12.1 Foundations of Quantum Theory

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Norman Roger Hudson
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1 1.1 Foundations of Quantum Theory Physics Tool box A blacbody of a given temperature emits electromagnetic radiation over a continuous spectrum of frequencies, with a definite intensity maximum at one particular frequency. As the temperature increases, the intensity maximum shifts to progressively higher frequencies. Plan proposed that molecules or atoms of a radiating blacbody are constrained to vibrate at discrete energy levels, which he called quanta. The energy of a single quantum is directly proportional to the frequency of the emitted radiation, according to the relationship =hf, where h is Planc s constant. Photoelectrons are ejected from a photoelectric surface when the incident light is above a certain frequency f0, called the threshold frequency. The intensity (brightness) of the incoming light has no effect on the threshold frequency. The threshold frequency is different for different surfaces. The cut-off potential is the potential difference at which even the most energetic photoelectrons are prevented from reaching the anode. For the same surface the cut-off potential is different for each frequency, and the higher the frequency of light, the higher the cut-off potential. The energy of light is transmitted in bundles of energy called photons, whose energy has a discrete, fixed amount, determined by Planc s equation, =hf. When a photon hits a photoelectric surface, a surface electron absorbs its energy. Some of the absorbed energy releases the electron, and the remainder becomes its inetic energy of the liberated electron, according to the photoelectric equations hf W. In the Compton effect, high-energy photons strie a surface, ejecting electrons with inetic energy and lower-energy photons. Photons have momentum whose magnitude is given by p h. Interactions between photons and matter can be classified into reflection, the photoelectric effect, the Compton effect, changes in electron energy levels within atoms and pair production. Blacbody Radiation If a piece of metal is placed into a flame, the metal begins to glow: fist a dull red, the brighter orange-red, the yellow, and finally white. If the metal gets even hotter, it can emit radiation in the ultraviolet spectrum. Thus as the temperature increases, the spectrum of the emitted electromagnetic radiation shifts to higher frequencies. It has been determined that the relative brightness of the different colours radiated by an incandescent solid depends mainly on the temperature of the material. The intensity of radiation emitted by a hot object at different temperatures. The curve below represents the radiation from an object that approximates an ideal emitter or absorber of radiation. Such an object would absorb all wavelengths of light striing it, reflecting none. It would, therefore appear blac under reflected radiation and hence is called a blacbody.

2 A detailed analysis of radiation absorption and emission shows that an object that absorbs all incoming radiation, of whatever wavelength, is liewise the most efficient possible emitter of radiation. The radiation emitted by a blacbody is called blacbody radiation. As scientists (1890 s) where trying to explain the dependence of blacbody radiation on temperate, they had a paradox. When the temperature increases, the frequency of the oscillations of electric charges in the atoms also increases, thus the corresponding frequency of the radiated light should increase as well. The there should be a predicted increase. But this is not the case, the radiation follows the solid line. The region of the graph where theory and experimental data disagree is in the ultraviolet portion of the spectrum, thus this paradox was called the ultraviolet catastrophe. Planc s Quantum Hypothesis In the 1900 s, a German physicist Max Planc, proposed a new radical theory to explain the data. His hypothesis was that the vibrating atoms in a heated material vibrated with

3 only specific quantities of energy. That is when energy is emitted, it is not emitted in a continuous form but in bundles, or pacets, which Planc called quanta. He proposed that the energy of a single quantum is directly proportional to the frequency of the radiation: hf Where is the energy in oules, f is the frequency in hertz, and h is a constant in joules seconds. By fitting his equation to the data, Planc approximated the value of h. Today we have 34 determined that h s (Planc s Constant) Plan further hypothesized that the emitted energy must be an integral multiple of the minimum energy, that is energy can only be hf, hf, 3hf, Thus nhf, n 1,,3,... This theory was revolutionary for two reasons: 1. it challenged the classical wave theory of light by proposing that electromagnetic waves do not transmit energy in a continuous manner but, instead transmit energy in small pacets.. It challenged the classical physics of Newton, since it proposed that a physical object is not free to vibrate with any random energy; the energy is restricted to certain discrete values. This was so radical, that it was not accepted until 1905 when instein demonstrated the process through the experiment nown as the photoelectric effect. xample Calculate the energy in joules and electron volts of a quantum of red light with a wavelength of 65 nm. Solution: First we need the frequency of the red light c f c f 8 m s m Hz

