you must find the period (T) of the wave
2) to find T, count how many horizontal squares one wavelength on the oscilloscope screen covers
3) then multiply this number by the timebase value you set on the oscilloscope - this gives you period
4) use f = 1/T to calculate the frequency of the wave being generated by the oscillator
how to measure the speed of sound using standing waves? (method)
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method:
1) you can create resonance tube by placing hollow tube into measuring cylinder of water
2) choose tuning fork + note down frequency of sound it produces (f will be stamped on side of tuning fork)
3) gently tap tuning fork + hold it just above hollow tube
⋅ sound waves produced by fork travel down tube + get reflected (+ form node) at air/water surface
4) move tube up + down until you find shortest distance between top of tube + water level that sound from fork resonates at
⋅ this will be when sound appears loudest
5) measure distance between water surface + tuning fork - just like with any closed pipe, this distance is quarter of wavelength of standing sound wave
6) once you know frequency + wavelength of standing sound wave, you can work out speed of sound (in air), v, using equation v = fλ
7) then, repeat this experiment using tuning forks w different frequencies
⋅ you could also move tuning fork higher above cylinder until you find next harmonic (equal to three quarters of wavelength)
when does refraction occur?
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refraction occurs when the medium a wave is travelling in changes
what does refraction describe?
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refraction is the way a wave changes direction as it enters a different medium
what happens when a ray of light meets a boundary between two mediums?
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when a ray of light meets a boundary between two mediums, some of its energy is reflected back into the first medium and the rest of its energy is transmitted through into the second medium
what happens if light meets a boundary at an angle to the normal?
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if light meets a boundary at an angle to the normal, the transmitted ray is bent or ‘refracted’ as it travels at a different speed in each medium
why does the wavelength of light entering a new medium change? (using the wave equation)
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⋅ when light enters a new medium, the wave speed changes - but the frequency stays the same - so the wavelength must then change too
⋅ bc v = fλ
what does it mean if a ray bends towards the normal (when the light is passing through a new medium)?
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⋅ if the ray bends towards the normal, the light is is slowing down
⋅ this is bc the ray is moving from a less optically dense material to a more optically dense material
⋅ so the wavelength also decreases
(n1sin(i) = n2sin(r) so if r is decreasing, n2 must be increasing so optical density of second medium must have increased to keep n2sin(r) constant and therefore keep equation true)
what does it mean if a ray bends away from the normal (when the light is passing through a new medium)?
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⋅ if the ray bends away from the normal, the wave is speeding up
⋅ this is bc the ray is moving from a more optically dense material to a less optically dense material
⋅ the wavelength also increases
what does the refractive index (n) of a material measure?
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the refractive index (n) of a material measures how much the material slows down light
what material does light travel the fastest in and what does this mean?
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⋅ light travels the fastest in a vacuum [air is assumed to be a vacuum]
⋅ so light will always slow down when travelling through other materials
why does light slow down in materials that are not a vacuum?
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light slows down in other materials bc light interacts with particles in other mediums
what is the relationship between how optically dense a material is and how much light slows down in that material?
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the more optically dense a medium is, the more light slows down when it enters the medium
what is the absolute refractive index in other words?
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the absolute refractive index of a medium (n) is a measure of optical density
how do you find the absolute refractive index (n)?
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⋅ n is found from the ratio between the speed of light in a vacuum (c), and the speed of light in the medium (c medium)
⋅ n = c/c medium
where c = 3.00 x10^8
what does snell’s law use to calculate the refractive index?
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snell’s law uses angles to calculate the refractive index
what happens when a ray of light is refracted at a boundary between two materials? (describe aspects of snell’s law)
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when a ray of light is refracted at a boundary between two materials:
⋅ the light is crossing the boundary, going from medium 1 with a refractive index n1 to medium 2 with a refractive index n2
⋅ the angle the incoming light makes to the normal is called the angle of incidence (i)
⋅ the angle the refracted ray makes with the normal is the angle of the refraction (r)
⋅ n1, n2, i and r are related by snell’s law:
n1sin(i) = n2sin(r)
⋅ the speed of light is only a tiny bit slower than c
what is snell’s law?
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n1sin(i) = n2sin(r)
where:
n1 = refractive index of 1st medium
n2 = refractive index of 2nd medium
i = angle of incidence
r = angle of refraction
how are c medium 1, c medium 2. sin(i), sin(r), n1 and n2 related?
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c medium1/c medium 2 = sin(i)/sin(r) = n2/n1
what can you assume about the refractive index of air?
