Quantum Mechanics Tutorial

Transcription

1 Quantum Mechanics Tutorial The Wave Nature of Matter Wave-particle duality and de Broglie s hypothesis. de Broglie matter waves The Davisson-Germer experiment Matter wave packets Heisenberg uncertainty principle Tutorial I: Concept of dependence of mass and velocity, mass-energy equivalence, energy-momentum relation (no deduction). Tutorial II: Blackbody radiation: Raleigh-Jeans law (derivation). Ultraviolet catastrophe. Wien s law, Planck s radiation law (calculation of average energy of oscillator). Wien s law and Stefan s law from Planck s radiation law, Rayleigh-Jenas & Wien s law-limiting case of Planck s law. Tutorial III: Compton Effect (calculation of Compton wavelength and Compton shift). Tutorial IV: Wave-particle duality and de Broglie s hypothesis, de Broglie matter waves, The Davisson-Germer experiment, Matter wave packets, Heisenberg uncertainty principle Wave Particle Duality: de Broglie s Hypothesis de Broglie s hypothesis can be stated as Every matter has dual characteristics particle like and wave like. The wavelength associated with the matter wave is given by λ = h/p, where h is Planck s constant and p is the linear momentum of the dynamical particle. Prepared by Dr. Rajesh Das, Applied Sciences-Haldia Institute of Technology 1

2 For a charged particle of rest mass m and charge q moving in a potential difference V, the kinetic energy (E k ) is given by E k =ev and the debroglie wavelength is given by h de Broglie 2mqV The de Broglie wavelength for a charged particle of rest mass m 0 and charge q moving in a potential difference V is given by h de Broglie qv 2m0qV (1 ) 2 2m c 0 Prepared by Dr. Rajesh Das, Applied Sciences-Haldia Institute of Technology 2

3 Experimental Verification of de Broglie s Hypothesis: Davisson and Germer Experiment Experimental Set-up In 1927 C. J. Davisson and L. H. Germer of the Bell Telephone Laboratory, USA performed an experiment on the diffraction of electrons from the surface of a solid crystal. The apparatus designed and built by them consisted of a vacuum chamber in which electrons were produced from a heated tungsten filament. They were collimated into a beam by passing through a narrow hole in a metal plate. The beam was accelerated by applying a potential difference between the filament and the collimator plate. It was directed to strike normally on the (111) face of a nickel crystal. Electrons were scattered in all directions from the nickel crystal which was held fixed throughout the experiment. Scattered electrons were collected in a box (called as Faraday box) mounted on an arc which could be rotated through an angle about an axis passing through the point of incidence of electrons on the crystal. The box was adjusted at different angles for collecting electrons and the electric current was recorded for each setting by means of a sensitive galvanometer as detector. The electric current was directly proportional to the number of scattered electrons per second collected in the box. Fig. Experimental Setup Prepared by Dr. Rajesh Das, Applied Sciences-Haldia Institute of Technology 3

4 Fig. Schematic of Experimental Setup Observations The data on the intensity of scattered electrons was obtained as a function of the scattering angle θ. Different sets of observations were taken corresponding to fixed potential differences (e.g. from 40V to 68V). For each potential difference a separate graph of electric current versus scattering angle was plotted in polar coordinates (r,θ). The curve produced from the data set for 40V was sooth. For the set using 44V, a slight hump appeared. As the potential difference was increased further, it increased and the spur became maximum in the set for 54V at scattering angle ϕ=50 0. For voltages greater than 54V, the peak decreased and almost vanished at 68 V. The peak indicates that (matter) waves (associated with electrons) scattered by the periodic arrangement of atoms in the crystal interfere constructively. It is similar to Bragg reflections of X-rays from atoms in the crystal and confirms the wave-like behavior of electrons. If the intensity of the electron beam is decreased so that only one electron at a time is allowed to go through the hole, the pattern of electron scattering is same as that of the electron beam. It indicates that different portions of the wave associated with a single electron are scattered from different parts of the crystal and interfere constructively to produce the peak. Prepared by Dr. Rajesh Das, Applied Sciences-Haldia Institute of Technology 4

5 Fig. Graph of Electric Intensity Vector vs Angle of Scattering Calculation and Explanation The theoretical value of the wavelength λ of matter waves of incident electrons having mass m and momentum p is calculated using the de Broglie s relation as follows λ=h/p=h/mv (h=planck s constant). The velocity v of electrons due to the potential difference V can be obtained from kinetic energy (non relativistic kinetic energy of the electron is: p 2 /2m=eV 1/2mv 2 =ev which gives λ=h/(2mev) 1/2 and on substitution of the value of (m,e and h) we have λ=12.27/v 1/2 λ=0.167nm for V=54volts The experimental value of λ is calculated as follows If d is the effective interplanar spacing in the crystal, then its value can be found by X-ray scattering from the same crystal and comes to be equal to 0.091nm. ϕ is the angle of scattering of electrons =50 0. Corresponding to the scattering angle of 50 0, the angle θ in the Bragg s law is given by = /2=65 0 Calculation and Explanation The wavelength of electrons is calculated using Bragg s Law: nλ=2dsinθ λ=0.165nm for n=1 The experimental value agrees very well with the theoretical value of the wavelength of electrons. Thus the experiment gives confirmation to the de Broglie s hypothesis. The concept of matter waves is not an imaginary idea but it can be shown to have real physical existence in the laboratory. Prepared by Dr. Rajesh Das, Applied Sciences-Haldia Institute of Technology 5

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11 According to Heisenberg s Uncertainty Principle, it is impossible to measure any two canonically conjugate observables preciously and simultaneously. Prepared by Dr. Rajesh Das, Applied Sciences-Haldia Institute of Technology 11