EJC Physics H219 (2024) - 1. Quantum Physics Notes (FULL)

Page 1 of 36 9749(2024) H2 Physics H219 Quantum Physics Notes H2 Topic 19 Quantum Physics The Nobel Prize in Physics 1921 was awarded to Albert Einstein " for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect." Content • Energy of a photon • The photoelectric effect • Wave-particle duality • Energy levels in atoms • Line spectra • X-ray spectra • The uncertainty principle Learning Outcomes Candidates should be able to: (a) show an appreciation of the particulate nature of electromagnetic radiation (b) recall and use the equation E = hf for the energy of a photon (c) show an understanding that the photoelectric effect provides evidence for the particulate nature of electromagnetic radiation while phenomena such as interference and diffraction provide evidence for the wave nature (d) recall the significance of threshold frequency (e) recall and use the equation 2 max s 1 2 mv eV= , where VS is the stopping potential (f) explain photoelectric phenomena in terms of photon energy and work function energy (g) explain why the stopping potential is independent of intensity whereas the photoelectric current is proportional to intensity at constant frequency (h) recall, use and explain the significance of the equation 2 max 1 2h mvf Φ+= (i) describe and interpret qualitatively the evidence provided by electron diffraction for the wave nature of particles (j) recall and use the relation for the de Broglie wavelength λ = h / p (k) show an understanding of the existence of discrete electronic energy levels in isolated atoms (e.g. atomic hydrogen) and deduce how this leads to the observation of spectral lines (l) distinguish between emission and absorption line spectra (m) recall and solve problems using the relation hf = E2 – E1 (n) explain the origins of the features of a typical X-ray spectrum (o) show an understanding of and apply ΔpΔx ≳ h as a form of the Heisenberg position-momentum uncertainty principle to new situations or to solve related problems.

Page 2 of 36 9749(2024) H2 Physics H219 Quantum Physics Notes 19.1 Particles and waves We often model matter using particles (small masses) that are hard, rigid and move according to Newton’s Laws of motion. The macroscopic phenomena can then be explained by the random motion and collisions of the distribution of particles. area microscopic model macroscopic phenomena typical diagrams electricity flow of electrons current gases Kinetic Theory pressure, volume, and temperature of gas solids lattice structure mechanical properties On the other hand, waves are used to explain the transport of energy without matter being transported. The medium of transport is often described as oscillating. area oscillating quantity typical diagrams sound air pressure light (electromagnetic radiation) electric field and magnetic field waves on a string displacement (of string)

Page 3 of 36 9749(2024) H2 Physics H219 Quantum Physics Notes 19.2 Light can behave like waves or like particles Light (or, in general, electromagnetic radiation) can undergo superposition and interference. These are strong evidences that electromagnetic radiation behaves like waves. However, in order to explain the photoelectric effect, e lectromagnetic radiation needs to be treated as if they were particles. 19.3 Energy of a photon The Planck constant, h, is 34106.6 J 3 sh −= × . Example 1 Determine the energy of one photon of frequency 5.5 x 1014 Hz. − − ××= = = × 34 1 1 4 9 10 )(5.5 10 ) 3.6 10 J (6.63 E hf Note: The energy in Joules for a photon is not a convenient order of magnitude, so the electronvolt (eV) is often used. One eV is the energy gained by an electron that is accelerated through the potential difference of 1 V: 1 eV = 1.6 x 10-19 C x 1 V = 1.6 x 10-19 J. energy per photon decreases as wavelength increases x-rays γ-rays visible microwaves infrared ultraviolet radio waves wavelength /m A photon is a discrete packet of energy of electromagnetic radiation. Energy of one photon = Planck constant × frequency E hf=

Page 4 of 36 9749(2024) H2 Physics H219 Quantum Physics Notes Example 2 Visible light has wavelengths spanning from 400 nm (violet) to 700 nm (red). Find the energy, in eV, of (i) a photon of red light and (ii) a photon of violet light. (i) For red light: Ered = hf = hc λ = �6.63 × 10-34��3 × 108� 700 × 10-9 = 2.84 × 10-19 J = 1.78 eV (ii) For violet light: Eviolet = hf = hc λ = �6.63 × 10-34��3 × 108� 400 × 10-9 = 4.97 × 10-19 J = 3.11 eV Note: Ultraviolet causes skin cancer while infrared causes heating. Per photon, “blue-er” photons are more energetic than “red-der” photons. Example 3 A 1.0 mW laser produces red light of wavelength 663 nm. (i) Calculate the number of photons that the laser produces in one second. (ii) The same 1.0 mW laser is then tuned to produce UV light of wavelength 221 nm instead. Calculate how many UV photons in a second that the laser now produces. (i) 34 8 9 19 3 tot 3 tot 19 15 1 energy per photon (6.63 10 )(3.0 10 ) 663 10 3.0 10 J energy released in 1s 1.0mW 1 s 1.0 10 J number of photons per second 1.0 10 3.0 10 3.33 10 s hcE hf E Pt E E λ − − − − − − − = = ××= × = × == ×= × ×= = × = × (ii) 34 8 9 19 3 tot 19 15 1 energy per photon (6.63 10 )(3.0 10 ) 221 10 9 10 J number of photons per second 1.0 10 9.0 10 1.11 10 s hcE hf E E λ − − − − − − = = ××= × = × ×= = × = × Note: 1. hcE hf λ= = , because fcλ = for photons. 2. For a fixed power output (or intensity), there can be either (i) more photons each of lower energy (lower frequency or longer wavelength) or (ii) fewer photons each of higher energy (higher frequency or shorter wavelength).

Page 5 of 36 9749(2024) H2 Physics H219 Quantum Physics Notes 19.4 The photoelectric effect Some characteristics of the photo-emission: • A single photon can only interact with (and pass on its energy to) a single electron. The energy transfer is all-or-nothing. • Not all photons (of sufficient energy) get to interact with electrons. There is probability involved for a successful interaction – some photons will “miss” the electrons in the metal. • The photoelectrons are emitted in all random directions with varying speeds. 19.5 The photoelectric equation The photoelectric equation is a statement of the principle of conservation of energy. A photon of energy hf is incident on the metal surface. Photon meets an electron. Electron absorbs all energy of the one photon. Some of the energy is used to escape from the metal. The rest remains as kinetic energy of the electron. Photoelectric effect is the emission of electrons when electromagnetic radiation of high-enough frequency is incident on a metal surfaces Photoelectric equation is 21 2 energy provided by work function energy maximum kinetic energy a single photon of a particular metal of a photoelectron maxh m f vΦ+       = =      +

Page 6 of 36 9749(2024) H2 Physics H219 Quantum Physics Notes 19.6 The work function energy Φ The work function energy Φ is unique to each type of metal. It is typically measured in eV. The work function energy can be affected by the surface condition of the metal, such as it being oxidised. Example 4 Light of wavelength 543 nm is incident on a clean sodium surface. The photoelectrons released are found to have negligible amount of kinetic energy. Determine the work function Φ for sodium metal in eV. 2 max Na ,max 1 2 K h hf mv c Eλ Φ+ = Φ+ = λ − − − Φ = = − ×× −× × = = Na ,max 34 8 9 19 (6.63 10 )(3 10 ) 543 10 10 J 2.29 eV 0 3.66 K hc E 19.7 The threshold frequency and threshold wavelength In other words, it is the frequency of electromagnetic radiation for electrons to be emitted from metal surface with zero