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Chapter 28 Quantum Physics (About quantization of light, energy and the early foundation of quantum mechanics)

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Blackbody Blackbody: A “perfect” absorber. For example, a hole in a cavity. It turns out a blackbody must also emit radiation, so a blackbody is not really “black”. The radiation from a blackbody depends only on the temperature of the cavity.

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Blackbody Radiation The radiation from a wide variety of sources can be approximated as blackbody radiation: Coal, sun, human body (infrared) As mentioned such radiation depends only on the temperature of the object, and is sometimes refer to as the thermal radiation.

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Material Independence It is observed that as an object gets hotter, the predominant wavelength of the radiation emitted by the object decreases (hence the frequency increases). Example: As temperature increases: Infrared Red Yellow White This is true regardless of the material that made up the blackbody. Objects in a furnace all glow red with the furnace walls regardless of their size, shape or materials.

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Temperature Dependence The peak of the wavelength distribution shifts to shorter wavelengths as the temperature increases:

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Conflict with classical physics Ultraviolet catastrophe

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Max Planck and Planck’s constant (1900) Proposed energy on the cavity wall: h becomes known as the Planck’s constant: All quantum calculations involves h. Sometimes it is more convenient to use:

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The idea behind Planck’s equation means it is now more difficult (or energy costly) to excite a mode of higher frequency. As a result less high frequency (low wavelength) radiations are produced, preventing ultraviolet catastrophe. Classical, the cost of a high frequency mode is the same as that of a low frequency mode.

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Quantization of Energy The energy emitted or absorbed by the energy transition of the cavity wall is therefore given by: The cavity cannot emit half of hf. Energy in the radiation only exists in packages (quanta) of hf.

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But why hf ? Even Planck himself could not give a more fundamental reason why the equation E=hf makes sense, except that it appeared to describe blackbody radiation perfectly. Planck continues to try to find a “better” explanation. Today physicists generally accept this equation as an observed fact of nature. Its introduction is regarded as the beginning of quantum mechanics.

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Photoelectric Effect When light shines on certain metals, electrons are sometimes released. The emitted electrons are sometimes referred to as photoelectrons. We can measure the energy of the photoelectrons using the setup below:

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The Setup When the external potential ξ is connected as shown, it helps the electrons to flow, generating a non-zero current when photoelectrons are produced.

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Reversing the potential Now the external potential ξ is reversed. It actually resists the flow of the electrons. When the potential is big enough, it can even stop the current completely. This is the stopping potential V s.

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The stopping potential and the number of photoelectrons Such an experiment measures the stopping potential V s, the external potential required to stop the flow of current completely. From V s one can deduce the maximum KE of the photoelectrons emitted by the metal, because by conservation of energy: By studying the KE and N e of the photoelectrons, further contradictions with classical physics were found. On the other hand, the current gives a measurement of the rate of electrons released. Roughly speaking, one can say:

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Photoelectric Effect, Results The maximum current increases as the intensity of the incident light increases When applied voltage is equal to or more negative than V s, the current is zero

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Photoelectric Effect Feature 1 Dependence of ejection of electrons on light frequency Classical Prediction Electrons should be ejected at any frequency as long as the light intensity is high enough Experimental Result No electrons are emitted if the incident light falls below some cutoff frequency, f c, regardless of intensity The cutoff frequency is characteristic of the material being illuminated No electrons are ejected below the cutoff frequency

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Photoelectric Effect Feature 2 Dependence of photoelectron kinetic energy on light frequency Classical Prediction There should be no relationship between the frequency of the light and the electric kinetic energy Experimental Result The maximum kinetic energy of the photoelectrons increases with increasing light frequency

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Photoelectric Effect Feature 3 Dependence of photoelectron kinetic energy on light intensity Classical Prediction Electrons should absorb energy continually from the electromagnetic waves As the light intensity incident on the metal is increased, the electrons should be ejected with more kinetic energy Experimental Result The maximum kinetic energy is independent of light intensity The current goes to zero at the same negative voltage for all intensity curves

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Photoelectric Effect Feature 4 Time interval between incidence of light and ejection of photoelectrons Classical Prediction For very weak light, a measurable time interval should pass between the instant the light is turned on and the time an electron is ejected from the metal This time interval is required for the electron to absorb the incident radiation before it acquires enough energy to escape from the metal Experimental Result Electrons are emitted almost instantaneously, even at very low light intensities Less than 10 -9 s

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Summary Action KENeNe Increase intensityNo effectsIncrease Increase frequencyIncrease Observation when f >f c : When f

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Frequency Dependence and Cutoff Frequency The lines show the linear relationship between KE max and f The slope of each line is h The absolute value of the y-intercept is the work function The x-intercept is the cutoff frequency This is the frequency below which no photoelectrons are emitted

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Some Work Function Values

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Einstein’s Explanation Energy in light comes in packages (photons). Each photon carries energy E=hf. You cannot get half a photon or 1/3 of a photon. The intensity of light is related to the number of photons present, but not to the frequency. Electrons are bind to the metal, so for an electron to escape, it needs to absorb a certain threshold amount of energy ϕ, called the work function. Each metal has a different value for ϕ. The stronger the binding to the metal, the larger is ϕ.