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⋅ the speed of light is only a tiny bit slower than c
⋅ this means that you can assume the refractive index of air is 1 (bc n air would be around c/c which = 1)
if you assume that the refractive index of air is 1, what does this do to snell’s law for an air-to-material boundary?
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if you assume that n air = 1, snell’s law becomes:
n = sin(i)/sin(r)
where:
n = refractive index of material
how do you calculate the refractive index of a transparent block? (practical)
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1) place glass block on piece of paper + draw around it
2) use ray beam to shine beam of light into glass block
⋅ turn off any other lights so you can see path of light beam through block clearly
3) trace path of incoming + outgoing beams of light either side of block
4) remove block + join up two paths you’ve drawn with straight line
⋅ line will follow path light beam took through glass block
⋅ you should be able to see from your drawing how path of ray bent when entering + leaving block
5) measure angles of incidence + refraction where light enters block
6) use snell’s law for air-to-material boundary to calculate refractive index of block
what is diffraction?
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diffraction is the way that waves spread out as they come through a narrow gap (aperture) or go around obstacles
do all waves diffract?
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yes, all waves diffract, but it’s not always easy to observe
what can you use to show the diffraction of water waves?
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you can use a ripple tank to show the diffraction of water waves
what does the amount of diffraction depend on?
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the amount of diffraction depends on the size of the wavelength of a wave compared to the size of a gap
what does the diffraction of a wave look like when a gap is a lot bigger than the wavelength of a wave?
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when the gap is a lot bigger than the wavelength, diffraction is unnoticeable
what does the diffraction look like when the gap is several wavelengths wide?
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⋅ you get noticeable diffraction when a wave passes through a gap several wavelengths wide
(⋅ here several does not mean lots and lots, ‘several’ just means n number of wavelengths wide (i.e. the width is multiple of wavelength), so like 2 or 3λ wide)
when do you get the most diffraction and what does this look like?
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you get the most diffraction when the gap is the same size as the wavelength
what does the diffraction look like when the gap is much smaller than the wavelength of the wave?
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⋅ if the gap is smaller than the wavelength, the waves are mostly reflected
⋅ however you still get lots of diffraction, but the diffracted waves have a lower amplitude than the wave before passing through the gap (bc most of the wave is reflected back)
why can you hear out-of-sight people in the next room through open doors?
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⋅ when sound passes through a doorway, the size of the gap (of the door) and the wavelength (of the sound) are usually roughly equal, so diffraction occurs
⋅ (diffraction means noise will spread around the door and around the room)
⋅ that’s why you have no trouble hearing someone through an open door to the next room, even if the other person is out of your line of sight
why can’t you see out-of-sight people in the next room through open doors?
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the reason that you can’t see an out-of-sight person in the next room through an open door, is that when light passes through a doorway, the light is passing through a gap around a hundred million times bigger than the wavelength of the light - so the amount of diffraction is tiny
how can you demonstrate the diffraction in light using laser light? (simple)
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1) the diffraction of light can be demonstrated by shining a laser light through a very narrow slit onto a screen
2) you can then alter the amount of diffraction by changing the width of the slit
how else can you demonstrate the diffraction of light, but using a white light source and a set of colour filters instead of laser light?
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1) diffraction of light can be demonstrated by shining the white light source through a very narrow slit onto a screen
2) the size of the slit can be kept constant while the wavelength is varied by putting different colour filters over the slit
what happens when a wave meets an obstacle?
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⋅ when a wave meets an obstacle, you get diffraction around the edges
⋅ behind the obstacle a ‘shadow’ forms, where the wave is blocked
⋅ the wider the obstacle compared with the wavelength of the wave, the less diffraction you get, and so the longer the shadow
what does the diffraction pattern look like for light waves?
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⋅ with light waves, you get a pattern of light and dark fringes
⋅the diffraction pattern has a bright central fringe with alternating dark and bright fringes on either side of the central bright fringe
⋅ the central fringe is the most intense
why is the central fringe of a light wave’s diffraction pattern the most intense?
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the central fringe is the most intense bc there are more incident photons per unit area in[/arriving at] the central fringe than in the other bright fringes
what is the relationship between the slit width and the width of the diffraction pattern?
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⋅ the slit width is inversely proportional to the width of the diffraction pattern
what is the relationship between slit spacing and fringe spacing?
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⋅ fringe spacing is inversely proportional to the slit spacing
⋅ bc x = Lλ/d
⋅ so x ∝ 1/d
where:
d = slit spacing
x = fringe spacing
what kind of light source do you have to use when demonstrating the diffraction of light?
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you have to use a monochromatic and coherent light source for this experiment
how do you explain diffraction patterns using phasors?
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⋅ the brightest point of the diffraction patterns is where the light passes in a straight line from the slit to the screen
⋅ so all the light waves that arrive there are in phase
⋅ at all other bright points where the light hits the screen, there is a constant phase difference between the waves arriving there, so the phasors point in slightly different directions and form a smaller resultant phasor
⋅ the dark fringes on the screen are where the phase difference between the light waves means their phasors add to form a loop, giving a resultant phasor of zero
why is it easier to demonstrate two-source interference in water and sound?
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sound and water have larger wavelengths that are easier to measure, so it’s easy to demonstrate two-source interference with either sound or water
do you need to use coherent sources for diffraction experiments?
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yes, you need to use coherent sources, which tells you that wavelength and frequency have to be the same for this experiment
what are ways to create coherent sources when investigating the diffraction of water and sound?
Q + W
⋅ a way that you can make coherent sources for investigating the diffraction of water and sound is to use the same oscillator to drive both sources
⋅ for water, one vibrator drives two dippers
⋅ for sound, one oscillator is connected to two loudspeakers
what are two methods to demonstrate two-source interference for light? (simple)
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⋅ to show two-source interference for light you can use two coherent light sources, or use a single light laser and shine it through two slits
⋅ shining a single light laser through two slits is young’s double-slit experiment
what are some important features of laser light?
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⋅ laser light is coherent and monochromatic
⋅ monochromatic means there’s only one wavelength [/frequency] present in the light
what are some things you have to consider when doing young’s double-slit experiment?
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1) the laser light used has to be coherent and monochromatic
2) the slits have to be about the same size as the wavelength of the laser light so that you get the most diffraction - then the light from the slits acts like two coherent point sources
what does the pattern of light and dark fringes depend on?
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you get a pattern of light and dark fringes, depending on whether constructive or destructive interference is taking place at that point on the wave
how can you do an experiment similar to young’s double-slit experiment with microwaves?
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1) to see interference patterns with microwaves, you can replace the laser slit with two microwave transmitter cones attached to the same signal generator
2) you also need to replace the screen with a microwave receiver probe
3) if you move the probe along the path of the green arrow, you’ll get an alternating pattern of strong + weak signals - just like the light and dark fringes on the screen
how do you relate the fringe spacing (x), wavelength (λ), spacing between slits [/slit spacing] (d), and the distance from the slits to the screen (L) together?
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⋅ you can relate all of them together in young’s double-slit experiment
⋅ x = λL/d
what knowledge can you deduce from the young’s double-slit formula?
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⋅ young’s double slit formula: x = λL/d
⋅ since the wavelength of light is so small, you can see from the formula that a high ratio of L/d is needed to make the fringe spacing big enough to see
what is young’s double-slit formula?
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x = λL/d
where:
x = fringe spacing
λ = wavelength of wave
L = distance between slits and screen
d = slit spacing
what is a way that you can lower the percentage uncertainty when doing young’s double-slit experiment?
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⋅ fringes are so tiny that it’s very hard to get an accurate value of x (the fringe spacing)
⋅ so it’s easier to measure across several fringes and then divide the length by the number of FRINGE WIDTHS between them
⋅ doing this helps to lower the percentage error
(⋅ number of fringe widths ≠ number of fringe spaces
⋅ as you can see below, there are 5 fringes but 4 fringe widths
⋅ so ensure that you do x = L/number of fringe WIDTHS and not L/number of fringes
⋅ so x = L/4 NOT x = L/5)
what two major theories of light were published by isaac netwon and huygen respectively?
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⋅ isaac netwon’s theory of light suggested that light is made up of tiny particles, which he called “corpuscles”
⋅ huygen’s theory suggested that light travels as waves
what properties of light could isaac newton’s corpuscular theory of light explain and what properties could it not?
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⋅ the corpuscular theory could explain reflection and refraction, but diffraction and interference are both uniquely wave properties
⋅ the photoelectric effect is a strictly particle property however
what properties could hyugen’s light theory explain?
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light travelling as a wave explains reflection, refraction, diffraction and interference
how did young’s double-slit experiment provide evidence for the theory that light travels as a wave?
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⋅ young’s double-slit experiment provided the necessary evidence to settle the debate (for a while) over whether light travelled as corpuscles or waves
⋅ by young’s double slit showing that light could diffract (through narrow slits) and interfere (to form an interference pattern on the screen), light had to travel as a wave (bc diffraction and interference are uniquely wave properties)
what happens to the interference patterns if you diffract the wave through more slits?