Now hf 6.63 10 34 s 4.8 10 14 Hz 3.1810 19 Now for electron Volts 19 1.6 10 1 ev 3.1810 1.6010 19 19 ev 1.99eV Thus the energy is 19 3.1810 or 1.

4 Now hf s Hz Now for electron Volts ev ev 1.99eV Thus the energy is or 1.99eV instein and the Photoelectric ffect Heinrich Hertz shone ultraviolet light on a zinc plate attached to a gold-leaf electroscope. Incident ultraviolet somehow caused the zinc plate to release electrons. Hertz s phenomenon was called the photoelectric effect, and the emitted electrons were called photoelectrons. We can duplicate the effect with a variety of apparatuses, and have found some significant findings. Photoelectrons are emitted from the photoelectric surface when the incident light is above a certain frequency 0 f, called the threshold frequency. Above this frequency, the more intense the light, the greater the current of photoelectrons. The intensity (brightness) of the light has no effect on the threshold frequency. In fact, it if it below the threshold frequency, not a single photoelectron is emitted.

5 The threshold frequency, at which photoelectric emission first occurs, is different for different surfaces. If you apply a retarding potential to the circuit, thus maing the anode negative, this has the effect of reducing the current flow, demonstrating that the photoelectrons are emitted with different inetic energies. When the potential is increase to the level when the current reaches zero, we say we have reached V the cut-off potential, 0 The cut-off potential is related to the maximum inetic energy with which photoelectrons are emitted. The inetic energy is related by: ev 0 The release of electrons is immediate. It appears that the electron absorbs the light energy immediately: no time is required to accumulate sufficient energy to liberate the electrons. When a photon hits a photoelectric surface, a surface electron absorbs its energy. Some of the energy is needed to release the electron, while the remainder becomes inetic energy of the ejected photoelectron. instein described this mathematically as:

6 W photon Where photon is the energy of the incident photon. W is the energy with which the electron is bound to the photoelectric surface, and ejected photoelectron. By rearranging we obtain the equation: W photon hf W is the inetic energy of the The value W is called the wor function of the metal xample Red light with a wavelength of function of 1.70eV Calculate m is directed at a metallic surface with a wor a) the maximum inetic energy, in joules, of the emitted electrons b) their maximum speed c) the cut-off potential necessary to stop these electrons Solution: a) hf W b) hc W 34 8 m s s 1.60eV mv v m g m s 19 7 m ev

7 c) V 0 ev 0 e V 0 19 C Momentum of a photon: The Compton ffect In 193 physicist A.H. Compton directed a beam of x-rays at a thin metal foil. He not only verified the photoelectric effect, but he also detected an emission of x-ray photons with a lower energy (thus frequency) than the x-rays he initially shot at the film. Compton discovered that x-ray photons collide elastically with the electrons in the metal xray xray electron 1 hf hf mv The problem was, is the conservation of momentum true, or simply could momentum be associated with a bundle of energy with no mass. Compton solved the problem with instein s mc p mv v c ; v c c Now a photon s energy is given by hf, and the wave equation c f Therefore hf hf h p c f This is the magnitude of the momentum of a photon.

8 xample What is the magnitude of the momentum of a photon with m? Solution: h p s m g m s

Radiation - Electromagnetic Waves (EMR): wave consisting of oscillating electric and magnetic fields that move at the speed of light through space.

Radiation - Electromagnetic Waves (EMR): wave consisting of oscillating electric and magnetic fields that move at the speed of light through space. Photon: a quantum of light or electromagnetic wave. Quantum: