The Variational Principle

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The Variational Principle. Scott Riggs. Expectation Values. Basic Statistics P ab = ∫ p(x) dx (limits a to b) The probability that x lies between a & b Where p(x) dx is the probability that individual (randomly chosen) lies between x and x+dx - PowerPoint PPT Presentation

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  • The Variational PrincipleScott Riggs

  • Expectation ValuesBasic StatisticsPab = p(x) dx (limits a to b)The probability that x lies between a & bWhere p(x) dx is the probability that individual (randomly chosen) lies between x and x+dx

    = x |(x,t)|^2 dx : is the average of measurements performed on particles all in the state .

  • Dirac Notation = f(x)*[g(x)] dx ketBracket notationket is just a vectorbra is a linear combination of vectors. = f(x)*(d/dx)[g(x)] dx

  • Variational PrincipleNeed to calculate the ground state energy for a system described by H, but unable to solve the time-independent Schrdinger equation.H = E also written (h2/2m)(d2 /dx2)+V= EEg Eg

  • ExampleTriangular trial for an infinite square well.(x) = Ax (if 0
  • Now H = -(h^2/2m) () + 0Use the variation principle = -h2A/2b [] (x) dx = 12h2/2ma2

    We know from exact E = (Pih)2/2ma2Theorem works because 12>Pi2

  • ConclusionsExpectation ValueBasic Dirac NotationVariation PrincipleFind an upper bound on the Eg of a system described by a Hamiltonian.