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⋅ you can repeat young’s double-slit experiment with more than two equally spaced slits (i.e. using a diffraction grating)
⋅ if you diffract a wave through more slits (so through diffraction grating) you basically get the same shaped pattern as for two slits
⋅ but the bright bands[/fringes] are brighter and narrower (they get sharper basically)
⋅ and the dark areas[/fringe widths] between the bright fringes are darker
why is the diffraction pattern much sharper when monochromatic light is passed through a grating with hundreds of slits per millimetre rather than two slits?
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when monochromatic light (monochromatic light = one wavelength, therefore one frequency) is passed through a grating with hundreds of slits per millimetre, the interference pattern is really sharp bc there are so many beams[/waves] reinforcing the patterns
why is it it better to have sharper fringes?
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sharper fringes make for more accurate measurements
what does the diffraction pattern look like for monochromatic light passing through a diffraction grating?
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⋅ for monochromatic light passing through a grating, all the maxima are sharp lines (it’s different for white light)
⋅ there’s a line of maximum brightness at the centre called the zero-order line
⋅ the lines just either side of the central line are called first-order lines - the next pair out is called the second-order lines and so on
for a grating (with slits a distance d apart by an angle θ between the incident beam) what is the nth order maximum given by?
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dsinθ = nλ
what does the equation dsinθ = nλ mean?
dsinθ = nλ means that by observing d, θ and n you can calculate the wavelength of the light passing through the grating
describe how you derive the equation dsinθ = nλ
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1) at each slit, the incoming waves are diffracted
⋅ these diffracted waves then interfere with each other to produce an interference pattern
2) consider the first-order maximum - this happens at the angle when the waves from one slit line up with the waves from the next slit that are exactly one wavelength behind (path difference = λ)
3) call the angle between the first-order maximum and the incoming light θ
4) now look at the triangle highlighted in the diagram - the angle is θ (using basic geometry), d is slit spacing, and the path difference = λ
⋅ measure the slit spacing from the middle of each slit
5) so, for the first maximum, using trig: dsinθ = λ
6) other maxima occur when the path difference is 2λ, 3λ, 4λ, etc - so to make the equation general, just replace λ with nλ, where n is an integer (the order of maximum)
what conclusions can you draw from dsinθ = nλ?
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1) if λ is bigger, sinθ, and so θ is bigger
⋅ this means that the larger the wavelength, the more the diffraction pattern will spread out
2) if d is bigger, sinθ is smaller
⋅ this means that the coarser the grating (the wider the slit spacing), the less the pattern will spread out
3) values of sinθ greater than 1 are impossible
⋅ so if for a certain n, you get a result of more than 1 for sinθ you know that order doesn’t exist
describe how you would derive young’s double-slit formula for fringe spacing from dsinθ = nλ?
W + Q
1) for first order maximum (n = 1) angle θ is small - this means you can use small angle approximations of tanθ ≈ θ and sinθ ≈ θ
2) tanθ is equal to opposite/adjacent, so in triangle shown tanθ = x/L (≈ θ ≈ sinθ)
3) this means you can substitute x/L into dsinθ = nλ to get xd/L = λ
⋅ remember n = 1 here
4) rearrange for x and you get x = Lλ/d
what happens if you shine white light through a diffraction grating?
W + Q
1) shining white light through a diffraction grating produces a spectra
2) white light is actually a mixture of colours (wavelengths) - so if you diffract white light through a grating, then the patterns due to the different wavelengths within the white light are spread out by different amounts
⋅ bc each wavelength will have a different diffraction pattern so diffracting white light will produce an amalgamation of all of the different diffraction patterns
3) each order in the pattern becomes a spectrum, with red on the outside and violet on the inside
⋅ the zero-order maximum stays white bc this is where all wavelengths just pass straight through the grating (so white light remains a mixture of all the different colours)
what do astronomers use spectra for?
W + Q
⋅ astronomers and chemists often need to study spectra to help identify elements
⋅ astronomers and chemists also use diffraction gratings rather than prisms bc gratings are more accurate
does light behave like a wave or a particle?
W + Q
light can actually act as a wave or a particle (photon)
why do we know light has to be able to travel as a particles (as well as a wave)?
W + Q
⋅ just like how young’s double-slit experiment provided evidence for light being able to irrefutably act as a wave, the photoelectric effect provided evidence for light acting as a particle (a photon)
⋅ diffraction and interference can only be explained by light acting as a wave, the photoelectric effect can only be explained by light acting as a particle
what is a photon?
W + Q
a photon is a quantum of em radiation
what is a quantum?
