3_Postulates-1 - III. The Postulates of Quantum Physics...

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Unformatted text preview: III. The Postulates of Quantum Physics Quoted from Bransden and Joachain 1 A. The Double Slit Experiment A.1 Postulate #1 • “To an ensemble of physical systems one can, in certain cases, associate a wave function or state function which contains all the information that can be known about the ensemble. This function is in general complex; it may be multiplied by an arbitrary complex number without altering its physical significance.” • From wave-particle duality observed in experiment 2 A. The Double Slit Experiment A.2 Postulate #2 • “The superposition principle.” ψ = c1 ψ1 + c2 ψ2 + · · · • From interference patterns observed in experiment. See board III.A.2 3 A. The Double Slit Experiment A.3 Postulate #3 • “With every dynamical variable is associated a linear operator.” • Dynamical variables are measurable (observables). • Position, momentum, angular momentum, energy, etc. • Measurement changes the wave function. • For example, the double slit experiment. • Wave-like properties in propagation until measurement is taken (hits screen), then point-like. Collapse of the wave function. system obser ved to o bey linear algebra • Had ψ, now have ψ’. • Phrase this mathematically: ψ’ = Aψ. • This is the only mathematical way to express the observed measurement properties. 4 measurements o bser ved to act as l inear operators B. Measurement and Expectation Values B.1 Postulate #4 • “The only result of a precise measurement of the dynamical variable A is one of the eigenvalues an of the linear operator A associated with A.” • The set of eigenvalues of A is called the spectrum of A. • The spectrum can be discrete or continuous. • The spectrum of an operator representing a dynamical variable must be real, since the results of measurements are real. Such operators are Hermitian operators. 5 See board III.B in a moment. B. Measurement and Expectation Values B.2 Postulate #5 • “If a series of measurements is made of a dynamical variable A on an ensemble of systems described by the wavefunction ψ, the expectation value, or average value of this dynamical variable is: ￿ drψ ∗ (r)Aψ (r) ” ￿ψ |A|ψ ￿ ￿A￿ = =￿ ￿ψ |ψ ￿ drψ ∗ (r)ψ (r) See board III.B 6 B. Measurement and Expectation Values B.3 Postulate #6 • “A wavefunction representing any dynamical state can be expressed as a linear combination of the eigenfunctions of A, where A is the operator associated with a dynamical variable.” • In other words, ∫dα|α><α| = I, where {|α>} are the set of eigenfunctions of A. could also be ∑ o ver discrete states instead o f integral Shown on board III.B previously. 7 C. The Schrödinger Equation C.1 Postulate #7 • “The time evolution of the wavefunction of a system is determined by the time dependent Schrödinger equation: ￿ ∂ψ (r, t) i￿ = Hψ (r, t) ∂t H=− 2 2m ∇2 + V (r, t) where H is the Hamiltonian, or total energy operator, of the system.” See board III.C.1 8 C. The Schrödinger Equation C.2 The Time-Independent Schrödinger ￿ Equation Griffiths Chapter 2 H=− ∇ 2m 2 2 ∂ψ (r, t) i￿ = Hψ (r, t) ∂t • When V(r) is independent of time t, the Schrödinger equation reduces to the time-independent Schrödinger equation. or Hϕn (r) = E ϕn (r) ￿2 2 − ∇ ϕn (r) + V (r)ϕn (r) = E ϕn (r) 2m 9 + V (r, t) C. The Schrödinger Equation C.2 The Time-Independent Schrödinger Equation ￿2 2 − ∇ ϕn (r) + V (r)ϕn (r) = E ϕn (r) 2m • The general solution to this equation is made up of a linear combination of stationary states and is, in one dimension: ψ (x, t) = ∞ ￿ cn ϕn (x)e−iEn t/￿ n=1 See board III.C.2 10 ...
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This note was uploaded on 09/28/2011 for the course PHYS 334 taught by Professor Resch during the Spring '08 term at Waterloo.

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