KMM 315E
Physical Chemistry For Chemical Engineers
Week 4
The Quantum Mechanical Postulates
Math for QChem
Wave Particle Duality
The essence of quantum mechanics is that particles and
waves are not really separate and distinct entities.
Waves can show particle-like behavior as illustrated by
the photoelectric effect.
Particles can also show wave-like properties as shown
by the diffraction of atomic beams from surfaces.
How can we develop criteria that tell us when a
particle description (classical) of an atomic or
molecular system is sufficient and when we need to use
a wave description (quantum mechanical)?
Two criteria are used:
1) the magnitude of the wavelength of the particle
relative to the dimensions of the problem
(wavelength << slit).
2) The degree to which the allowed energy values
form a continuous energy spectrum.
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Schrödinger Equation
https://www.youtube.com/watch?v=TUFC9V0sA_U
The Schrödinger equation is a second-order differential equation used to calculate the wavefunction of a system.
the wavefunction contains all the dynamical information about the system it describes
In 1926, the Austrian physicist Erwin Schrödinger proposed an equation for finding the wavefunction of any system
The time-independent Schrödinger equation for a particle of mass m moving in one dimension with energy E in a system that
does not change with time (for instance, its volume remains constant) is
time-dependent Schrödinger equation
time-independent Schrödinger equation
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Sinusoidal Wave
Frequency 𝒗 = 𝟏/𝑻 where T is the
period and 2π for the function on the LHS
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WaveFunction
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The Born interpretation of the wavefunction
Born’s Probability Interpretation (1926): Max Born proposed
that the square of the wave function’s magnitude represents
the probability density for a particle’s position, a fundamental
aspect of quantum mechanics.
(a) A wavefunction is normalized if the integral of its square is
equal to 1.
(b) The quantization of energy stems from the constraints that
an acceptable wavefunction must satisfy
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For a particle free to move in three dimensions (for
example, an electron near a nucleus in an atom)
If the wavefunction of a particle has the value ψ
at some point x,then the probability of finding
the particle between x and x + dx is
proportional to | ψ |2 dx.
The Born interpretation of the wavefunction in threedimensional space implies that the probability of finding
the particle in the volume element d = dxdydz
at some location r is proportional to the
product of d  and the value of | ψ |2 at that
location.
For systems with spherical symmetry it is best to work in
spherical polar coordinates
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What is happening in 3D?
Spherical Coordinates
r, the radius, ranges from 0 to ∞
θ, the colatitude, ranges from 0 to π
φ, the azimuth, ranges from 0 to 2π
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Probability Density Function
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POSTULATES
1. Physical Meaning Associated with the Wave Function Is Probability
The association of the wave function with the probability places an important requirement on a wave function called
normalization.
Such a definition is obviously meaningless if the integral does
not exist. Therefore, Ψ (x,t) must satisfy several mathematical
conditions to ensure that it represents a possible physical state.
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The wave function must be a single-valued function of the spatial coordinates. If this were not the case, a particle would have
more than one probability of being found in the same interval.
The second derivative must exist and be well behaved. If this were not the case, we could not set up the Schrodinger equation.
This is not the case if the wave function and/or its first derivative are discontinuous.
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Normalization and An Acceptable Wavefunction
N
A need for multiplication factor N, any constant that must be REAL
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EXAMPLE
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POSTULATES
2. Every Observable Has a Corresponding Operator
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Operators, Eigenvalues and EigenFunctions
Operator: A rule that transforms one function into another function
To formulate a systematic way of extracting information from the wavefunction, we first note that any Schrödinger equation can
be written in the succinct form,
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Properties of Operators
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POSTULATES
3. The Result of an Individual Measurement
Postulate 3 says that if we measure the energy of a particle in a box, we will find one of these
energies and no others.
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Observables
The importance of eigenvalue equations is that the pattern
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EXAMPLE
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All operators in QM are linear
All operators in QM are Hermitian type
The Hermitian matrix is pretty much comparable to a
symmetric matrix. The symmetric matrix is equal to its
transpose, whereas the Hermitian matrix is equal to its
conjugate transpose, sometimes referred to as tranjugate.