W + Q
a quantum is a single discrete packet of em radiation
what did planck suggest whilst investigating black body radiation?
W + Q
when planck was investigating black body radiation, he suggested that em waves can only be in discrete packets, called quanta
what is the equation for the energy carried by a photon?
W + Q
E = hf
or
E = hc/λ
where:
h = planck’s constant, h = 6.63 x10^-34
f = frequency
λ = wavelength
c = speed of light in a vacuum
what can you deduce from the equation for the energy carried by a photon?
W + Q
⋅ E = hf
⋅ so the higher the frequency of the em radiation, the more energy its quanta [/photons] carry
what two things did einstein suggest about photons?
W + Q
⋅ einstein suggested that em waves (and the energy that they carry) can only exist in discrete packets
⋅ einstein also believed that a photon acts like a particle, and will either transfer all or none of its energy when interacting with another particle (eg, electron)
do photons have charge?
W + Q
⋅ no
⋅ photons have no charge - they are neutral, like neutrons
what units are the energies of photons usually given in?
W + Q
photon energies are usually given in electronvolts (eV)
what is an electronvolt and why is it used?
W + Q
⋅ an electronvolt is used as the unit when talking about photons[/electrons/most other small particles]
⋅ because the energies involved when talking about photons are so tiny that it makes sense to use a more appropriate unit than the joule
what happens when you accelerate an electron between two electrodes and what does this mean?
W + Q
⋅ when you accelerate an electron between two electrodes, the electron transfers some electric potential energy (electric potential energy = eV) into kinetic energy
⋅ E = qV
⋅ hence eV = 0.5 mv^2 (when all electric potential energy is converted to KE, eg) finding max v)
what is the electronvolt defined as and what does this mean?
W + Q
⋅ the electronvolt is defined as: “the kinetic energy gained by an electron when it is accelerated through a potential difference of 1 volt”
⋅ so 1 electronvolt = e x V = (1.60 x10^-19 C) x (1 J C^-1)
⋅ so 1 eV = 1.60 x 10^-19 J
how can you use the threshold voltage of LEDs to find planck’s constant? (theory)
W + Q
⋅ you can find planck’s constant by doing a simple experiment with some light-emitting diodes (LEDs)
1) current will only pass through LED after the minimum[/threshold] voltage is placed across it - the threshold voltage V0
2) threshold voltage (V0) is the voltage needed to give electrons the same kinetic energy as a photon emitted by the specific LED
⋅ all of the electron’s KE after the electron is accelerated over this pd (threshold voltage) is transferred into the photon
3) so by finding the threshold voltage for a particular LED’s wavelength, you can estimate the planck constant
⋅ E = hc/λ = eV0
⋅ therefore h = (eV0)λ/c
how can you use LEDs to estimate the planck constant? (practical)
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1) connect LED of known wavelength in the circuit shown [on front of flashcard]
2) close any blackout blinds + place a shaded tube over the LED to look through. the room should be as dark as possible so you can see when the LED first begins to emit light
3) start off with no current flowing through the circuit, then adjust the variable power source until the current just begins to flow through the circuit + LED lights up
4) record voltage (V0) across LED
5) repeat this experiment with a number of LEDs with different colours that emit light at different wavelengths
6) plot graph of threshold voltages (V0) against frequency (f = c/λ) (where λ is the wavelength of light emitted by LED in meters)
⋅ if your straight line doesn’t go through the origin, there could be some systematic errors you need to account for, you can do this by adding or taking away the difference between origin + vertical intercept from all of your data
7) you should get a straight line graph with a gradient of h/e - which you can then use to find h (by multiplying gradient by e [charge of electron])
8) repeat the experiment to find an average value of h
what does the photoelectric effect show about the behaviour of light?
the photoelectric effect shows the particle behaviour of light
what does the photoelectric effect involve? (simple)
W + Q
⋅ the photoelectric effect is when light with a high enough frequency is shone onto the surface of metal, and causes electrons to be emitted
⋅ for most metals, this frequency falls in the UV range
what significance did the photoelectric effect have in physics?
W + Q
the photoelectric effect was one of the first experiments with light which couldn’t be explained with the wave theory, and supported planck’s theory that light is quantised
what happens in the photoelectric effect?
W + Q
in the photoelectric effect:
1) free electrons on the surface of the metal absorb the energy from the light, making the electrons vibrate
2) if the electron absorbs enough energy, the bonds holding the electron to the metal are broken and the electron is released
3) this is called the photoelectric effect and the electrons emitted are called photoelectrons
what three main conclusions were drawn from the photoelectric effect experiment?
