ACJC Quantum Physics Lecture Notes

Anglo-Chinese Junior College Lecture Notes Quantum Physics H2 (9478) JC2 2026 Page 1 of 47 Quantum Physics Guiding Questions Learning Objectives 1. How does light behave like particles? (a) show an understanding that the existence of a threshold frequency in the photoelectric effect provides evidence that supports the particulate nature of electromagnetic radiation while phenomena such as interference and diffraction provide evidence that supports its wave nature. (b) state that a photon is a quantum of electromagnetic radiation, and recall and use the equation E hf= for the energy of a photon to solve problems, where h is the Planck constant. 2. How do particles display wave-like properties? (c) show an understanding that while a photon is massless, it has a momentum given by p E c= and ph = , where c is the speed of light in free space. (d) show an understanding that electron diffraction and double-slit interference of single particles provide evidence that supports the wave nature of particles. (e) recall and use the equation hp = for the de Broglie wavelength to solve problems. 3. How does a wavefunction explain where a particle can be found and why its energy is quantised? (f) show an understanding that the state of a particle can be represented as a wavefunction  , e.g. for an electron cloud in an atom, and that the square of the wavefunction amplitude 2  is the probability density function (including calculation of normalisation factors for square and sinusoidal wavefunctions). (g) show an understanding that the principle of superposition applies to the wavefunctions describing a particle’s position, leading to standing wave solutions for a particle in a box and phenomena such as single -particle interference in double -slit experiments. (i) show an understanding of standing wave solutions n for the wavefunction of a particle in a one -dimensional infinite square well potential. (j) solve problems using 2 2 2 8 n h n mL E = for the allowed energy levels of a particle of mass m in a one -dimensional infinite square well of width L.

Anglo-Chinese Junior College Lecture Notes Quantum Physics H2 (9478) JC2 2026 Page 2 of 47 4. Why is the simultaneous measurement of position and momentum impossible? (h) show an understanding that the Heisenberg position - momentum uncertainty principle xp ≳ h relates to the necessity of a spread of momenta for localised particles, and apply this to solve problems. 5. How do spectral lines provide evidence for the quantisation of energy? (k) show an understanding of the existence of discrete electronic energy levels for the electron’s wavefunction 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) solve problems involving photon absorption or emission during atomic energy level transitions. Nature of Science Moment – Why Quantum Physics? For much of the 19th century, light was successfully described as a wave. This model explained reflection, refraction, diffraction, and interference so well that few scientists doubted it. However, experiments soon revealed behaviour that the wave model could not explain. To account for this, physicists were forced to treat light as quantised particles called photons, each carrying a fixed amount of energy. This single idea resolved contradictions that classical physics could not. Quantum physics did not arise because classical physics was wrong in everyday situations —it arose because classical ideas had limits. When observations no longer matched existing models, physicists did not ignore the data. Instead, they changed the model. The study of quantum physics begins with a fundamental question: What happens when our best ideas no longer match what we observe?

Anglo-Chinese Junior College Lecture Notes Quantum Physics H2 (9478) JC2 2026 Page 3 of 47 1. Light as a Particle In the earlier topics on Wave Motion and Superposition, we saw strong experimental evidence that light behaves as a wave, with diffraction and interference patterns such as those observed in Young’s double-slit experiment. However, further experiments showed that the wave model of light could not explain all observed behaviour. To account for these results, physicists proposed that light, and more generally all electromagnetic radiation, transfers energy in discrete packets rather than continuously. This led to the idea that light can be treated as quantised particles called photons, each carrying a fixed amount of energy. 1.1 Energy of a Photon Since each quantum of electromagnetic radiation behaves like a particle , the minimum amount of energy of electromagnetic radiation would be the energy of a single photon. For electromagnetic radiation with larger amounts of energy, there will be multiple photons, and the total energy will be an integer multiple of hf. Example 1.1 Determine the energy of a photon of wavelength 440 nm. ( )( ) 34 8 9 19 19 6.63 10 3.00 10 440 10 4.520 10 4.52 10 J E hf hc  − − − − = =  =  =  How does light behave like particles? A photon is a quantum of electromagnetic radiation. The energy of one photon, E, is given by the equation E hf= where h : Planck’s constant, 346.63 10 J sh −= f : frequency of electromagnetic radiation

Anglo-Chinese Junior College Lecture Notes Quantum Physics H2 (9478) JC2 2026 Page 4 of 47 1.2 The Electron Volt From Example 1.1, it can be seen that the typical energies of a single photon (and other fundamental particles which we will encounter in Nuclear Physics) are very small when expressed in joules. The electron volt (eV) is a unit of energy which is more suitable for such contexts. One electron volt (eV) is defined as the work done by an electric field to accelerate an electron across a potential difference of 1 V. ( )( ) 19 19 gain in kinetic energy of electron = loss in electric potential energy 1 eV 1.60 10 C 1 V 1 eV 1.60 10 J W q V − − = = = In Example 1.1, the photon with wavelength 440 nm would have 19 19 4.520 10 2.83 eV1.60 10 − −  = of energy. Example 1.2 (a) A source emits monochromatic light of wavelength λ at power P. Given that h is Planck’s constant and c is the speed of light, show that the rate of emission of photons is given by the expression P hc  . (b) An FM radio transmitter has a power output of 150 kW and operates at a frequency of 99.7 MHz. Determine the number of photons emitted by the transmitter in one second. Solution (a) ( ) number of photons emittedrate of emission time number of photons emitted energy of each photonpower time shown N t Nhf N hcP tt NP t hc   == =   ==      = (b) ( )( ) 3 34 6 30 From (a): 150 10 6.63 10 99.7 10 2.27 10 photons per second NhfP t N t N t − =  =   =

Anglo-Chinese Junior College Lecture Notes Quantum Physics H2 (9478) JC2 2026 Page 5 of 47 1.3 The Ph