The Hermitian matrix has complex numbers; however, its
diagonal entries are real. The Eigenvalues of a Hermitian matrix
are always real.
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Math for QCHEM
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Quantum-Mechanical Operators Must Be Hermitian
Operators
Hermitian operators are enormously important by virtue of two
properties:
1.Their eigenvalues are real
2. Their eigenfunctions are ‘orthogonal’.
Wavefunctions corresponding to different eigenvalues of an Hermitian operator are orthogonal.
For example, the hamiltonian operator is hermitian (it corresponds to an observable, the energy).
Therefore, if 1 corresponds to one energy, and 2 corresponds to a different energy, then we know at
once that the two functions are orthogonal and that the integral of their product is zero.
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Commutator Property
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Commutator Property
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Ex: Evaluate the Commutator
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DIY
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POSTULATES
The Expectation Value
Postulate 4 shows how we can do measurements: For the case in which is normalized, the
denominator in this expression has the value 1.
Wave functions are usually normalized!!
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EXAMPLE
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Probability Density (Where is the PARTICLE?)
Suppose that your wave equation is this:
Suppose that B = 0
Where is the particle?
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Probability Density (Where is the PARTICLE?)
Suppose that A=B Where is the particle?
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What is the linear momentum of a particle for the wavefunction given below?
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EXAMPLE
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POSTULATES
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Superpositions and Expectation Values
Suppose now that the wavefunction is the one given in (with A = B). What is the linear
momentum of the particle it describes?
When the wavefunction of a particle is not an eigenfunction of an
operator, the property to which the operator corresponds does not
have a definite value. However, in the current example the momentum
is not completely indefinite because the cosine wavefunction is a linear
combination, or sum, of eikx and e−ikx, and these two functions, as we
have seen, individually correspond to definite momentum states. We
say that the total wavefunction is a superposition of more than one
wavefunction.
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Superposition
Animation of two waves, the green wave moves to the right
while blue wave moves to the left, the net red wave amplitude
at each point is the sum of the amplitudes of the individual
waves.
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Superpositions and Expectation Values
suppose the wavefunction is known to be a superposition of many different linear momentum eigenfunctions
and written as the linear combination
1. When the momentum is measured, in a single observation one of the
eigenvalues corresponding to the ψk that contribute to the superposition will
be found.
2. The probability of measuring a particular eigenvalue in a series of
observations is proportional to the square modulus (| ψ k |2) of the
corresponding coefficient in the linear combination.
3. The average value of a large number of observations is given by the
expectation value, <Ω>, of the operator corresponding to the observable of
interest.
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2𝑐 2 ℏ2 𝑥 2
A certain one particle 1D system has the V=
and is stationary with
𝑚
2
Ψ 𝑥 = 𝑏𝑥𝑒 −𝑐𝑥 where b is constant, c=2.00 nm-2 and m=1.00x10-27 g.
Find the total E of the particle.
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The uncertainty principle
https://www.youtube.com/embed/TQKELOE9eY4
https://youtu.be/a8FTr2qMutA?si=n_GI4EaCTMOqktNs
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The uncertainty principle
It is impossible to specify simultaneously, with arbitrary precision, both the momentum and the position of a particle.
If we know that the particle is at a definite location, its
wavefunction must be large there and zero everywhere else
In other words, we can create a sharply localized wavefunction,
called a wave packet, by forming a linear combination of
wavefunctions that correspond to many different linear
momenta. Now the particle is perfectly localized. However, we
have lost all information about its momentum because, as we
saw above, a measurement of the momentum will give a result
corresponding to any one of the infinite number of waves in
the superposition, and which one it will give is unpredictable.
https://www.youtube.com/watch_popup?v=TQKELOE9eY4
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If we locate an electron to within 20 pm then what is
the uncertainity on its speed?
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SUMMARY
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A Proper Wavefunction Should be
https://www.youtube.com/watch?v=sOI4DlWQ_1w
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The Information in a Wavefunction
With zero
potential energy
The solutions of
this equation
have the form
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