W + Q
⋅ conclusion 1: for a given metal, no photoelectrons are emitted if the radiation has a frequency below a certain value - this certain value is called the threshold frequency
⋅ conclusion 2: photoelectrons are emitted with a variety of KEs ranging from zero to some maximum value. this value of maximum KE increases with the frequency of the radiation, and is unaffected by the intensity (photons per unit area) of radiation (bc E = hf - Φ)
⋅ conclusion 3: number of photoelectrons emitted per second is proportional to the intensity of radiation
what would happen if you attempted to explain the photoelectric effect using wave theory, and why does this not work?
W + Q
according to the wave theory:
1) for a particular frequency of light, the energy carried is proportional to the intensity of the beam
2) energy carried by the light would be spread evenly over the wavefront
3) each free electron on the surface of the metal would gain a bit of energy from each incoming wave
4) so gradually, each electron would gain enough energy to leave the metal
so…:
⋅ if the light had a lower frequency (i.e. was carrying less energy) it would take electrons longer to gain enough energy - but it would happen eventually - so there is no explanation for the threshold frequency when using the wave theory
⋅ higher the intensity of the wave, the more energy it should transfer to each electron - so the kinetic energy should increase with the intensity - so there’s no explanation for the kinetic energy depending only on the frequency
how does the photon model explain the photoelectric effect?
W + Q
according to the photon model:
1) when the light hits the metal’s surface, the metal is bombarded with electrons (bc electrons are liberated using energy from light?)
2) if one of these atoms[/photons] is absorbed by one of the free electrons, the electrons will gain energy equal to hf - so a higher frequency will result in a higher kinetic energy for the electron
⋅ each electron only absorbs one photon at a time, so all the energy an electron needs to gain before it can be released must come from that one photon
4) so an increase in the intensity of the light (i.e. more photons) won’t affect the kinetic energy of the electrons - only frequency will
⋅ before an electron can leave the surface of the metal, it needs enough energy to break the bonds holding it in place - this energy is called the work function energy (symbol Φ, phi) and its value depends on the metal
how does the photon model also explain the threshold frequency in the photoelectric effect?
W + Q
1) if the energy gained by an electron (on the surface of a metal) from a photon is greater than the work function, the electron is emitted
2) if the energy gained by the electron from the photon isn’t greater than the work function, the metal will heat up but no electrons will be emitted
3) since, for electrons to be released, hf ≥ Φ, the threshold frequency must be: f = Φ/h
⋅ h is planck’s constant: 6.63 x10^-34
what should you not do in a calculation when the energy in the questions is initially given in electronvolts?
W + Q
⋅ do NOT use electronvolts in calculations, convert the electronvolts to joules for calculations first, and then when presenting your answer convert the energy back to electron volts
⋅ eV -> J -> eV
how do electrons exist in an atom?
W + Q
• electrons in an atom can only exist in certain well-defined discrete energy levels
⋅ each level is given a number, with n = 1 representing the ground state
how do electrons move between energy levels [in an atom]?
W + Q
⋅ electrons can move down energy levels by emitting a photon
⋅ electrons can move up energy levels by absorbing energy
why can the photons emitted by electrons moving down energy levels only have energies of certain values?
W + Q
⋅ since the transitions of the electrons are between definite energy levels, the energy of each photon emitted can only take a certain allowed value
⋅ so the energy carried by each photon is the difference in energies between two energy levels
what is the equation to show the energy of the photon emitted by an electron when transitioning between energy levels n = 2 and n = 1 (or just two consecutive levels)
ΔE = E2 - E1 = E = hf = hc/λ
what kind of spectra do hot gases produce?
hot gases produce line emission spectra
what happens when you heat a gas to a high temperature?
W + Q
1) if you heat a gas to a high temperature, many of its electrons move to higher energy levels (this is known as excitation - the atoms become excited)
2) as they fall back to ground state, these electrons emit energy as photons (electrons must rid of extra energy to be able to fall down energy levels)
3) if you split the light from a hot gas with a prism or diffraction grating, you get a line spectrum
what does a line emission spectra look like?
W + Q
⋅ a line spectrum is seen as a series of bright lines against a black background
⋅ each line on the spectrum corresponds to a particular wavelength of light emitted by a source - since only certain photon energies are allowed [/emitted], you only see their corresponding wavelengths
what happens when you shine white light through a cool gas? (simple)
W + Q
⋅ shining white light through a cool gas gives absorption spectra
⋅ these absorption spectra are continuous spectra and contain all possible wavelengths
what does the spectrum of white light look like?
