Quantum Physics

1. Quantum Physics • 26. Quantum physics • Content • 26.1 Energy of a photon • 26.2 Photoelectric emission of electrons • 26.3 Wave-particle duality • 26.4 Energy levels in atoms • 26.5 Line spectra • Learning outcomes • Candidates should be able to: • (a) show an appreciation of the particulate nature of electromagnetic radiation • (b) recall and use E = hf • (c) show an understanding that the photoelectric effect provides evidence for a particulate nature of electromagnetic radiation while phenomena such as interference and diffraction provide evidence for a wave nature • (d) recall the significance of threshold frequency • (e) explain photoelectric phenomena in terms of photon energy and work function energy

2. (f) explain why the maximum photoelectric energy is independent of intensity, whereas the photoelectric current is proportional to intensity • (g) recall, use and explain the significance of hf = Φ + ½mv2max (h) describe and interpret qualitatively the evidence provided by electron diffraction for the wave nature of particles • (i) recall and use the relation for the de Broglie wavelength λ = h/p • (j) show an understanding of the existence of discrete electron energy levels in isolated atoms (e.g. atomic hydrogen) and deduce how this leads to spectral lines • (k) distinguish between emission and absorption line spectra • (l) recall and solve problems using the relation hf = E1– E2.

3. Bedtime stories • The particle theory(copuscular theory) predicted that light should speed up when passing from air to water. The wave theory predicted the opposite. In 1862, the speed of light in water was measured and found to be less than in air • So far we have learnt about properties of waves and that light is a wave motion and introduced concepts of interference and diffraction • In the last 100 years, new phenomena has revolutionized physics • A number of experiments in 1900 showed that light behaves more like a stream of particles than as a wave • Then later, it was shown experimentally that electrons, which had always been thought of as particles, could also behave as waves • This wave-particle duality is one of the most important concepts of modern physics • The phenomena that light had a particle nature is the explanation of Albert Einstein in 1905 for which he received the 1921 Nobel Prize • Then, we shall find that diffraction which is a characteristic of waves is exhibited by electrons which have always been considered to be particles!

4. Wave-particle duality of light In 1924 Einstein wrote:- “ There are therefore now two theories of light, both indispensable, and … without any logical connection.” • Evidence for wave-nature of light • Diffraction and interference • Evidence for particle-nature of light • Photoelectric effect • Compton effect • Light exhibits diffraction and interference phenomena that are only explicable in terms of wave properties • Light is always detected as packets (photons); if we look, we never observe half a photon

5. Symmetry in physics

7. Photoelectric emission • In a metal, some electrons are free to move around it which forms the electric current when a potential difference is applied across the ends of a wire • To remove free electrons from a metal or to make an electron escape from the surface of a metal requires energy, because they are held together by the electrostatic attraction of the positively charged nuclei • When this energy that is required to remove an electron is in the form of light energy, the phenomenon is called photoelectric emission • Hence photoelectric emission is the release of electrons from the surface of a metal when electromagnetic radiation is incident on its surface

8. Demonstration of photoelectric emission using gold-leaf electroscope • When a clean zinc plate is placed on a gold-leaf electroscope and the electroscope then charged negatively, the gold leaf deflects proving that the zinc plate is charged • When visible light of any color is shone on to the plate, the leaf does not move even if the intensity (brightness) of the light is increased • However, when ultra-violet radiation is shone on the plate, the leaf immediately falls, showing that it is losing negative charge • This means that electrons are being emitted from the zinc plate • These electrons are called photoelectrons and the process is called photoelectric emission • If the intensity of the uv radiation is now increased, the leaf falls more quickly, showing that the rate of emission of photoelectrons has increased • The difference between uv radiation and visible light is that uv radiation has a shorter wavelength and hence a higher frequency than visible light

9. Further investigations with apparatus like these lead to further conclusions • If photoelectric emission takes place, it does so instantaneously. There is no delay between illumination and emission (10-9 s) • Photoemission takes place only if the frequency of the incident radiation is above a certain minimum value called the threshold frequency f0 • Different metals have different threshold frequencies • Whether or not emission takes place depends only on whether the frequency of the radiation used is above the threshold for that surface. It does not depend on the intensity of the radiation. Even very dim light gives some electrons with high KE • For a given frequency (above the threshold frequency), the rate of emission of photoelectrons is proportional to the intensity of the radiation

