Quantum I Notes

 2014 Yeow Kok Han Page 1 QQuuaannttuumm PPhhyyssiiccss II 11 HHiissttoorriiccaall BBaacckkggrroouunndd Hundreds of years back, scientists were speculating about whether light was made up of particles or waves. By 1800s, it was generally accepted that light was a wave. The definitive feature of waves is their abil ity to diffract, suppose and produce interference patterns. Particles on the other hand, cannot superpose. Instead, they collide and bounce off each other. In 1887 Hertz discovered the photoelectric effect. It was found that electrons, called photoelectrons, were emitted from a clean metal surf ace when high frequency light such as ultraviolet light fell on it. The experimental data could not be explained by treating light as a wave but they could be explained by treating light as particles. Particles of lig ht or p ackets of electromagnetic radiation are called photons. Each packet of energy or radiation is called a quantum. Since then, many quantities other than energy have been found to be quantised under certain conditions. This quantisation of physical qua ntities and the probabilistic nature of their measurements are two of the features of Quantum Physics that we will study. In Quantum Physics, many phenomena seem very bizarre. This is because quantum behaviour is mostly encountered in the microscopic realm , of which we would normally have very little experience of. 22 PPhhoottooeelleeccttrriicc EEffffeecctt A positive p.d is applied when collector is at higher potential while negative p.d. is when collector is at lo wer potential than emitter . The data in Fig. 2.2 is obtained when p.d. V is varied while light intensity is kept fixed. A positive p.d creates an E-field that accelerates the electrons towards the collector. It also steers some electrons that would other wise have collided with the casing and be absorbed towards the collector. When all the electrons reach the collector, the current is called saturation current. Photon is a packet of electromagnetic radiation energy. Photoelectric effect is the emission of electrons from metals by electromagnetic radiation of sufficiently high frequency. Saturation current is the maximum photoelectric current which occurs when all the emitted electrons are able to reach the collector. Photoelectric effect is the emission of electrons from metals by electromagnetic radiation of sufficiently high frequency. Fig. 2.1 The incident light falls on the emitter metal, ejecting electrons (photoelectrons), some of which are collected by the collector, giving rise to a current (photoelectric current). The vacuum inside is needed to prevent formation of oxide layer on metal and to facilitate the movement of emitted electrons from emitter to collector. variable voltage source incident radiation V A quartz window emitter collector vacuum V saturation current i is Vs Fig. 2.2 + _ Fig. 2.3

 2014 Yeow Kok Han Page 2 2 max 1 2 eSm v eV A negative p.d. causes some emitted electrons to slow down and even turn back to the emitter. Some fast enough electrons may still be able to reach the collector. A more negative p.d. will cause more electrons to get turned back and thus the current to decrease. Eventually, when p.d. is -Vs, the fastest electrons just fail to reach the collector and the current goes to zero . These fastest electrons lost all their KE just before reaching the collector and the lost KE becomes electric PE (from ‘Electric Field’, ∆EPE or ∆UE = Q∆V where ∆V is p.d.). Thus where me & e are electron mass and charge respectively , vmax is speed of the fastest electrons and Vs is the magnitude of the negative p.d. known as stopping potential. MMoorree DDaattaa aanndd CCoommppaarriissoonn ooff WWaavvee aanndd PPaarrttiiccllee TThheeoorriieess Intensity is fundamentally the ‘rate of energy transfer’ per unit area. In our situation, it would be the rate of arrival of energy per unit metal surface area. Wave Theory Einstein’s Particle Theory For electromagnetic wave s, intensity is proportional to the squared of the wave amplitude a. Greater intensity means greater oscillating E and B -field amplitudes. For a stream of particles or photons , intensity is logically proportional to the ‘energy in each packet’ Ep, and the ‘rate of arrival of packets’ Np/t, per unit area: Both theories expect that if I increases, the rate of arrival of energy on metal increases. Hence more electrons per unit time can gain energy to escap e. Thus current should increase as in Fig. 2.6 and 2.7. Stopping potential is the minimum magnitude of the p.d. required to bring the photoelectric current to zero by just stopping all electrons from reaching the collector. When electrons just fail to reach collector: 2 max 1 2 eSm v eV Data 1 Photoelectric current i  light intensity I at a fixed frequency. Intensity for light as wave is I  a2 Intensity for light as photons is 1p p NE tA    I where Ep = hf Both theories can explain Data 1 V Vs i All electrons prevented from reaching collector. On the way to collector, electrons decelerate. Slower ones will not arrive. Electrons accelerate towards collector, but some will miss. All electrons reach collector. Fig. 2.5 1. The photoelectric current i is proportional to the intensity I of the light at a fixed frequency (Fig. 2.6). I i V intensity I2 > I1 I2 i I1 i2 i1 Vs Fig. 2.6 Fig. 2.7 0 _ Fig. 2.4 + I  a2 1p p NE tA    I

 2014 Yeow Kok Han Page 3 Wave Theory Einstein’s Particle Theory No matter how low I is, with enough time for electrons to absorb the wave energy, they should be ejected eventually. Hence wave theory expects electron emission to depend on I only. However, for electromagnetic waves, I does not depend on frequency . Thus, wave theory cannot explain why there exists a threshold frequency fmin. As for dependence of KEmax on frequency, wave theory similarly cannot account for it. Einstein postulated that: 1 a photon’s energy, Ep = hf where Plank constant h = 6.63  10-34 J s. 2 an electron either absorb all the energy of a photon or not at all. 3 probability of electron absorbing more than one photon is negligible. 4 each metal has a minimum energy  needed for electrons to escape from the metal’s attraction.  is called work function and it is characteristic of the metal. A model and explanation based on Einstein’s postulates: Electrons in a metal are like tennis balls in a depression in the ground (Fig. 2.10). For a tennis ball to get out of the depression and thus further from Earth, it needs to overcome gravity and gain gravitational PE. Similarly, an escaping electron must overcome the metal’s attraction and gain electric PE. Using imaginary numbers, the topmost electron needs the least energy (work function ) of 6 J to get out. After absorb ing 10 J from a photon, 6 J is converted to electric PE, the rest becomes KE upon escape, hence: Data 2 There is a threshold frequency or minimum frequency fmin of incident radiation for electrons to be emitted, regardless of the intensity of radiation. Data 3 Electrons are emitted with a range of KE and the maximum KE(=eVs) is dependent on the frequency but not intensity. Wave theory cannot but particle theory can explain Data 2 & 3. Work function  is the minimum energy required to remove an electron from a metal’s surface. It is a characteristic property of a metal. 3. Electrons are emitted with a range of KE and the maximum KE(=eVs) is dependent on the frequency but not intensity. 2. There