Modern Physics Part 1: Quantization & Photons

Modern Physics Part 1: Quantization & Photons Last modified: 15/12/2017

Contents Links Contents Introduction Classical Physics Modern Physics Quantization Definition & Examples Photons Black Body Radiation Black Body Radiation: The Ultraviolet Catastrophe Planck: Light is Quantized Example Electronvolts (a Non-SI unit) Electromagnetic Spectrum Isn t Light a Wave? Photoelectric Effect More Problems with Classical Physics The Experiment Einstein: Photons Are Particles! Threshold Frequency Cutoff Wavelength Summary Example 1 Example 2 Light: Wave or Particle?

Classical Physics Contents After Isaac Newton developed calculus and then his Laws of Motion in the 1660s, Physics (together with Chemistry, Engineering etc.) developed very quickly and by the 1880s, there were broadly three very successful areas of what is now known as Classical Physics: Mechanics (Newton s Laws, Energy, Momentum etc) Electromagnetism (Coulomb, Gauss etc.) Thermodynamics (the study of heat, and how matter responds to it - not covered in this course) Classical Physics describes most of the world around us very well and is the basis for the development of inventions such as the steam engine and electric motor etc.

Modern Physics Contents However beginning about 1880, a variety of unexpected and puzzling experimental observations were made that despite many attempts, could not be satisfactorily understood using Classical Physics. Among others, these include: photoelectric effect atomic spectral lines radioactivity predictions of the age of the Sun, which had it much younger than the Earth! Investigating and understanding these observations lead on to Modern Physics which is the basis of much 20th century technology - computers, ipads, lasers, nuclear reactors etc.

Contents Classical Physics is still useful for dealing with most of our everyday world, but as soon as we start to deal with very large speeds and/or very small objects, we need to use Modern Physics. Often, these two situations occur at the same time, as it is relatively easy to accelerate a very small, light object such as a proton or electron up to a high speed. A (very!) simplified summary of Modern Physics: The speed of light is constant and cannot be exceeded (Special Relativity) Many traditionally continuous physical properties are QUANTIZED (Quantum Physics)

Quantization Contents A physical quantity is said to be QUANTIZED if it can only take certain fixed values. The smallest unit is called the QUANTUM (plural: QUANTA) of that quantity. A day-to-day example: Money is quantized. At an Australian supermarket, every marked price is an integer number of cents. e.g. $1.53 = 153c, $2.16 = 216c etc. The quantum of money is 1 cent. However, the quantum of cash is 5 cents.

Contents A physics example: Electric charge is quantized. The smallest possible charge (the quantum) is the charge of a proton, e = 1.602 10 19 C (negative for an electron). Any charged object must have a charge equal to a positive or negative integer times this value. Another physics/chemistry example: Matter is quantized. The smallest unit of an element is an atom, or in the case of a compound, a molecule.

Contents If we are dealing with a collection of a very large number of quanta, then the quantization isn t always obvious. If Bill Gates bank made a 1 cent error in his account, he probably wouldn t notice. If we have an object with 1 mc of charge then removing one charge of 1.6 10 19 C will not make a measurable difference. If we remove 1 water molecule from a glass of water we will not notice. Typically quantization will only be significant if we are considering a situation where there are only a small number of quanta present.

Contents As we ll see, many physical quantities such as energy and momentum which in Classical Physics are able to take any value at all (i.e. are continuous) are actually quantized, with the size of the quanta being very small. When we are studying the motion of large objects such as cats, tennis balls, cars etc we do not notice this quantization and can successfully use our familiar formulas to describe the motion. However when we are looking at the behaviour of very small objects such as atoms or the circuits on computer chips we must use Quantum Physics.

Black Body Radiation Contents One of the most puzzling of the late 1800s experimental observations was that of Black Body Radiation. This refers to the light emitted by a heated object. A range of wavelengths of emitted light are measured, with intensities depending on temperature, as seen below. 10 5000 K Black Body Radiation Measurements intensity 5 4000 K 3000 K 0 0 1,000 2,000 3,000 4,000 wavelength (nm)

The Ultraviolet Catastrophe Contents Attempts using Classical Physics (where light is thought of as a wave with speed c = λf ) to explain the observed distribution of wavelengths failed dismally. (Note this is a complex thermodynamics calculaton - only the result is shown in the graph below) 20 intensity 15 10 measurement for T=5000K Classical Physics calculation for T=5000K 5 0 0 1,000 2,000 3,000 4,000 wavelength (nm) Especially for lower wavelengths (the ultraviolet end of the spectrum), the calculated curve is wildly wrong, and as λ 0, it asymptotes to

