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Quantum Mechanics: Historical Perspective Spring 2012 Mark F. Horstemeyer Amitava Moitra ICME 4990/6990

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Reference Texts Principles of Quantum Mechanics by R Shanker Modern Quantum Mechanics by J.J. Sakurai Other Books: D. J. Griffiths ， Introduction to Quantum Mechanics, Sara M. McMurry. Quantum mechanics. Paul Roman, Advanced Quantum Theory-An outline of the fundamental ideas, J. J. Sakurai ， Modern quantum mechanics. R. Shankar. Principles of quantum mechanics. Feynman, The Feynman Lectures on Physics, Vol. 3; Quantum Mechanics Cohen-Tannoudji ， Quantum Mechanics, Vol. 1&2 ； Leonard I. Schiff. Quantum mechanics. L. D. Landau and E. M. Lifshitz, Quantum Mechanics R. Robinett’s Quantum mechanics: classical results, modern systems and visualized examples J. Townsend, A modern approach to quantum mechanics, D. Park, Introduction to quantum theory, P.A.M.Dirac, The Principles of Quantum Mechanics Leslie E. Ballentine ， Quantum Mechanics: A Modern Development ter Haar, Problems in Quantum Mechanics

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The Beginning What is a Physical Law? Based upon Bacon’s (1260) Scientific Method: observation, hypothesis, experiments A statement of nature that must be experimentally validated Experiments done in different frames must yield same results Describes the physical world Why should truth be a function of time? Laws formulated with observations Observations depends on accuracy of the instruments Advancement of technology leads to better instrumentation Laws that remain true gain in stature, those which don’t may lose their impact (example of Newton’s Laws of Gravity and Relativity) Domain of physical law and the general abstraction

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Matter & Radiation Classical Mechanics Formulated by Galileo, Newton, Euler, Lagrange, Hamilton Remained unaltered for three centuries Some History Beginning of 1900’s two entities--- – Matter & Radiation Matter described by Newtons laws (mass and particles) 1687 Radiation wave equations by Maxwell’s equation (Field Theory) 1864 (light consists of transverse undulations which cause the electric and magnetic phenomenon) It was thought that we now understood all… First breakthrough came with radiations emitted by a black body

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The Black Body Radiation What was the observation m T = Constant Total Power radiated T 4 Raleigh Jeans Law Why Black Body?

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Black Body… continued At a more fundamental level why should there be three laws that apparently have no relation with each other and yet describe one physical phenomenon? Why is it a physical phenomenon? Planck (1899)solved the mystery by enunciating that it emitted radiation in quantas of photons with energy h

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Enter Einstein (1905) Nobel Prize Idea! If radiation is emitted in quantas, they should also be absorbed in quantas He could explain photo electric effect using this… Light is absorbed in quanta of h If it is emitted in quantas of h Must consist of quantas

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What was the importance of Compton effect? Collision between two particles – Energy-momentum must both be conserved simultaneously Light consist of particles called photons What about phenomenon of Interference & diffraction? Logical tight rope of Feyman Light behaves sometimes as particles (Newton) sometimes as waves (Maxwell) Enter Compton (1923)

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Enter de Broglie (1923) Radiation behaved sometimes as particles sometimes as waves What about matter? De Broglie’s hypothesis Several questions cropped up! What is it? --Particle or Waves What about earlier results? What is a good theory? Need not tell you whether an electron is a wave or a particle if you do an experiment it should tell you whether it will behave as a wave or particle. Second Question brings us to the domain of the theory

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Domain of a theory Domain D n of the phenomenon described by the new theory Subdomain D 0 where the old theory is reliable. Within the sub domain D 0 either theory may be used to make quantitative predictions It may be easier and faster to apply the old theory New theory brings in not only numerical changes but also radical conceptual changes These will have bearing on all of D n velocity Inverse size RelativityClassical Mechanics Quantum Mechanics Quantum Field Theory

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Experimental Observations Thought Experiments Stern-Gerlach Experiments (1921); measurements of atomic magnetic moments Analogy with mathematics of light Feynman’s double slit thought experiment

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Thought Experiments We are formulating a new theory! Why are we formulating a new theory? Already motivate you why we need a new theory? How? Radiation sometimes behaves as Particles Waves Same is true for Matter

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Thought Experiments We must walk on a logical tight rope What is Feyman’s logical tightrope? We have given up asking whether the electron is a particle or a wave What we demand from our theory is that given an experiment we must be able to tell whether it will behave as a particle or a wave. We need to develop a language for this new theory. We need to develop the Mathematics which the language of TRUTH which we all seek What Kind of Language we seek is the motivation for next few lectures.

