NYJC EJC 2026 Quantum Physics Tutorial

9814 H3 Physics (2026) Quantum Physics Tutorial 1 Particulate nature of photon 1 (a) Describe the nature of a γ-ray photon. (b) When a beam of γ-rays or X-rays passes through matter, it may interact with an electron and be scattered through an angle θ, as show in Fig. 1.1. Fig. 1.1 According to a theory, the energy E ’ of the scattered radiation is given by 2 11 1 ' (1 cos ) eEE mc θ−−= , where E is the energy of the incident radiation, me is the mass of the electron and c is the speed of light. The electron, originally at rest, recoils in a direction defined by the angle φ. (i) A 60 Co source emits γ-ray of energy 1.33 MeV. 1. Find the largest possible energy loss this γ-ray can suffer in an interaction with a stationary electron. 2. In what direction does the recoil electron move after this interaction? [6] (ii) The reduction in energy of the γ-radiation can be detected as a change in wavelength. 1. Is this wavelength change an increase or a decrease? 2. Show that the magnitude ∆λ of the wavelength change corresponding to the energy reduction (E − E’ ) is given by ∆λ = A(1− cos θ ), where A is a constant. Find A in terms of m e, c and the Planck constant h. [5] incident radiation scattered radiation recoil electron θ φ

9814 H3 Physics (2026) Quantum Physics Tutorial 2 (iii) In an experiment to measure ∆ λ for X -rays of incident wavelength 7.09 ×10−11 m, scattered from graphite at various angles θ , the following results were obtained: θ / ° 45 75 90 120 135 ∆λ / 10−11 m 0.07 0.17 0.23 0.35 0.41 Each of the readings of θ was subjected to an uncertainty of ± 1°, and each of the value of ∆λ values to an uncertainty of ± 0.01×10−11 m. By plotting a suitable linear graph using graph paper, investigate the extent to which these results support the theoretical relationship. State your conclusion. [7] [Special Paper Nov 1994 Q9] Photoelectric effect 2 An evacuated tube contains two parallel metal electrodes, one of which is an emitter of electrons and the other a collector. When the emitter is illuminated with electromagnetic radiation of photon energy 4.7 eV at a power of 3.8 mW, photoelectrons are emitted. The potential difference V between collector and emitter is adjusted, and the photocurrent I is measured. Fig. 2.1 is a graph of the variation with V of I. Fig. 2.1 (i) Calculate the rate at which photons are incident on the emitter. [2] (ii) Calculate the maximum rate at which electrons leave the emitter. [1] (iii) Explain the difference between your answers in (i) and (ii). [2] (iv) Calculate the maximum speed at which electrons leave the emitter. [3] I / 10−8A − 0.5 0 V / V 0.5 1.0 0.8 A B

9814 H3 Physics (2026) Quantum Physics Tutorial 3 (v) Determine the work function of the material of the emitter. [1] (vi) Suggest an explanation for the shape of the graph in region AB of Fig. 2.1[2] [Special Paper Nov 1999 Q11] 3 Fig. 3.1 illustrates a device called an image intensifier. Fig. 3.1 A thin layer of photoemissive material is deposited on the inner side of a glass window in an evacuated tube. A fluorescent screen is mounted parallel to the photoemitter. An optical image I1 of an object O is cast on the photoemitter. Electrons from the emitter are accelerated to the fluorescent screen, producing an image I2 which is brighter than I1. In a particular image intensifier, the primary image I1 is formed with light of wavelength 700 nm and power 5.0 × 10−9 W. For light of this wavelength, one photoelectron is emitted, on average, for every 85 photons incident on the emitter. (a) (i) Calculate the number of photons incident on the emitter in one second. (ii) Hence calculate the rate at which electrons are emitted. (iii) Assume that all emitted electrons reach the fluorescent screen, calculate the corresponding current between emitter and screen. [5] O I1 I2 fluorescent screen photoemitter image intensifier electrons electrons

9814 H3 Physics (2026) Quantum Physics Tutorial 4 (b) (i) Because the final image is brighter than the primary image, light energy has been created. Given the principle of conservation of energy, how do you account for this? [2] (ii) Image intensifier tubes may be coupled in series so that the final image of the first intensifier acts as the primary image of the second, and so on. An arrangement of three intensifiers is illustrated in Fig. 3.2. Fig. 3.2 The optical gain (the ratio of the brightness of the final image to that of the primary image) of each intensifier is 1000, and the efficiency of the optical coupling between each is 80 %. Calculate the overall gain of this composite intensifier. [2] (c) Perfect reproduction of the primary image is achieved only if electrons leaving the emitter travel in straight lines perpendicular to the emitter, directly to the screen. In fact, electrons leave the emitter with small initial speeds in all directions. This means that the electrons from one point on the primary image fall on a circular area on the fluorescent screen, called the disc of confusion. The diameter of this disc is proportional to the square root of the ratio of the initial kinetic energy to the final kinetic energy of the electrons. (i) By means of a sketch showing electron trajectories, illustrate the formation of the disc of confusion. [1] (ii) Explain how the size of the disc of confusion limits the quality of the secondary image. [1] (iii) In a certain image intensifier, operating at an accelerating potential difference of 2.0 kV, the diameter of the disc of confusion is estimated to be about 1.0 mm. Estimate the diameter of the disc if the accelerating potential were increased to 4.0 kV. Assume that the initial kinetic energy of the emitted electrons is 1.0 eV in both cases. [2] [Special Paper Nov 2001 Q12] gain 1000 gain 1000 gain 1000 incident image 80% 80% output image intensifiers

9814 H3 Physics (2026) Quantum Physics Tutorial 5 Line spectrum 4 According to a theory of the atom, the frequencies f of the corresponding spectral lin es of the elements are related to their proton numbers Z by the equation: f = a(Z − b) where a and b are constants. The wavelength λ of some of these spectral lines are tabulated below: Element sodium potassium manganese zinc strontium rhodium Z 11 19 25 30 38 45 λ / pm 1200 375 210 144 87.5 61.4 Use a graphical method to determine the values of the constants a and b. [11] [Special Paper Nov 1997 Q5] de Broglie wavelength 5 In a simple model of the hydrogen atom, the electron is allowed to move only along a straight line of length a . The associated wave motion must always show a standing wave pattern along this line, with nodes at the ends. (a) Sketch two possible modes of the standing waves. [2] (b) From the condition that the standing waves must have nodes at the ends of the line of length a, find the allowed values of the momentum of the electrons. [2] (c) Hence show that the allowed val ues En of the energy of the electron in this model of the hydrogen atom are given by 22 28 nhEn mae = where n = 1, 2, 3, …. [3] (d) The approximate diameter of the hydrogen atom is 1.1 × 10−10 m. Take this as the value of a . Calculate the values of E 1, E 2 and E3, and draw an energy level diagram showing the n = 1, 2 and 3 states of this model of the hydrogen atom. [3]

9814 H3 Physics (2026) Quantum Physics Tutorial 6 (e) A photon is emitted when the electron in the hydrogen atom makes a transition from a higher energy state to a lower energy state. Calculate the longest wavelength of radiation