Download Physical Chemistry - Angelo Raymond Rossi

Physical Chemistry
Why Study Quantum Mechanics?
C HAPTER 8 - Q UANTUM T HEORY:
I NTRODUCTION AND P RINCIPLES
Classical physics is unable to explain why
• an atom with a positively charged nucleus surrounded by electrons is stable.
• graphite conducts electricity but diamond does not, why the light emitted by a
hydrogen discharge lamp appears only at a small number of wavelengths, and
why the bond angle in H2 O is different from that in H2 S.
Professor Angelo R. Rossi
Department of Chemistry
Chemistry is a molecular science, and the goal of chemists is to understand
macromolecular behavior in terms of the properties of individual molecules.
For example, H2 is a good fuel
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H2 (g) + O2 (g) −→ H2 O (g)
because the energy required to break the H-H and O-O bonds is much less than the
energy that is released in forming two O-H bonds.
Quantum Chemistry will provide the underlying explanation for the differences in bond
strengths of these molecules.
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Quantum Mechanics: an Interplay of Experiment and Theory
Distinguishing Between Classical Physics and Quantum
Mechanics
At the end of the 19th century,
Experimental evidence points to two key properties that distinguish
classical physics from quantum physics:
Maxwell’s electromagnetic theory unified existing knowledge in the
areas of electricity, magnetism and waves; and the well-established field
of classical mechanics described the motion of particles.
1. quantization: energy at the atomic level is not a continuous
variable but comes in discrete packets called quanta.
But, a number of key experiments showed that the predictions of
classical physics were inconsistent with experimental outcomes, and
this stimulated scientists to formulate quantum mechanics.
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2. wave-particle duality: at the atomic level, light waves have
particle like properties, and atoms as well as electrons have
wave-like properties.
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Blackbody Radiation
Blackbody Radiation: Spring Model of a Metal
A red-hot block of metal, with a spherical cavity in its interior emanates
radiation through a hole small enough that the conditions inside the
block are not perturbed.
An ideal blackbody is shown below
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Blackbody Radiation: Predictions from Classical Theory
Blackbody Radiation: Predictions from Classical Theory
Under the conditions of equilibrium between the radiation field inside the cavity and
the glowing piece of matter, classical electromagnetic theory can predict the what
frequencies (ν) of light are radiated and their relative magnitudes.
Classical theory predicts that the average energy of an oscillator is
simply related to the temperature:
The spectral density is the energy at frequency ν per unit volume and unit frequency
stored in the electromagnetic field of the blackbody radiator and is given by the
equation:
8πν 2
ρ(ν, T )dν = 3 Ēosc dν
c
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Ēosc = kB T
where kB is the Boltzmann constant.
Combining the two equations results in an expression for ρ(ν, T )dν,
• ρ, the spectral density in units of E × V −1 × ν −1 , is a function of the
temperature (T ) and the frequency (ν).
8πkB T ν 2
dν
c3
the amount of energy per unit volume in the frequency range between ν
and ν + dν within a blackbody at temperature T
ρ(ν, T )dν =
• The speed of light is c, and Ēosc is the average energy of an oscillating dipole in
the solid.
• The factor dν provides the energy density observed in the frequency interval of
width dν centered at frequency ν.
The basis of the model is that the atomic nuclei and corresponding electrons act as
oscillating dipoles.
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Blackbody Radiation: Experimental Results
Blackbody Radiation: Quantum Hypothesis
It is possible to measure the spectral density of the radiation emitted by a blackbody as shown in the adjacent
figure.
The first person to offer a successful explanation for blackbody radiation was the
German physicist Max Planck in 1900.
The experimental curves have a common behavior:
Planck used a quantum hypothesis to derive the blackbody radiation law.
• Implicit in the classical theory of radiation is the assumption that the energies of
the electronic oscillators responsible for the emission of radiation could have any
value whatsoever.
1. The spectral density is peaked in a broad maximum and falls off to higher and lower frequencies.
2. The shift of the maxima to higher frequencies with
increasing temperature is consistent with our experience that if a block of metal is heated to higher
temperatures, the color will change from red to
yellow to blue (i.e a trend toward increasing frequency and shorter wavelength).
• Planck found that he could obtain agreement with theory and experiment only if
he assumed that the energies of the oscillators were discrete and proportional
to an integral multiple of the frequency
E = nhν
3. The two curves show similar behavior at low frequencies (longer wavelengths), but the theoretical
curve keeps on increasing with higher frequencies (shorter wavelengths).
where h is Planck’s constant initially an unknown proportionality constant, and
n is a positive integer (n = 0, 1, 2, . . . ).