W + Q
⋅ the spectrum of white light is continuous
⋅ if you split [white] light up with a prism, the colours all merge into each other - meaning there also aren’t any gaps in the spectrum
what spectrum do hot things emit?
W + Q
hot things emit a continuous spectrum in the visible and infrared parts of the em spectrum
what do cool gases do if you shine white light through them? (simple)
W + Q
cool gases remove certain wavelengths from a continuous spectrum
why do cool gases absorb certain wavelengths when you shine white light through them?
W + Q
1) at low temperatures, most of the electrons in gas atoms will be in their ground states
2) photons of the correct wavelength are absorbed by electrons to excite them to higher energy levels
3) these wavelengths are missing from the continuous spectrum when the white light comes out the other side of gas
4) you see a continuous spectrum with black lines in it corresponding to the absorbed wavelengths
how many paths do any subatomic particles (quanta) try when moving from one point to another?
W + Q
photons try every possible path [simultaneously]
what did feynman theorise about the path of a photon from a source to a detector?
W + Q
feynman theorised that instead of taking one route to the detector from the source, a photon will take all of the possible paths to the detector in one go
how do you keep track of a photon (or any quanta) whizzing along every possible route?
W + Q
you can keep track of the photon (or any quanta) whilst it’s travelling along every possible route using phasors
what features of a wave do phasors show?
W + Q
a phasor shows the amplitude of a point on a wave (as the size of the phasor) and the phase of the point on the wave (as the direction of the phasor)
what can you use phasors for in young’s double-slit experiment for?
W + Q
with young’s double slit experiment, you can use phasors to show how light or dark a certain point on a screen will be
how are phasors and probability linked in quantum mechanics?
W + Q
in quantum mechanics, you can use phasors to tell you how probable it is that a quantum (in the case of the double-slit experiment, a photon) will arrive there
how do you explain f = E/h?
W + Q
1) as a photon travels, its phasors will rotate (anticlockwise) until it reaches the detector
2) by knowing energy of the photon, you can work out the frequency of the phasors rotation (f) by rearranging planck’s formula (E = hf)
3) therefore f = E/h
how do you work out the resultant phasor of a photon [or any quanta] from a source to a detector, and what can you ignore in this calculation?
W + Q
1) you want to record the position of a phasor at end of every path - you could then sum these phasors to find the resultant phasor for a photon making a journey from source to detector
2) you can’t find the final phasor for every path as there’s an infinite number of them - however, nearly all the phasors cancel each other out (when doing calculations), so you only need to consider straightest/quickest possible paths
what do phasors look like throughout young’s double-slit experiment?
W + Q
1) imagine that a photon is emitted by source + hits point X on screen. take two of its possible paths + say it follows both of them, as shown on the front of the flashcard
2) the phasor of each photon along each path rotates at the same rate (bc it’s the same photon so phasors will have the same frequency)
3) bc the photon has to travel slightly further on the green path, it takes slightly longer to reach point X. this means the final phasor of the green path will have rotated slightly further than that for the blue path
4) you can find the resultant phasor arrow for a photon reaching point X by adding the final phasor position for each path, tip-to-tail (just like normal vector sum, as shown in the diagram)
how do you find the probability that a quantum will arrive at a point?
W + Q
you can find the probability that a quantum will arrive at a point from squaring the resultant phasor amplitude
what relationship is there between probability of an area and the number of particles that arrive there?
W + Q
the higher the probability, the more likely a particle will arrive there
if the quantum you are using is a photon, what can you do with probability and the brightness of an area?
W + Q
⋅ if a photon is a quantum you are using, you can think of the probability and the brightness of an area as pretty much the same thing
⋅ bc the more probable it is that a photon will arrive at the point, the brighter it will appear (bc more photons have probably landed there, making it brighter)
which path of a quantum gives the quickest route?
W + Q
the path that gives the highest probability is the quickest route
what does the sum over paths rule (where you get the path with the highest probability) predict?
W + Q
the sum over paths rule predicts that: “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”
what does the sum over paths rule also predict with how light travels?
W + Q
⋅ the sum over paths rule also predicts that light travels in a straight line (one of the most fundamental light behaviours)
⋅ as a straight line is the shortest (and therefore the quickest) path between two points, it provides the largest probability of a photon arriving at a particular point
how does quantum also predict refraction?
