3_Postulates-1

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

This note was uploaded on 09/28/2011 for the course PHYS 334 taught by Professor Resch during the Spring '08 term at Waterloo.

Ask a homework question - tutors are online