Particle nature of light & Quantization

Transcription

1 Particle nature of light & Quantization A quantity is quantized if its possible values are limited to a discrete set. An example from classical physics is the allowed frequencies of standing waves on a stretched string. Only integer multiples of the fundamental frequency produce standing waves. Two experimental low-energy observations that could not be properly explained in the wave picture gave rise to the idea of quantization: (1) The black body radiation and (2) the photoelectric effect. Later, mid-energy effects (Compton scattering) and high-energy effects (pair production & annihilation) were explained with the particle picture as well. 1

2 A blackbody emits a continuous spectrum of radiation. The spectrum is determined only by the temperature of the blackbody. Blackbody Radiation To correctly explain the shape of the blackbody spectrum Planck proposed that the energy absorbed or emitted by oscillating charges came in discrete bundles called quanta. The energy of the quanta are where h = Js is called Planck s constant, f is frequency. The quantum of EM radiation is the photon. 2

3 The Photoelectric Effect (Einstein 1905) Hertz observed in 1887 that EM radiation incident on a metal will eject electrons from the metal. This is the photoelectric effect. Classical theories that use the wave picture fail to explain the effect. Einstein in 1905 explained the effect in a non-classical particle picture (Nobel prize 1921). Experiments show that: 1. Brighter light causes more electrons to be ejected (but not with more kinetic energy as expected in wave picture). 2. The maximum KE of ejected electrons depends on the frequency of the incident light (not on intensity of incoming light as in wave picture). 3. The frequency of the incident light must exceed a certain threshold, otherwise no electrons are ejected (in wave picture increased intensity would lead to more electrons). 4. Electrons are ejected with no observed time delay regardless of the intensity of the incident light. 3

4 Einstein proposed a particle theory of light to explain the photoelectric effect. 4

5 Observation 1: Brighter light causes more electrons to be ejected. Wave theory predicts a more intense beam of light, having more energy, should cause more electrons to be emitted and they should have more kinetic energy. Particle theory predicts a more intense beam of light to have more photons so more electrons should be emitted, but since the energy of a photon does not change with beam intensity, the kinetic energy of the ejected electrons should not change. The particle theory is consistent with observation 1. 5

6 Observation 2 The maximum KE of ejected electrons depends on the frequency of the incident light. Wave theory cannot explain the frequency dependence of the maximum kinetic energy. Particle theory predicts the maximum kinetic energy of the ejected electrons to show a dependence on the frequency of the incident light. Each electron in the metal absorbs a whole photon: some of the energy is used to eject the electron and the rest goes into the KE of the electron. The maximum kinetic energy E max of an ejected electron is E max = hf φ where φ is called the work function and is the energy needed to break the bond between the electron and the metal. The particle theory is consistent with observation 2. 6

7 Observation 3 The frequency of the incident light must exceed a certain threshold, otherwise no electrons are ejected. Wave theory can offer no explanation. Particle theory predicts a threshold frequency is needed. Only the incident photons with f > f threshold will have enough energy to free the electron from the metal. The electron is ejected from the metal when the energy supplied by the photon exactly equals the work function. This defines the threshold frequency. The particle theory is consistent with observation 3. 7

8 Observation 4 Electrons are ejected with no observed time delay regardless of the intensity of the incident light. Wave theory predicts that if the intensity of the light is low, then it will take some time before an electron absorbs enough energy to be ejected from the metal. Particle theory predicts a low intensity light beam will just have a low number of photons, but as long as f > f threshold an electron that absorbs a whole photon will be ejected; no time delay should be observed. The particle theory is consistent with observation 4. 8

9 Wave-particle duality The particle theory of light is needed to explain the photoelectric effect (as well as mid-energy effect Compton scattering and and high-energy effect pair production). A wave theory of light is needed to explain interference phenomena. Note that both pictures are correct - this is referred to as the wave-particle duality. The nature of the experiment determines whether light shows its wave nature (like in interference) or its particle nature (like in the photoeffect). 9

10 Examples: (1) A 200 W infrared laser emits photons with a wavelength of m while a 200 W ultraviolet laser emits photons with a wavelength of m. (a) What is the energy of a single infrared photon and the energy of a single ultraviolet photon? (b) How many photons of each kind are emitted per second? (2) The photoelectric threshold frequency of silver is Hz. What is the minimum energy required to remove an electron from silver? 10

11 X-ray Production When high energy electrons impact a target x-ray photons can be emitted as the electrons are slowed. This process is called bremsstrahlung (German for breaking radiation ). There is a continuous spectrum of radiation emitted up to a cutoff frequency. The cutoff frequency is defined by the maximum kinetic energy of the electron K = hf max. The spikes in the spectrum are called characteristic x- rays. These peaks depend on the target material and are due to fluorescence transitions 2p 1s, which is referred to as Kα radiation. They are called characteristic because the energy levels of the electrons are characteristic for the material. 11

12 Compton Scattering Before Collision After Collision y Photon (E 1, p 1 ) x Photon (E 0, p 0 ) Free electron at rest θ φ Free electron (K, p) 12

13 Conserve momentum and energy during the collision: E=pc for a photon 13

14 Manipulating the previous expressions gives Δλ is the Compton shift. The Compton wavelength 14

15 Spectroscopy and Early Models of the Atom Examples of emission spectra: An absorption spectrum (bright background with dark lines) is seen if a hot source is viewed though a gas. 15

16 The Thomson model of the atom had a volume of positive charge with the negatively charged electrons embedded within the volume. Scattering experiments by Rutherford led to the conclusion that an atom had a very small nucleus of positive charge (10-5 times the size of the atom containing nearly all of the mass) that was surrounded by the electrons. 16

17 It was thought that the electrons in their orbits should radiate (they are accelerated) causing the electron s orbit to decay, implying that atoms are not stable. This is obviously false. Any model of the atom must also explain the line spectra of the elements. 17

18 The Bohr Model of the Hydrogen Atom The Bohr model assumes: The electron is allowed to be in only one of a discrete set of states called stationary states. The electron orbits have quantized radii, energy, and angular momentum. 18

19 Newtonian physics applies to an electron in a stationary state. The electron can transition between one stationary state and another provided it can absorb/emit a photon of energy equal to the energy difference between the states. ΔE = hf. The stationary states have quantized angular momentum in the amount n is an integer. 19

20 The allowed radii are where a 0 = 52.9 pm is the Bohr radius. 20

21 The energy levels are given by where E 1 = ev is the energy of the ground state, the lowest possible energy of the electron. When n > 1 the electron is in an excited state. The quantity n is an integer and is the principal quantum number. 21

22 Energy level diagram for hydrogen 22

23 The energy of a photon emitted (absorbed) by an electron during a transition is where is the Rydberg constant. When n f =2, the above result reduces to the Balmer formula. 23

24 The Bohr model correctly predicted the wavelengths of the spectral lines of hydrogen in the visible. There are several problems with the Bohr model. 24

25 Bohr s model, while successful at predicting the spectrum of hydrogen, fails at predicting the spectra of most other elements. Only hydrogenic atoms (atoms that only have one electron; Li 2+ for example) can have their spectra computed using the Bohr model. The allowed radii are The energy levels are where Z is the atomic number of the atom and E 1H = ev. 25