Download Postulate 1

Heisenberg Uncertainty Principle
• Heisenberg (1926) thought about measuring
simultaneously the position and momentum
(velocity) of an electron. Realization →
measurement of both with precision is
impossible and, in fact, the measurement
process perturbs the system (electron, etc).
Heisenberg, Light and Electrons
• Measuring the position of a very small particle
such as an e- requires the use of short
wavelength light (high frequency, high
momentum light (Compton effect)). High
frequency light will alter the electron’s
momentum. Use low frequency (long λ) light
and the position of the electron is not well
determined.
Heisenberg Principle – Equations:
• The Heisenberg uncertainty principle has
several forms. We denote the uncertainty in
momentum, for example, by Δp. In one
dimension we have
• ΔpxΔx ≈ h
• If q denotes a three dimensional position
coordinate then
• ΔpΔq ≥ h/4π
Heisenberg and Mass
• Since momentum p = mu (u = velocity) the
Heisenberg expression can be written as
• ΔuΔq ≥ h/4πm
• This equation tells us that the product of the
uncertainties in velocity and position decrease
as the mass of the particle increases. Ignoring
history, Heisenberg tell us that as the mass of
a particle/system increases we move towards
a deterministic description (Newton!).
Heisenberg and Spectroscopy
• The Heisenberg expression can be recast, in
terms of energy and time, as
• ΔEΔt ≥ h/4π
• Relates in “real life” to widths of spectral lines.
In electronic emission spectra Δt can be small
(short lifetime in the excited state). This
means that ΔE will be “large”. The transition
energy covers a range of values – the
observed spectral “line” is broadened.
de Broglie, Electrons and Waves
• Einstein’s theory of relativity tells us that the
energy of a particle having a momentum p
and rest mass m is given by
• E2 = p2c2 + m2c4
• If our interest lies in photons (zero rest mass)
then
• Ephoton = pc = pλν
De Broglie – continued:
• Using the familiar expression from Planck, E =
hν, we get for photons ( i.e., light or
electromagnetic radiation).
• λ = h/p
• De Broglie suggested that this last expression
could apply to beams of particles with a finite
rest mass, m, and a velocity u (momentum p =
mu).
De Broglie and Diffraction:
• For finite rest mass particles de Broglie gives
us
• λ = h/p = h/mu
• This expression does have meaning in our
physical world. Wave properties of subatomic
particles and light molecules have been
demonstrated though , for example, a variety
of diffraction experiments.
de Broglie – examples:
• 1. Electrons are easily accelerated by an
electric field. Find the de Broglie wavelength
for an electron accelerated by a voltage of
12.5 kilovolts.
• 2. Would the de Broglie picture be more
useful in describing the behaviour of a
neutron or an NFL lineman?
De Broglie and Molecular Structure:
• Aside: Highly accelerated and collimated
electrons are diffracted when they move
through a gas. Knowing the energy or de
Broglie wavelength of the collimated electrons
one can, for simple gas phase molecules, use
the diffraction pattern obtained to determine
a three dimensional structure for the
molecules. (Why are protons and neutrons
less useful for such experiments?).
de Broglie, Waves and Electrons:
• The work of de Broglie and others showed
that light and subatomic particles had
something in common – wave properties. This
led Schrodinger, in particular, to wonder
whether equations used to describe light
waves could be modified to describe the
behaviour of electrons. This led to quantum
mechanics and a probabilistic description of
the behaviour of electrons.
Operators:
• A brief class discussion of operators,
eigenvalues, eigenfunctions and eigenvalue
equations is needed before moving to, for
example, the postulates of quantum
mechanics. Some of this material may be
familiar from mathematics courses. The
“eigenfunctions” that are useful in describing
particles with wave properties are of familiar
form (and, in part, predictable?).
Postulates of Quantum Mechanics:
• The development of quantum mechanics
depended on equations that are not, in the
normal sense, derivable. This development
was based on a small number of postulates.
The reasonableness of these postulates will
become clear through application of the
postulates.
Quantum Mechanics – Postulates:
• Postulate 1: A quantum mechanical system or
particle can be completely described using a
wave function, ψ. (Wave functions were
introduced briefly in Chemistry 1050).
• In different examples the problems of interest
will have varying dimensionality.
Correspondingly, one sees wave functions
described in terms of one or more
coordinates/variables.
Postulate 1 – Dimensionality:
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Possible “forms” of wave functions:
One dimensional problems: ψ(x), ψ(r), ψ(θ)
Two dimensional problems: ψ(x,y) etc.
Three dimensional problems: ψ(x,y,z), ψ(r,θ,φ)
If the time dependant evolution of the system
needs to be treated (or, appears interesting!)
the wave functions will have the form ψ(x,y,z,
t) or ψ(r,θ,φ,t).
Postulate 1 – Wave function
Properties:
• The wave function ψ(x,y,z) (for example) must
be continuous, single valued (not a new
requirement!) and square integrable. Both
real and imaginary wave functions are
encountered. Thus, if ψ(x) is imaginary (for
example, eimx , where m is an integer) then we
will need the complex conjugate wave
function ψ*(x) = e-imx.