Planck: Light is Quantized Contents Max Planck (1900) successfully explained the observed distribution of wavelengths of a Black Body by assuming that the energy of the emitted light is quantized. Instead of a gradual (continuous) transfer of energy into light, as expected for a wave, the energy is instead emitted in (very small) distinct chunks (i.e. quanta). The quantum of light is called the photon. The amount of energy E contained in one photon depends on the frequency f (or wavelength λ) of the light: E = hf = hc/λ where h = 6.626... 10 34 Js is called Planck s constant, (Remember for any wave: v = f λ).

Example Contents The laser in a CD player produces infrared light with wavelength of 780 nm. (a) What is the frequency of this light? (b) What is the energy of one photon of this light? (a) f = c/λ = (3 10 8 )/(780 10 9 ) = 3.85 10 14 Hz (b) E = hf = (6.63 10 34 ) (3.85 10 14 ) = 2.55 10 19 J (2.55 10 19 )/(1.6 10 19 ) = 1.59 ev

Electronvolts (a Non-SI unit!) Contents The electronvolt (or electron volt ) is a non-si unit for energy which is often used in quantum (and nuclear) physics. It corresponds to the work done on a charge of e = 1.602 10 19 C (i.e. the charge of a proton/electron) moving through a potential difference of 1 V. The standard abbreviation is ev. 1 ev = (1.602 10 19 C) (1 V) = 1.602 10 19 J This unit is used because it is convenient, as we often use voltages to accelerate or decelerate charged particles. For example if a proton is accelerated through 83.5 V, we know straight away that the increase in kinetic energy of the proton is 83.5 ev. As always, if we needed to use this energy in a calculation, perhaps to find the velocity of the proton, we must first convert ev to J.

Contents Here are some further examples of photon energies, across the electromagnetic spectrum: λ f E (J) E (ev) FM radio 3 m 100 MHz 6.6 10 26 4.1 10 7 microwave 1 cm 3 THz 2.0 10 23 1.2 10 4 red light 650 nm 4.6 10 14 Hz 3.1 10 19 1.9 green light 500 nm 6.0 10 14 Hz 4.0 10 19 2.5 violet light 400 nm 7.5 10 14 Hz 5.0 10 19 3.1 X-ray 1 nm 3 10 17 Hz 2.0 10 16 1200 gamma ray 0.01 nm 3 10 19 Hz 2.0 10 14 1.2 10 5 Even the most energetic of these examples the gamma ray has a very tiny photon energy, and so much of the time we would expect this quantization of light to not be noticable.

Isn t Light a Wave? Contents Young s Double Slit experiment in 1800 was thought to have settled the debate over the nature of light. It is a wave. This was confirmed in a variety of other experiments over the next 100 years or so. In 1900, Planck s explanation of Black Body Radiation requires light to have quantized energy. This is a property we expect from a particle rather than a wave. Because Planck s Black Body calculation was quite complex, most physicists at the time (including Planck himself) felt that this quantization of light into photons was some sort of mathematical approximation that, though useful, didn t reflect the real world. It was generally thought that someone clever would eventually come along and explain things properly treating light as a wave. This belief was wrong!

More Problems with Classical Physics Contents Another of the confusing observations of the late 1800 s was the Photoelectric Effect. When light shines on a metal surface, electrons (often called photoelectrons) are seen to be emitted from the metal, with a range of different kinetic energies. light electrons Metals are conductors with many electrons that move freely about inside a piece of the metal. The Classical Physics explanation of the Photoelectric Effect is that a wave (the light) adds energy gradually to these electrons until they accumulate enough energy to escape from the surface of the metal. We should recall from our previous study of waves, that the power P transferred by a wave depends (among other things) on the frequency f of the wave: P f 2 This power is related to the brightness (i.e. intensity) of the light. (Remember intensity is power/area) metal

Contents Using this relationship we can make some predictions about how light waves will cause electrons to be emitted from a metal. Similarly to the Black Body calculations, the predicted and observed behaviours are found to be very different. Classical Prediction 1 Using a dim light, there will be a short time delay before electrons are seen. Observation 1 Some electrons are always seen immediately, even for a dim light. This is because the electrons will need to gradually accumulate the energy required to escape.

