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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 waveparticle 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.
• Wavelike properties in propagation until measurement is taken
(hits screen), then pointlike. 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 Aψ
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, ∫dαα><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 TimeIndependent 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
timeindependent 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 TimeIndependent 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.
 Spring '08
 RESCH
 Quantum Physics

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