Introduction to Quantum Mechanics I
Introduction to Quantum Mechanics I PHY 567
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This 4 page Class Notes was uploaded by Clement Bernier on Wednesday October 21, 2015. The Class Notes belongs to PHY 567 at Syracuse University taught by Staff in Fall. Since its upload, it has received 21 views. For similar materials see /class/225636/phy-567-syracuse-university in Physics 2 at Syracuse University.
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Date Created: 10/21/15
Lecture 2 History Lest go back and discuss some of the speci c problems which forced this radical departure from classical physics BlackBody radiation 1901 Consider a cavity in an oven at uniform temperature Once everything has come into equilibrium we can sample the radiation emerging from the cavity We nd the distribution of energy WT f with frequency f initially rises like f2 but then turns over and falls to zero with large The initial rise is to understand the number of modes of the radiation eld between f and f Af is simply 47rf2Af To understand this remember that a em wave is a vector quantity it has both a magnitude and direction Thus a given mode is described by a wavelength and a direction in space When we calculate the number of modes of a given or qu39 y fre quency we nd a result analogous to the surface area of a sphere but now a sphere in frequency space In classical thermodynamics each such mode carries the same energy basically kgT so the net energy at frequency f rises like f 2 Notice not only does this disagree with the measured distribution it gives the total radiated energy in nite l However Planck able to t the distribution with the function 87rhf3 1 03 EhfkBT 1 1 n u By drawing on ideas in classical statistical physics and recasting the above expression an in nite sum over different energy states he led to a very unusual hypothesis the radiation energy of a single mode of the em eld could only come in units of hf He had no explanation for this Photoelectric effect 1905 It noticed that when UV light incident on a metal plate electrons are ejected When the energy of the electrons is measured a function of light frequency it found that below a certain threshold frequency there were no electrons and above this the energy of the electrons rose linearly with frequency The light intensity had no effect on the maximum electron energy it only affected the number emitted This completely at variance with classical ideas which would have yielded electrons whose energy intensity dependent essentially the larger amplitude waves would case the driven electrons to wiggle more vigorously which would lead to an increase in their kinetic energy Einstein explained the effect by extending Planck s idea to suppose that light consists of photons whose energy varies linearly with frequency A given electron is kicked out from the metal when it collides with a single photon of sufficient frequency A photon imagined to be a packet of wavepacket with a certain particular energy given by the Planck formula Rutherford and Bohr 191113 In 1911 Rutherford performed an historic experiment in which he fired a beam of alpha particles type of radiation at a gold foil He found that most of the alpha particles suffered only small deflections while just a few were scat tered through very large angles He interpreted the results of this scattering experiment indicating that the gold atoms consisted of a smalldense core of positive charge surrounded by a much larger and more diffuse cloud of negative charge the electrons Unfortunately this planetary model of the atom in conflict with classical physics if the electrons were in a circu lar orbit they would be accelerating and because of Maxwell s theory they should radiate light energy But this loss of energy would lead to a spiraling of the electron into the nucleus atoms would not be stable Furthermore the spectrum of light emitted by such an atom would contain light of all fre quencies which not observed In fact the light emitted by heated atoms shows a discrete structure characteristic of that particular atom a socalled line emission spectrum Bohr tried to fix this in an ad hoc fashion Specifically he assumed that for hydrogen only certain states were stable those in which the angular momentum were a multiple of In those states the electron does not radiate Furthermore when an electron moves from one such state to another lower state it emits the difference in energy a photon whose frequency is related to its energy via PlanckEinstein s relation Using classical physics it is then to see that the allowed radii are 47mm Tn 77182 Similarly the allowed possible energies can be found 1mc4 1 E if quot 2 47mm2 712 This explanation accounted well for the experimentally observed line spec trum of hydrogen but what justi cation could there be for the quantization of momenta that Bohr had assumed or the resulting stability at those mo menta de Broglie 1925 The situation lay fallow for some years before de Broglie started a new line of reasoning If light could sometimes behave a particle photon could not matter behave sometimes a wave The energy relation of Planck could be written E E2hw and phk 1 Perhaps a similar relation governed material particles Notice that this iden ti cation of wavelength with momenta allowed a possible interpretation of the Bohr quantization condition via 27Wquot 71A the condition for standing waves In this case electrons should be able to exhibit phenomena character istic of waves such as interference and diffraction Such behavior looked for in a famous experiment of Davisson and Germer 1927 in which a beam of electrons scattered off a crystal surface in which the interatom separation comparable to the de Broglie wavelength of the electrons Lo and behold an interference pattern observed Peaks in intensity were observed whenever the path difference between wave reflected from the rst and second atomic layers matched an integral number of wavelengths 20 cos 9 71A We may imagine generalizing this setup to the classic Young s double slit experiment used for light Electrons are shot at a screen possessing two closely spaced slits A screen is placed a large distance beyond the slits and is used to record the arrival of electrons which have passed through one of other of the slits In practice we detect electrons with a detector which flashes when an electron hits it If we were to do this experiment and record the intensity of electrons recorded by the detector we would nd a surprising thing at certain places on the screen we would never see electrons while at others we 3 would see always a maximum electron intensity Furthermore these maxima and minima occur at regular intervals along the screen we see interference fringes just as we would with light So the electrons must be associated with a wave de Broglie had suspected Clearly the particle character of the electrons emerges a statistical thing any individual electron can land anywhere the wave just gives the probability of nding it at one place or another It rst thought that the associated wave must somehow describe the aggregate behavior of a bunch of mutually interacting electrons a given electron will through one or other of the slits for certain But consider the following variation we can turn down the intensity of the beam until just one electron passes through at a time If the electron has to go through just one slit then we would predict that the interference pattern would disappear but it does not we still see an interference pattern In some sense the electron passes through both slits l The associated wave describes the behavior of just a single electron Equivalently we can that the electron in passing through the apparatus behaves a wave but when we come to record it it behaves a particle This is the basis for wave particle duality In effect the electron is represented by an abstract state which depending on what kind of measurement we choose to make make look alternately particle or wavelike in character By 1926 the stage set quantum matter should be describable in terms of a wave theory where the momentum of a free particle is just p hA The interference experiment hints that the intensity of the wave gives the probability for nding the electron But what is the equation that describes the wave evolution Enter Schroedinger
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