10. PHOTOELECTRIC EFFECT Light, frequency ν Vacuum chamber Collecting plate Metal plate I Ammeter Potentiostat

11. Another expt to investigate photoelectric emission • Pg 239 fig 9.2 Physics by Chris Mee • A is the metal surface and B is the collector across which there is a variable potential difference • Without any p.d. applied, when uv radiation of a fixed frequency above the threshold, is shone on to Athe metal surface, it emits photoelectrons • These electrons travel from A to B causing current to flow which is detected by the micro-ammeter • If now a p.d. is applied between A and B, with B negative wrt A, any electron going from A to B will gain potential energy as it moves against the electric field. The gain in potential energy is at the expense of the kinetic energy of the electron • i.e. loss in k.e. = gain in p.e. = charge of electron x potential difference (E = eV) • If the voltage is now gradually increased, the current registered decreases and finally falls to zero. This minimum value of the p.d necessary to stop electron flow completely is the stopping voltage and it measures the max k.e. with which the electrons are emitted • The fact that there is current flow detected at voltages less than the stopping voltage indicates that there is a range of kinetic energies for these electrons

12. cont.. • When the experiment is repeated with radiation of greater intensity but the same frequency, the max current increases but the value of the stopping potential is unchanged. The graph of max k.e. vs frequency of radiation gives a straight line • Conclusions to be drawn: • The photoelectrons have a range of k.e. from zero up to some max value • If the frequency of the incident radiation is increased, the max k.e. of the photoelectrons also increases • For constant frequency, the max k.e. is unaffected by the intensity of radiation • If the graph of max k.e. vs frequency of radiation is extrapolated to the point where the max k.e. of the electrons is zero, the min frequency required to cause emission i.e.the threshold frequency can be found • The above 2 experiments on photoemission showed conclusively that radiation of frequency below the threshold, no matter how intense or for how long, does not produce photoelectrons • This cast a major doubt as to whether light is a continuous wave (which was until then fully accepted due to interference and diffraction effects)

13. Classical expectations Electric field E of light exerts force F=-eE on electrons. As intensity of light increases, force increases, so KE of ejected electrons should increase. Electrons should be emitted whatever the frequency ν of the light, so long as E is sufficiently large For very low intensities, expect a time lag between light exposure and emission, while electrons absorb enough energy to escape from material PHOTOELECTRIC EFFECT J.J. Thomson Hertz When UV light is shone on a metal plate in a vacuum, it emits charged particles (Hertz 1887), which were later shown to be electrons by J.J. Thomson (1899). Light, frequency ν Vacuum chamber Collecting plate Metal plate I Ammeter Potentiostat

14. Einstein’s interpretation (1905): Light comes in packets of energy (photons) Actual results: Maximum KE of ejected electrons is independent of intensity, but dependent on ν For ν<ν0 (i.e. for frequencies below a cut-off frequency) no electrons are emitted An electron absorbs a single photon to leave the material There is no time lag. However, rate of ejection of electrons depends on light intensity. PHOTOELECTRIC EFFECT Einstein Millikan Verified in detail through subsequent experiments by Millikan

15. Wave theory shaken and quantum theory is born! • In classical wave theory : • When an em wave of amplitude and frequency interacts with an electron, the electron will absorb energy from it and when it absorbs enough energy it will escape free from the metal • Thus even a low frequency wave whose energy can be boosted by increasing the amplitude (i.e. increasing the intensity or brightness) can be used to cause photoemission • Alternatively, we can use less bright light but shine it on the metal for a longer time until it has enough energy to cause emission • Classical wave theory of em radiation predictions: • Whether an electron is emitted or not should depend on the power of the incident wave i.e. on its intensity. A very intense wave of any frequency should cause photoemission • The max k.e. of the photoelectrons should be greater if the radiation intensity is greater • There is no reason why photoemission should be instantaneous • Hence a new concept of the quantum theory arose

16. Albert Einstein’s theory of photoelectric emission • In 1901, Max Planck put forward the theory that energy cannot be divided into smaller and smaller amounts and had suggested that the energy carried by em radiation might exist as discrete packets called quanta and gave his expression E = hf where f is the frequency of the radiation and h is a constant called the Plank’s constant (h = 6.63 x 10-34 J s) • In 1905 Albert Einstein developed the theory of quantized energy to explain all the observations associated with photoelectric emission. • Einstein proposed that light radiation consists of a stream of energy packets called photons. This is a special name given to a quantum of energy when the energy is in the form of em radiation • When a photon interacts with an electron, it transfers all its energy to the electron • It is only possible for a single photon to interact with a single electron; the photon cannot share its energy between several electrons • This transfer of energy is instantaneous

17. Photon theory of photoelectric emission • If the frequency of the incident radiation is less than the threshold frequency for the metal, the energy carried by each photon is insufficient for an electron to escape the surface of the metal • If the photon energy is insufficient for an electron to escape, it is converted to thermal energy in the metal • The minimum amount of energy required for an electron to escape from the surface is called the work function energy, φ • If the frequency of the incident radiation is equal to the threshold frequency, the energy carried by each photon is just sufficient for electrons at the surface to escape • If the frequency of the incident radiation is greater than the threshold frequency, surface electrons will escape and have surplus energy in the form of kinetic energy. These electrons will have the max k.e. • If a photon interacts with an electron below the surface, some energy is used to take it to the surface, so that it is emitted with less than the max k.e. This gives rise to a range of values of k.e.