Contents Classical Prediction 2 Increasing light intensity (brightness) should cause higher energy electrons to be emitted. i.e. More energy in = more energy out Classical Prediction 3 Some electrons will always be emitted, no matter what colour (wavelength) of light is used. Observation 2 Increasing the intensity leads to a greater number of electrons being emitted, but there is an observed maximum electron energy, independent of light intensity. Observation 3 Above a particular wavelength, no electrons are seen. This wavelength is different for different metals, and is independent of light intensity.

Photoelectric Effect Experiment Contents To fully understand (and explain) the Photoelectric Effect, we need to look in more detail at the experimental findings. Some of you will already have seen this in lab classes. A typical apparatus is shown below, consisting of a photocell (a glass tube containing two metal plates in a vacuum) connected in a circuit with a power supply, a voltmeter and an ammeter. photocell V variable DC voltage A

Contents To begin, the power supply is adjusted down to zero. Monochromatic light (i.e light with a single fixed wavelength) is directed on to one of the photocell s plates (the cathode), causing electrons to be emitted. These electrons are collected by the second plate (the anode), and flow through the circuit, back to the original plate. The ammeter measures a non-zero current i. light DC Power Supply electrons anode V=0 Voltmeter + cathode i Ammeter

Contents The voltage of the power supply is adjusted upward to a value V = V. This voltage causes a potential difference, and thus an electric field between the plates of the photocell. This field will do work of ev on the electrons, reducing their kinetic energy by this amount. The current i recorded by the ammeter will decrease, as the less energetic electrons no longer reach the anode of the photocell. light DC Power Supply electrons + + + + + + + + anode V Voltmeter + cathode i Ammeter

Contents The external voltage is continually increased, and the measured current decreases until the point where the current just reaches zero. The voltage when this occurs is called the stopping potential, V 0. light DC Power Supply electrons +++++++++++++++ anode V 0 Voltmeter + cathode i=0 Ammeter V 0 is the smallest external voltage which stops all the photoelectrons reaching the anode of the photocell. This indicates that there is a maximum kinetic energy for the photoelectrons: KE max = ev 0.

Contents This same procedure is repeated for a range of wavelengths, and for different metal surfaces. Plotting the measurements of V 0 against λ will look something like the graph below (where different colours = different metals). V 0 10 9 8 7 6 5 4 3 2 1 0 100 200 300 400 500 wavelength (nm) For each material, there is a maximum wavelength, above which no photoelectrons are seen.

Contents If we instead plot V 0 against frequency f, the plotted data forms straight lines, as senn below. This is a strong clue that there is a simple connection between V 0 and f. V 0 10 9 8 7 6 5 4 3 2 1 0 5 10 15 20 25 30 frequency (10 14 Hz) The data is not affected by the brightness of the light. Brighter light will cause larger currents to flow initially, but the measured stopping potential will be the same.

Einstein: Photons Are Particles! Contents As we ve already mentioned, Planck s idea of quantized light suggests that the quanta of light have a definite energy, a property which is more associated with particles than waves. At the time this concept was not really accepted to make sense, but only a few years later, in 1905, Albert Einstein took the idea seriously and was able to explain the photoelectric effect observations, for which he was later awarded the Nobel Prize. Einstein s basic insight was to treat the photons as if they were normal particles. The first step is to observe the behaviour of electrons inside a metal. Individual electrons will move freely around the metal, similarly to a gas. The electrons do not escape the metal because they have negative energy (compared to free electrons, outside the metal, which have a positive (or zero) energy).

Contents The value of this negative electron energy is called the work function φ, of the metal (i.e. the work required to remove the electron from the metal). The value of φ is a property of a particular metal, generally in the range 1 5 ev. (The term work function is historical and a little confusing - it is NOT a mathematical function. Something like work value would be a better term.) An incoming photon is absorbed by an electron and the energy of this photon is added to the electron. The photon ceases to exist at this point. It is wrong to think of it hiding inside the electron. All of the photon energy is given to one electron, it can t be shared amongst multiple electrons. metal E electron = φ photon (γ) E γ = hf metal E electron = φ metal E electron = hf φ

Contents If hf < φ, then the electron still has an overall negative energy, and so remains confined to the metal. In this case, no photoelectric effect is observed. The added energy will cause a slight heating of the metal. If on the other hand hf > φ, then the electron will now have a total positive energy and is able to escape from the metal. The excess positive energy is kinetic energy of the emitted electron. KE electron = hf φ metal In reality, the situation is not always quite this simple. Some electrons will not come directly out of the metal, but will collide with other electrons or nuclei on the way out, and so lose energy. We will observe electrons with a range of kinetic energies from KE = 0 (where the electron only just makes it out of the metal), up to the maximum value KE max = (hf φ).