W + Q
1) imagine spotting an object (eg, pineapple) at bottom of swimming pool. what route does light take from object to your eye? it takes all of them
2) when light travels in water, it slows down, but its frequency stays same. this means photons will have same energy, + photon’s phasor will still have same amplitude + frequency of rotation whatever material it’s travelling through
⋅ if you add up all phasors for all possible paths, it’s path that takes shortest time that contributes most to resultant phasor amplitude (+ so gives highest probability that photon will get to your eye)
how do you focus quanta?
W + Q
to focus photons (or any other quanta), you need to make sure all the straight line paths (that follow the reflection or refraction rule) from the source to the focus point take the same amount of time - so the final phasors for every path will be in the same direction
how does a convex lens focus quanta passing through it?
W + Q
⋅ paths toward the edges of the lens are longer than those that go through the middle
⋅ so you make the time taken for each path the same by increasing the amount of glass in the middle part of the lens to increase the time it takes to travel along the shorter paths between the source and the detector
what did de broglie suggest?
W + Q
⋅ de broglie suggested that electrons were quantum objects
⋅ to elaborate: de broglie suggested that “if ‘wave-like’ light showed particle properties (photons), ‘particles’ like electrons would be expected to show wave-like properties
in what equation does de broglie relate a wave property (wavelength) to a moving particles property (momentum)?
W + Q
⋅ λ = h/p
OR
⋅ λ = h/mv
where λ = wavelength, h = planck’s constant, p = momentum, m = mass, v = velocity
how can the de broglie wave of a particle be interpreted?
W + Q
⋅ the de broglie wave of a particle can be interpreted as a ‘probability wave’
⋅ you can use the de broglie wave to find the probability of finding electrons at a particular point [/of electrons arriving at particular point]
what is the evidence for electron interference and superposition?
W + Q
⋅ you can repeat experiments like young’s double-slit experiment with electrons
⋅ in the experiment, the electrons show the same kind of interference and superposition effects as you get with photons
how do you show electron interference and superposition?
W + Q
⋅ you usually show interference and superposition patterns of electrons using a fluorescent screen
⋅ as electrons hit the screen, it causes a photon to be released, so you can see the location of the electron
⋅ and just like photons, electrons try every path
⋅ the bright fringes in the electron interference pattern show where the probability of an electron arriving is high - and the dark fringes show where the probability of electrons hitting the screen is low
what does electron diffraction support?
W + Q
electron diffraction also supports electrons being quantum objects
when do you observe electron diffraction patterns?
W + Q
electron diffraction patterns are observed when accelerated electrons in a vacuum tube interact with the spaces in a graphite crystal (that are comparable to the electrons de broglie wavelength)
what can you do with electron diffraction patterns?
W + Q
⋅ you can think of electron diffraction in the exactly the same way as photon diffraction
⋅ by summing the final phasor for ever possible path, you can find how likely it is that an electron will hit the fluorescent screen at a particular point
⋅ the higher the probability, the brighter the point on the screen
what is the only difference when comparing electron diffraction and photon diffraction?
W + Q
⋅ the only difference is that when you are finding the frequency and the amplitude of the electrons’ phasor, the E is the kinetic energy of the electron
⋅ this also confirms that electrons show quantum behaviour
what does increasing the accelerating voltage in the electron diffraction experiment do?
W + Q
⋅ increasing the accelerating voltage also increase electron speed
⋅ this then causes the diffraction pattern circles to squash together towards the middle - fitting de broglie’s wavelength equation: λ = h/p = h/mv
⋅ bc if the velocity is higher, the wavelength is shorter and the spread of the lines is smaller
do electrons show quantum behaviour all the time?
W + Q
⋅ electrons don’t show quantum behaviour all the time
⋅ as particles only diffract if the particle interacts with an object/gap of comparable size to the particle’s de broglie wavelength
how do microscopes make use of different diffraction effects?
W + Q
⋅ a shorter wavelength gives less diffraction effects
⋅ diffraction effects blur detail on an image
⋅ so if you want to resolve tiny detail on an image, you need a shorter wavelength
⋅ so microscopes use electrons, hence electron microscopes
⋅ light blurs out detail more than electrons do (because it has a larger wavelength than electrons), so an electron microscope can resolve finer detail than a light microscope
how do we know that electrons exhibit wave-like properties as de broglie theorised?
W + Q
⋅ particle theory says that particles are physical objects that cannot superpose with other particles - however, to have interference patterns, you need superposition
⋅ you also need at least two slits to create an interference pattern - classic particles would either go through one slit or the other, not both
⋅ however, interference patterns can be seen when only a single electron is sent through narrow slits
⋅ so electrons exhibit wave-like properties
what behaviours can be explained by wave theory or particle theory?
W + Q