• For a given ν, energy is quantized.
It is clear that the theoretical prediction is incorrect!!!
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Blackbody Radiation: Quantum Hypothesis
The Photoelectric Effect: Experimental Observations
Planck obtained the following relationship for a quantum oscillator:
The ejection of electrons from the surface of a
metal by radiation is called the photoelectric
effect.
hν
Ēosc =
hν
e kB T − 1
Incident light on a copper plate held in a vacuum can be absorbed, leading to the excitation of electrons into unoccupied energy levels.
Using the equation,
ρ(ν, T )dν =
8πν 2
Ēosc dν
c3
Planck obtained the following general formula for the spectral radiation density from a blackbody:
8πhν 3
×
ρ(ν, T ) dν =
c3
1
hν
Sufficient energy can be transferred to the
electrons such that some of them leave the
metal and are ejected into the vacuum and
collected.
dν
e kB T − 1
The following trend is obtained:
• At higher temperatures,
hν
( kT
<< 1), the term e
and Ēosc → kB T . The leads to ρ(ν, T )dν =
predictions.
hν
kB T
According to conservation of energy, the absorbed light energy must be balanced by the
kinetic energy of the emitted electrons and
the energy required to eject an electron at
equilibrium.
can be expanded in a Taylor-Mclaurin series,
8πkB T ν 2
dν
c3
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and approximately follows classical
hν
• At lower temperatures, ( kT
>> 1), the denominator in the above equation becomes very large, and
Ēosc → 0 which produces a drop in the spectral density.
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Thus,
Planck’s Quantum Hypothesis seems
to accurately
predict
behavior of blackbody radiation.
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The Photoelectric Effect: Experimental Observations
The Photoelectric Effect: Predictions of Classical Theory
1. The kinetic energy per electron increases with the light intensity.
1. The number of emitted electrons is proportional to the light intensity, but their
kinetic energy is independent of the
light intensity.
2. Any one electron can absorb only a small fraction of the incident light.
3. Light is incident as a plane wave over the entire metal surface and is absorbed
by many electrons in the solid.
2. No electrons are emitted unless the
frequency ν is above a threshold frequency ν0 even for high light intensities.
4. Electrons are emitted for all light frequencies, provided that the light is
sufficiently intense.
3. Electrons are emitted even at such low
intensities such that all the light absorbed by the entire metal surface is
barely enough to eject a single electron
based on energy conservation considerations..
4. The kinetic energy of the emitted electrons depends linearly on the frequency
after the threshold frequency has been
attained.
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The Photoelectric Effect: Quantum Hypothesis
Louis de Broglie: Matter Has Wavelike Properties
Planck applied the concept of energy quantization (E = nhν) to the emission and
absorption mechanism of the atomic electronic oscillators but believed that once the
light energy was emitted, it behaved like a classical wave.
de Broglie reasoned that, if light can display wave-particle duality,
then matter which appears particle-like, might also display wave-like
properties under certain conditions.
In 1905, Einstein proposed that the radiation itself existed as small packets of energy
(E = hν) now known as photons.
Einstein had shown from relativity theory that the wavelength, λ, and
the momentum, p, of a photon are related by
Einstein showed that the kinetic energy (KE) of an ejected electron is equal to the
energy of the incident photon (hν) minus the minimum energy required to remove an
electron from the surface of a particular metal (Φ).
KE =
λ=
1
mv 2 = hν − Φ
2
Because the momentum of a particle is given by mv, the above
equation predicts that for a particle of mass, m moving at velocity v will
have a de Broglie wavelength given by
The minimum frequency that will eject an electron is just the frequency to overcome
the workfunction of the metal: hν0 = Φ.
Combining the above two equations gives KE = hν − hν0 where ν > ν0 .
λ=
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h
p
de Broglie argued that both light and matter obey this equation.
where Φ is called the work function.
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h
mv
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de Broglie Wavelength of the Electron
de Broglie Wavelength of the Electron
What is the de Broglie Wavelength λ of an electron that has been
accelerated through a potential difference of 100 V?
The energy of the electron is (1.602x10−19 C)(100V) = 1.602x10−17 J
Thus, the momentum is
p
p = (2)(9.109 × 10−31 kg)(1.602 × 10−17 J) = 5.403 × 10−24 kg m s−1
The momentum after the acceleration process must be calculated first.