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Stern-Gerlach Experiment (1921) Oven containing Ag atoms Collimator Slits Inhomogeneous Magnetic Field detector Classically one Would expect this Nature behaves this way

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Stern Gerlach Experiment unplugged Silver atom has 47 electrons where 46 electrons form a symmetrical electron cloud with no net angular momentum Neglect nuclear spin Atom has angular momentum –solely due to the intrinsic spin of the 47 th electron Magnetic moment of the atom is proportional to electron spin If z 0) atom experiences an upward force & vice versa Beam will split according to the value of z

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Stern-Gerlach Experiment (contd) One can say it is an apparatus which measures the z component of S z If atoms randomly oriented No preferred direction for the orientation of Classically spinning object z will take all possible values between & - Experimentally we observe two distinct blobs Original silver beam into 2 distinct component Experiment was designed to test quantisation of space

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Two possible values of the Z component of S observed S Z UP & S Z down Refer to them as S Z + & S Z - Multiples of some fundamental constants, turns out to be Spin is quantised Nothing is sacred about the z direction, if our apparatus was in x direction we would have observed S x + & S x - instead What have we learnt from the experiment SG Z This box is the Stern Gerlach Apparatus with magnetic Field in the z direction

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Thought Experiments start Z+Z+ Z-Z- Source SG Z Z+Z+ Z-Z- Source SG Z Blocked Z+Z+ Source SG Z Blocked SG Z SG Z

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Thought Experiment continues No matter how many SG in z direction we put, there is only one beam coming out Silver atoms were oriented in all possible directions The Stern-Gerlach Apparatus which is a measuring device puts those atoms which were in all possible states in either one of the two states specific to the Apparatus Once the SG App. put it into one of the states repeated measurements did not disturb the system

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Conclusions from our experiment Measurements disturb a quantum system in an essential way The boxes are nothing but measurements Measurements put the QM System in one of the special states Any further measurement of the same variable does not change the state of the system Measurement of another variable may disturb the system and put it in one of its special states.

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Complete Departure from Classical Physics Measurement of S x destroys the information about S z We can never measure S x & S z together – Incompatible measurements How do you measure angular momentum of a spinning top, L = I Measure x, y, z No difficulty in specifying L x L y L z

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Consider a monochromatic light wave propagating in Z direction & it is polarised in x direction Similarly linearly polarised light in y direction is represented by A filter which polarises light in the x direction is called an X filter and one which polarises light in y direction is called a y filter An X filter becomes a Y filter when rotated by 90 Analogy

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An Experiment with Light The selection of x` filter destroyed the information about the previous state of polarisation of light Quite analogous to situation earlier Carry the analogy further – S z x & y polarised light – S x x` & y` polarised light SourceX FilterY Filter SourceX’ FilterY FilterX’ Filter No LIGHT No LIGHT

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PREFACE Newton’s mechanics Maxwell’s electrodynamics Einstein’s relativity Quantum mechanics Not by one individual

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Characteristic of Quantum mechanics No general consensus Can “do”, but can’t tell what we are doing. Niels Bohr:: “If you are not confused by quantum physics then you haven’t really understood it”. Richard Feynman: ”I think I can safely say that nobody understands quantum mechanics”.

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rudiments of linear algebra Knowledge requirements for students complex numbers calculus and partial derivatives Fourier analysis Elementary classical mechanics Maxwell’s electrodynamics mathematical physics Dirac delta function Math Physics

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(1) Classical mechanics: Initial conditions: The Schrödinger Equation (1926) Classical mechanicsQuantum mechanics

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(2) Quantum mechanics Wave function Schrödinger Equation

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The Statistical Interpretation What is the Wave function? Born’s (1926) statistical interpretation: gives the probability of finding the particle at point x, at time t － or more precisely, Probability of finding the particle between a and b, at time t.

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The shaded area represents the probability of finding the particle between a and b, at time t. The particle would be relatively likely to be found near A, and unlikely to be found near B.

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The statistical interpretation introduces a kind of indeterminacy into quantum mechanics. Quantum mechanics offers statistical information about the possible results. Question? Where was the particle just before I made the measurement? Do measure the position of the particle Is it a fact of nature or a defect of theory??

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Which is the right answer? In 1964, John Bell Showing that it will make an observable difference The experiments show that the orthodox answer is right. Immediate repetition of measurement Zeno effect = remains on its initial state Continuous evolution == no measurement discontinuous collapse == measurement

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Probability Discrete Variables AgeNumber 141 151 163 222 242 255 Number(age) The total number of people:

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10111412131516171819202123 22 272625 24 (1) The probability that the person’s age would be j ? (2) What is the most probable age ?

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(3) What is the median age ? (4) What is the average (or mean) age ? 10111412131516171819202123 22 272625 24

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the average value of age j is (5) What is the average of the squares of the ages ? Discussions:

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12534678910125346789 The statistical quantities: Average value the most probable value Median value How to discriminate them ? the number of elements

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Deviation from the average

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Continuous Variables age16 years, 4 hours, 27 minutes,……Continuous Value interval : [ 16, 17 ] The probability in a sufficiently short interval is proportional to the length of the interval. Infinitesimal intervals dx Probability that an individual (chosen at random) lies between x and x+dx The proportionality factor,, is called probability density.

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The probability that x lies between a and b is given by the integral

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Normalization is the probability density for finding the particle at point x, at time t. A complex constant We can choose complex constant A to meet above equation Normalization

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Proof: The Schrödinger Equation automatically preserves the normalization of the wave function

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Normalizable

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Momentum For a particle in state Ψ, the expectation value of x is The expectation value is the average of repeated measurements on an ensemble of identically prepared systems.

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Using integration-by-parts method or

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momentum operator position

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All classical dynamical variables can be expressed in terms of position and momentum. For example: Kinetic energy: Angular momentum: Classical dynamical variablesQuantum operators

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The Uncertainty Principle (Heisenberg 1926) wavelengthposition The wavelength of Ψ is related to the momentum of the particle by the de- Broglie formula: The Uncertainty Principle wavelengthposition nonlocal localnonlocal local

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