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The Picture The picture: An electron absorbs energy hf from the radiation, spends ϕ to escape from the metal, leaving only hf - ϕ as the KE : This explains why the slope of each line is h. Increase f Increase KE max Increase intensity Increase number of e -

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Photon Model Explanation of the Photoelectric Effect Dependence of photoelectron kinetic energy on light intensity KE max is independent of light intensity KE depends on the light frequency and the work function The intensity will change the number of photoelectrons being emitted, but not the energy of an individual electron Time interval between incidence of light and ejection of the photoelectron Each photon can have enough energy to eject an electron immediately

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Photon Model Explanation of the Photoelectric Effect, cont Dependence of ejection of electrons on light frequency There is a failure to observe photoelectric effect below a certain cutoff frequency, which indicates the photon must have more energy than the work function in order to eject an electron Without enough energy, an electron cannot be ejected, regardless of the light intensity

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Photon Model Explanation of the Photoelectric Effect, cont Dependence of photoelectron kinetic energy on light frequency Since KE max = hf – ϕ As the frequency increases, the kinetic energy will increase Once the energy of the work function is exceeded There is a linear relationship between the kinetic energy and the frequency

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The cutoff frequency and wavelength

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Rewriting hc

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Photoelectric Effect Features, Summary The experimental results contradict all four classical predictions Einstein extended Planck’s concept of quantization to electromagnetic waves All electromagnetic radiation can be considered a stream of quanta, now called photons A photon of incident light gives all its energy hf to a single electron in the metal

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Compton Scattering The loose electrons recoils from the momentum of the photon. The classical wave theory of light failed to explain the scattering of x-rays from electrons. The results could be explained by treating the photons as point-like particles having energy hƒ and momentum hf / c.

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Conservation of P and E

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Photons and Waves Revisited Some experiments are best explained by the photon model Some are best explained by the wave model We must accept both models and admit that the true nature of light is not describable in terms of any single classical model Light has a dual nature in that it exhibits both wave and particle characteristics The particle model and the wave model of light complement each other

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Louis de Broglie 1892 – 1987 Originally studied history Was awarded the Nobel Prize in 1929 for his prediction of the wave nature of electrons Pronounced “de broy”

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Wave Properties of Particles Louis de Broglie postulated that because photons have both wave and particle characteristics, perhaps all forms of matter have both properties The de Broglie wavelength of a particle is

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Frequency of a Particle In an analogy with photons, de Broglie postulated that particles would also have a frequency associated with them These equations present the dual nature of matter: particle nature, E and p wave nature, f and λ

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Particle / Wave Duality Summary The two equations can be rewritten as: Defining the angular frequency ω and the wave number k :

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Relativistic or Non-relativistic

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Use Non-relativistic Version in HW Make sure your KE is in J !

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Example Find the wavelength of a non-relativistic electron traveling with KE = 20eV.

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Example Determine the wavelength of an electron traveling with KE = 10MeV.

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Electron Diffraction, Set-Up

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Electron Diffraction, Experiment Parallel beams of mono-energetic electrons are incident on a double slit The slit widths are small compared to the electron wavelength An electron detector is positioned far from the slits at a distance much greater than the slit separation

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Electron Diffraction, cont If the detector collects electrons for a long enough time, a typical wave interference pattern is produced This is distinct evidence that electrons are interfering, a wave-like behavior The interference pattern becomes clearer as the number of electrons reaching the screen increases

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Electron Diffraction Explained An electron interacts with both slits simultaneously If an attempt is made to determine experimentally which slit the electron goes through, the act of measuring destroys the interference pattern It is impossible to determine which slit the electron goes through In effect, the electron goes through both slits The wave components of the electron are present at both slits at the same time

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Werner Heisenberg 1901 – 1976 Developed matrix mechanics Uncertainty Principle Noble Prize in 1932

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The Uncertainty Principle, Introduction In classical mechanics, it is possible, in principle, to make measurements with arbitrarily small uncertainty Quantum theory predicts that it is fundamentally impossible to make simultaneous measurements of a particle’s position and momentum with infinite accuracy The inescapable uncertainties do not arise from imperfections in practical measuring instruments The uncertainties arise from the quantum nature of matter

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Heisenberg’s Uncertainty Principle (1D) You cannot tell the position and the momentum of a particle simultaneously.

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Example

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Another Uncertainty Principle Another Uncertainty Principle can be expressed in terms of energy and time: A particle that have a short life-time Δt will have large uncertainty with its energy ΔE.

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Erwin Schrödinger 1887 – 1961 Best known as one of the creators of quantum mechanics His approach was shown to be equivalent to Heisenberg’s

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Schrödinger Equation The (time-independent) Schrödinger equation of a particle of mass m in a potential energy well V(x) is given by: The complex function ψ is called the wave function. It determines the probability of all experimental outcomes.

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Wave function and probability

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Potential Energy for a Particle in a Box The picture on the right represents the potential energy V(x) of a “box” (or a square well). A particle inside the box cannot go beyond 0 and L because of the infinitely high energy. Solution to the Schrödinger equation gives:

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Derivation of the wave function

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Continuity of wave function

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Graphical Representations for a Particle in a Box Energy is quantized.

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Laser Light Amplification by the Stimulated Emission of Radiation. Based on three processes: a)Absorption b)Spontaneous emission c)Stimulated emission

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Stimulated Emission A photon of frequency f passes and it triggers an excited electron to fall to the lower level. Same energy, same phase, polarization, direction.

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Summary

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Population Inversion When there are more excited atoms than atoms at ground state, it is said to be population inverted. This can only happen when the system is not in thermal equilibrium.

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He-Ne Laser Selection rules forbid He 2s level from decaying via radiation, so a population inversion is created. It can decay via collision with Ne, hence creating a population inversion in Ne between the 5s and 3p levels. The decay from 5s to 3p is the laser beam.

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