18. The photoelectric equation • Albert Einstein used the principle of conservation of energy to derive the photoelectric equation • Photon energy = work function energy + maximum k.e. of the photoelectron i.e. hf = φ + ½mevmax2 • For radiation incident at the threshold frequency, hf0 = φsince½mevmax2 = 0 • Hence the complete photoelectric equation can be written as hf = hf0 + ½mevmax2 • Materials with a low work function emit electrons in visible light whereas those with a higher work function require the higher energy photons of ultraviolet

19. Work function energies φ and threshold frequencies of some materials • Metal φ/eVf0/Hz Sodium 2.4 5.8 x 1014 Calcium 2.9 7.0 x 1014 Zinc 3.6 8.8 x 1014 Silver 4.3 1.0 x 1014 Platinum 5.6 1.4 x 1014 (Remember; 1 eV = 1.6 x 10-19 J) A quantum is an extremely small amount of energy E.g. quantum of red light = 2 eV quantum of violet light = 4 eV

20. Example The work function energy of platinum is 9.0 x 10-19 J. Plank’s constant h = 6.63 x 10-34 J s Calculate a) the threshold frequency for the emission of photoelectrons from platinum b) the maximum k.e. of a photoelectron when radiation of frequency 2.0 x 1015 Hz is incident on a platinum surface Solution a) using hf0 = φ, f0 = φ/h = 1.4 x 1015 Hz b) using hf = hf0 + ½mevmax2 ½mevmax2 = hf - hf0 = h(f – f0) = 4.0 x 10-19 J

21. Compton X-ray source Collimator (selects angle) Crystal (selects wavelength) θ Target COMPTON SCATTERING Compton (1923) measured intensity of scattered X-rays from solid target, as function of wavelength for different angles. He won the 1927 Nobel prize. Detector Result: peak in scattered radiation shifts to longer wavelength than source. Amount depends on θ(but not on the target material). A.H. Compton, Phys. Rev. 22 409 (1923)

22. Incident light wave Emitted light wave Oscillating electron Before After scattered photon Incoming photon θ Electron scattered electron COMPTON SCATTERING Classical picture: oscillating electromagnetic field causes oscillations in positions of charged particles, which re-radiate in all directions at same frequency and wavelength as incident radiation. Change in wavelength of scattered light is completely unexpected classically Compton’s explanation: “billiard ball” collisions between particles of light (X-ray photons) and electrons in the material

23. Before After scattered photon Incoming photon θ Electron scattered electron COMPTON SCATTERING Conservation of energy Conservation of momentum

24. Electron diffraction • If light waves can behave like particles (photons) perhaps particles can behave like waves? • If a beam of X-rays of a single wavelength is directed at a thin metal foil, a diffraction pattern is produced. • This is a similar effect to the diffraction pattern produced when light passes through a diffraction grating. • The foil contains many crystals and the gaps between neighboring planes of atoms in the crystals act as slits, creating a diffraction pattern • If a beam of electrons is directed at a thin metal foil, a similar diffraction pattern is produced • The electrons which we normally consider to be particles, are exhibiting a property which we would normally associate with waves • To observe diffraction, we know that the wavelength of the radiation should be comparable to the size of the aperture • The fact that diffraction is observed with electrons suggests that they have a wavelength of about the same magnitude as the separation of planes of atoms in crystals of the order of 10-10 m

25. ELECTRON DIFFRACTIONThe Davisson-Germer experiment (1927) Davisson G.P. Thomson The Davisson-Germer experiment: scattering a beam of electrons from a Ni crystal. Davisson got the 1937 Nobel prize. θi θi At fixed angle, find sharp peaks in intensity as a function of electron energy Davisson, C. J., "Are Electrons Waves?," Franklin Institute Journal 205, 597 (1928) At fixed accelerating voltage (fixed electron energy) find a pattern of sharp reflected beams from the crystal G.P. Thomson performed similar interference experiments with thin-film samples

26. ELECTRON DIFFRACTION θi Path difference: θr Constructive interference when a Note θiandθrnot necessarily equal

27. Double-slit experiment for electrons • When electrons fired toward a slit for which d < , a diffraction pattern is seen • If current is lowered so that only 1 electron is allowed at a time - pattern is STILL SEEN!

