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 their application.
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.
Quantum Mechanics and Probability:
• No fisherman would expect that the
probability of finding a salmon would be the
same in every pool on a river. It is perhaps not
entirely surprising that, for a particular
molecule, the probability of finding an e- will
be different depending on what part of the
molecule is considered. Probability
information for quantum mechanical systems
follows from the form of the wave function, ψ.
Probability and Classical Waves:
• Imagine making a wave by throwing a rock
into a swimming pool. If one considers only
the water molecules displaced upwards from
their equilibrium position then the probability
of finding an upward displaced water
molecule is highest where the waves have the
highest positive amplitude. Similar
considerations apply to downward displaced
water molecules.
Probability and Quantum Mechanics:
Probability and QM – continued:
Probability and QM – continued:
Probability and Normalized ψ’s:
• If the integrals discussed are equal to one we
say that our wave function is normalized. A
similar idea may have been encountered
previously when you considered the dot
product of two vectors. If the integral is not
equal to one we can normalize the wave
function after first evaluating an integral of
the form discussed.
Normalizing Wave Functions:
Normalized and Un-normalized Wave
Functions:
• The use of un-normalized wave functions in
quantum mechanical problems does enable us
to calculate from the Schrödinger equation
the correct values for eigenvalues which
specify energy and momentum values.
However, the use of normalized wave
functions does afford both mathematical and
conceptual advantages. In particular,
“probabilities” are more readily calculated.
Class Example – Simple Normalization:
• The simplest normalizations involve a one
dimensional integration. Normalizations
involving imaginary wave functions can
appear ominous – until one tries them!
• Example: Normalize the wave function eimθ
where r is a scalar (it could be a quantum
number!) and the allowed θ values are
defined by 0 ≤ θ ≤ 2π.
Simple Normalization – continued:
• Hint: in the example to be considered the trial
and possibly un-normalized wave function is
• ψTrial(θ) = eimθ
• This wave function will “reappear” when we
move to discussions of the “particle on the
ring” and angular momentum. ( Sadly (?), for
Canadians, there are no known wave
functions for the “particle (puck) on a rink”!)
Hamilton – Another Wise Guy:
• The famous mathematician Hamilton found
that, in classical mechanics, energy
calculations were often simplified if some
familiar equations were recast in terms of
momentum. In this way, momentum terms
served to specify kinetic energy (potential
energy is described separately).
Hamilton & “Classical” Kinetic Energy:
Class Example – Angular Momentum:
• In a number of cases we will need to consider
angular momentum. Common examples are
the treatment of “spinning” electrons (spin
angular momentum) and the rotation of
molecules (molecular framework angular
momentum). Starting with the EKinetic for the
rotational energy of a one dimensional rod,
derive an expression for EKinetic which employs
an angular momentum term.