Contents Remember, that in the experiment this maximum kinetic energy is measured via the stopping voltage V 0 : KE max = ev 0 So, our final equation is: ev 0 = KE max = hf φ = hc λ φ Dividing both sides of the equation by e gives: V 0 = ( h e )f + ( φ e ) which explains the linear relationship between V 0 and f observed in experiments. V 0 gradient = h e y-intercept = φ e f

Threshold Frequency Contents The threshold frequency is the frequency where the photon energy is just sufficient to remove an electron from the metal. i.e. hf threshold = φ f threshold = φ h Below this frequency, no photoelectrons are seen. At the threshold frequency, V 0 = 0 V 0 f threshold is the x intercept of the V 0 vs f graph. f f threshold

Cutoff Wavelength Contents The cutoff wavelength is the wavelength where the photon energy is just sufficient to remove an electron from the metal. i.e. hc λ cutoff = φ λ cutoff = hc φ Above this wavelength, no photoelectrons are seen. At the cutoff wavelength, V 0 = 0 λ cutoff is the x intercept of the V 0 vs λ graph. V 0 The cutoff wavelength and threshold frequency are of course connected: λ cutoff = c f threshold λ cutoff λ

Contents Einstein s picture of the Photoelectric Effect in terms of quantized photons, is the first really convincing result of Quantum Physics. All of the previously problematic experimental observations can be explained simply: Observation 1 Electrons are always seen immediately, even for a dim light. Quantum Explanation 1 Energy is given to the electrons in one step, not gradually. If there are any photons with sufficient energy, then there will be electrons emitted.

Contents Observation 2 Increasing the intensity leads to a greater number of electrons being emitted, but there is an observed maximum electron energy, independent of light intensity. Observation 3 Above a particular wavelength, no electrons are seen. This wavelength is different for different metals, and is independent of light intensity. Quanum Explanation 2 The maximum electron energy is determined by the photon energy. The gain in energy of one electron = energy of one photon. Quantum Explanation 3 If the photon energy is less than the work function of the metal, then there is not enough energy for an electron to escape the metal.

Photoelectric Effect Summary Contents The stopping potential V 0 for a photocell is related to the frequency f or wavelength λ of incident light by: ev 0 = KE max = hf φ = hc λ φ where φ is the work function of the metal in the cell. This is a characteristic property of the type of metal - lead, tin, copper etc. all have different values of φ, typically in the range 1-5 ev. When V 0 = 0 this equation gives: f threshold = φ h λ cutoff = hc φ minimum frequency for electron emission maximum wavelength for electron emission

Photoelectric Effect Example 1 Contents Sodium has a work function of 2.36 ev. What is its cutoff wavelength? λ c = hc φ = (6.63 10 34 ) (3.0 10 8 ) 2.36 (1.6 10 19 ) = 5.27 10 7 m = 527 nm This is in the green part of the visible spectrum. For λ < 527 nm, photoelectrons will be seen, with increasing kinetic energy as λ decreases. For λ > 527 nm, e.g. red light, NO photoelectrons will be seen.

Photoelectric Effect Example 2 Contents A sodium surface is exposed to light of wavelength 450 nm. (a) What is the stopping potential for this light? (b) What will be the maximum speed of the emitted electrons? (a) ev 0 = hc λ φ V 0 = hc eλ φ e = (6.63 10 34 ) (3.0 10 8 ) (1.6 10 19 ) (450 10 9 ) 2.36 = 0.40 V

Contents (b) To find the maximum speed v max : ev 0 = KE max = 1 2 mv 2 max And using the electron mass of 9.1 10 31 kg, 2eV0 v max = m 2 (1.6 10 19 ) 0.40 = 9.1 10 31 (using V 0 from part (a)) = 3.76 10 5 m/s

Light: Wave or Particle? Contents Einstein s explanation of the photoelectric effect suggests that a beam of light directed at a sheet of metal will behave like a stream of particles (photons). The same beam of light directed at two thin slits will behave like a wave as demonstrated in Young s double slit experiment. They can t both be correct. Or can they...? This split personality of light is known as wave-particle duality, and will be discussed further in coming weeks.