28. Diffraction

29. Electron diffraction

30. Electron Microscopes

32. Wave-particle duality • In 1924 Louis de Broglie suggested that all moving particles have a wave-like nature. • Using ideas based upon the quantum theory and Alert Einstein’s theory of relativity, he suggested that the momentum p of a particle and its associated wavelength λ are related by the equation λ = h/p where h is the Planck constant. λ is known as the de Broglie wavelength

33. Example Calculate the de Broglie wavelength of an electron traveling with a speed of 1.0 x 107 m/s. (Planck constant h = 6.6 x 10-34 J s; electron mass me = 9.1 x 10-31 kg) Solution Using λ = h/p , p= h/λ= 7.3 x 10-11 m

34. Spectra • White light is not a single colour. If white light from a tungsten filament is passed through a prism, the light due to refraction is spread out or is dispersed into its component colors and the band of different colors is called a continuous spectrum • In the case of white light it ranges from 400 nm to 700 nm (800 THz to 400 THz) (VIBGYOR) • Since this spectrum has been produced by the emission of light from the tungsten filament lamp, it is referred to as an emission spectrum • When a high potential difference is placed across 2 electrodes of a discharge tube (which is a transparent tube containing a gas at low pressure) light is emitted • Examination of the light with a diffraction grating shows that the emitted spectrum is no longer continuous but consists of a number of bright lines. Such a spectrum is known as a line spectrum • The wavelengths corresponding to the lines of the spectrum are characteristic of the gas which is in the discharge tube

35. How are line spectra produced? • In order to understand this, we need to understand how electrons in atoms behave • Bohr’s allowed-orbit analysis only works for the simplest atom, hydrogen. This model has now been replaced by a mathematical wave mechanics model of the atom in which allowed orbits are replaced by allowed energy levels • Electrons in an atom can only have certain specific energies called the electron energy levels of the atom • The energy levels are usually represented as a series of lines against a vertical scale of energy and the electron can have any of these energy levels but cannot have energies in between them • Normally electrons occupy the lowest energy levels available and under these conditions the atom and its electrons are said to be in the ground state • However, if the electron absorbs energy perhaps by being heated or by collision with another electron, it may be promoted to a higher energy level • The energy absorbed is exactly equal to the difference in energy of the 2 levels and under these conditions the atom is described as being in an excited state • Light waves come from atoms. A burst of wave energy is given off whenever an electron loses energy by dropping to a lower orbit. This burst of energy which acts as a particle is what we call a photon

36. Electron energy levels in atoms • Pg 247 fig 9.9 Physics by Chris Mee

37. Electron energy levels…. • An excited atom is unstable and hence after a short time, the excited electron will return to a lower level • In order to achieve this it must lose energy • It does so by emitting a photon of electromagnetic radiation whose energy is given by hf = ΔE = E2 – E1 where E2 is the energy of the higher level and E1energy of the lower level and h is the Planck constant • In terms of the speed of light c, λ = hc/ΔE • This movement of an electron between energy levels is called an electron transition • The larger the energy of the transition, the higher the frequency (and hence shorter the wavelength) of the emitted radiation • This downward transition results in the emission of a photon • The atom can be raised to an excited state by the absorption of a photon, but the photon must have the right energy corresponding to the difference in energy of the excited state and the initial state • Hence a downward transition corresponds to photon emission and an upward transition to photon absorption

38. Spectroscopy • Because all elements have different energy levels, the energy differences are unique to each element, consequently each element produces a different and characteristic line spectrum • Spectra can be used to identify the presence of a particular element and the study of spectra is called spectroscopy and instruments used to measure wavelengths of spectra are called spectrometers • Spectrometers for accurate measurements of wavelength make use of diffraction gratings to disperse the light

39. Continuous spectra • Whilst the light emitted by isolated atoms such as those in low-pressure gases in tubes produce line spectra, the light emitted by atoms in a solid, liquid or gas at high pressure produces a continuous spectrum • This is due to the proximity of the atoms to each other which causes interaction between the atoms resulting in a broadening of the electron energy levels seen as a continuous spectrum

40. Absorption spectra • If white light passes through a low-pressure gas and the spectrum of the white light is then analyzed, it is found that light of certain wavelengths is missing and in their place are dark lines • This type of spectrum is called an absorption spectrum • As the white light passes through the gas, some electrons absorb energy and make transitions to higher energy levels • The wavelengths of the light they absorb correspond exactly to the energies needed to make the particular upward transition • When these excited electrons return to lower levels, the photons are emitted in all directions rather than in the original direction of the white light, hence some wavelengths appear to be missing • The wavelengths missing from an absorption spectrum are exactly those present in the emission spectrum of the same element

41. Example Calculate the wavelength of the radiation emitted when the electron in a hydrogen atom makes a transition from the energy level at -0.54 x 10-18 J to the level at -2.18 x 10-18 J (Planck constant h = 6.62 x 10-34 J s, c = 3.00 x 108 m/s) Solution Using ΔE = E2 – E1 = 1.64 x 10-18 J And using λ = hc/ΔE = 121 nm