RI Chap 19 Quantum Physics Lecture Notes

19 Quantum Physics H2 Physics 9478 Content Page Introduction 2 19.1 The Photoelectric Effect 3 19.2 Wave-Particle Duality 5 19.3 Wavefunction of a Particle 8 19.4 Energy Levels of Atoms and Line Spectra 18 19.5 Heisenberg Uncertainty Principle 27 19.6 Appendix 29 Learning Outcomes Candidates should be able to: (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 (c) show an understanding that while a photon is massless, it has a momentum given by p Ec= and ph λ= , 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 hpλ = for the de Broglie wavelength to solve problems (f) show an understanding that the state of a particle can be represented as a wavefunction ψ, e.g. for an electron cloud in an atom, and that the square of the wavefunction amplitude 2 ψ 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 (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 (i) show an understanding of standing wave solutions 𝜓𝜓 n for the wavefunction of a particle in a one- dimensional infinite square well potential

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT Page | 2 (j) solve problems using 2 2 28 n hEn mL= for the allowed energy levels of a particle of mass m in a one- dimensional infinite square well of width L (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. Introduction Wave theory of light has been highly successful in describing phenomena such as diffraction, interference and polarisation, leading to the creation of radio waves, radar and optical filters, just to name a few. However, wave theory of light is unable to explain phenomena such as blackbody radiation, photoelectric effect and atomic line spectra. The search for an explanation for these experiments led to breakthrough at the beginning of the 20 th century, resulting in the birth of modern physics – quantum mechanics and relativity. While quantum mechanics deals with the “lows” (low temperature, small sizes), Einstein’s relativity deals with the “highs” (high velocities, large distances). In this chapter, we will discuss important aspects of quantum theory – quantisation, duality and uncertainty. The concept of quantisation was first suggested by Max Planck during his study of blackbody radiation in 1900. Inspired by Planck’s work, Einstein theorised that electromagnetic (EM) radiation itself is quantised – that EM radiation transports energy in quanta called photons – and successfully explained the photoelectric effect in 1905. James Franck and Gustav Hertz then showed conclusively that ene rgy levels in atoms are quantised in 1914, which then provided an explanation for atomic line spectra. The bizarre behaviour of EM radiation – that it has both wave and particle properties – led de Broglie to propose the wave-particle duality for EM radiation and matter in his doctoral dissertation in 1924. His idea was way ahead of its time and there was a real possibility that de Broglie failed his doctoral interview, if not for the intervention by Einstein. Experiments conducted by Clinton Davisson and Lester Germer in 1927 showed that electrons, indeed, have wavelike properties.

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT Page| 3 19.1 The Photoelectric Effect  Theories of Light Two apparently contradictory theories of the nature of light were competing in the 17 th century. The corpuscular theory proposed by Sir Isaac Newton (English, 1643− 1727) regarded light as a stream of tiny particles travelling at high speed in a straight line. The corpuscular theory did account for rectilinear propagation, reflection and refraction – the latter by assuming that on entering an optically denser medium t he corpuscles are attracted, thereby causing bending towards the normal. On the other hand, the theory proposed by Christiaan Huygens (Dutch, 1629−1695) in 1678 considered light as a wave. His wave model can also account for reflection and refraction. Opponents of the wave theory argued that waves require a medium for transmission and there did not appear to be one for light which was able to travel in vacuum. Subsequently a medium, called the ether, was invented but it defied all attempts at detection. An apparently crucial difference between the two theories was that whereas the corpuscular theory predicted light to have a greater speed in any medium denser than air while the wave theory predicted the opposite. Thomas Young (English, 1773− 1829) demonst rated interference of light through his famous Young’s double -slit experiment in 180 1 which were readily explicable in terms of waves. The wave theory prediction was further confirmed in 1862 when Jean Bernard Léon Foucault (French, 1819−1868) found the speed of light in water was less than that in air. In 1865, Maxwell derived theoretically that electric and magnetic fields travel through space as waves moving at the speed of light. He proposed that visible light is a type of electromagnetic wave, and his equations can be used to derive the laws of reflection and refraction that were well established in geometrical optics. This firmly establishes that light is a wave phenomenon. The photoelectric effect was first discovered in 1887 by Heinrich Hertz (German, 1857 −1894). In his experiments to prove the existence of electromagnetic radiation, he discovered that electrodes illuminated with ultraviolet light create electric sparks more easily. This phenomenon, known as the photoelectric effect , can be easily demonstrated with the gold-leaf electroscope shown in Fig. 19.1. The electroscope is given a negative charge. When ultraviolet (UV) light of wavelength 254 nm is incident on the zinc plate , the gold leaf falls rapidly. No change in the position of the leaf is observed when UV light of wavelength 365 nm or a much brighter (higher intensity) white light is incident on a negatively charged electroscope.

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT Page | 4 Fig. 19.1 (a) Leaf falls rapidly as electrons are repelled away. (b) No change in position of leaf since no electrons are emitted. Based on these observations, we can conclude that negatively charged particles escaped from the surface of the zinc plate when light is incident on it but this effect is dependent on frequency or wavelength but not on intensity of light. Specific charge (qm ) of these ejected particles confirms that they are indeed electrons. According to the wave theory, one plausible explanation of the photoelectric effect is that