This can be done by first calculating the energy of the electron.
The energy of an electron of mass m moving with a velocity v well
below the velocity of light is given by
and the wavelength is
1
p2
E = mv 2 =
2
2m
λ=
λ = 1.226 × 10−10 m = 0.1226 m
Thus, the momentum is given by
p=
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6.626 × 10−34 J s
h
=
p
5.403 × 10−24 kg m s−1
√
Note that λ is of the same order of magnitude as the distance between
the atoms in a crystal
2mE
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The Atomic Spectrum of Hydrogen
The Atomic Spectrum of Hydrogen
For some time scientists had known that each atom has a characteristic
line spectrum when subjected to high temperatures or electrical
discharge.
The atomic spectrum of hydrogen consists of several series of lines.
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• The atomic spectra consist of only discrete frequencies and are
called line spectra.
• Hydrogen has the simplest spectrum because of its simplicity.
The Swiss spectroscopist Johannes Rydberg accounted for all the lines
in the hydrogen atomic spectrum by obtaining the equation
1
1
1
−
= RH
λ
n21 n22
where RH is the Rydberg constant whose value is 109667.57 cm−1 .
Hydrogen Discharge Tube
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The Line spectrum of Hydrogen
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What Determines if a System Needs
to be Described Using Quantum Mechanics
The Atomic Spectrum of Hydrogen
It is important to realize that classical mechanics and quantum
mechanics are not two competing ways to describe the world around us.
Each theory has its usefulness in a different regime of physical
properties that describe reality: quantum mechanics merges
seamlessly into classical mechanics in the limit in which allowed energy
values are continuous rather than discrete.
It is not correct to state that whenever one talks about atoms, a
quantum mechanical description must be used.
The above image is shows
• a continuous spectrum (top),
• an absorption spectrum (middle),
• and an emission spectrum (bottom), for some of the Balmer
lines of hydrogen, i.e., those that
lie within the visible part of the
spectrum.
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What Determines if a System Needs
to be Described Using Quantum Mechanics
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First Criterion for Classical or Quantum Mechanical Behavior:
Magnitude of Particle Wavelength Relative to Dimensions of Problem.
For example, the origin of pressure for a container filled with argon
atoms at low pressure can be described by the collision of rapidly and
randomly moving argon atoms with the container walls.
This is the first criterion which is used to provide an understanding for
when a particle or classical description of an atomic or molecular
system is sufficient, or for when a wave or quantum mechanical
description must be used.
Classical mechanics provides a perfectly good description of the origin
of pressure of a single argon atom colliding with the wall.
For the diffraction of light of wavelength λ passing through a slit of width
a, diffraction is observed only when the wavelength is comparable to the
width of the slit: λ << a.
However, if light of very short wavelength is passed through this same
gas, a quantum mechanical description must be used to describe how
much energy can be taken up by the argon atom.
For the H2 molecule at room temperature, λ =
The essence of quantum mechanics is that particles and waves are not
really separate and distinct entities. Waves can show particle-like
behavior.
The localization of an electron to a small volume around the nuclei
brings out the wave-like behavior of the electrons and protons making
up a molecule.
h
p
≈ 1 × 10−10 m.
Also, crystalline solids have atomic spacings that are appropriate for the
diffraction of electrons as well as light atoms and molecules.
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Second Criterion for Classical or Quantum Mechanical Behavior:
Degree to which Allowed Energies Form a Continuous Energy Spectrum
The Boltzmann Distribution
Because all values of the energy are allowed for a classical system, it has a continuous energy spectrum.
In a quantum mechanical system, only certain values of the energy are allowed, and it has a discrete
energy spectrum.
Consider a one-liter container filled with an ideal gas of atoms at a pressure of 1 bar and temperature of
298.15 K.
The broad distribution of kinetic energy in the gas is for the individual molecules is given by the Boltzmann
Distribution:
ni
gi − (ǫi −ǫj )
kT
=
e
nj
gj
which states that the number of atoms with energy ǫi relative to the number with energy ǫj depends on
• The atoms have no rotational or vibrational degrees of freedom, and all of the energy is in the form of
translational kinetic energy.
1. the difference in energies with an exponential dependence.
The ratio of differences in energies relative to kT , the average energy that an atom at temperature
T , is important.
Fraction of Molecules
• At equilibrium, there is a broad distribution of energies of atoms at p = 1 bar and T = 298.15 K.
3. the ratio of degeneracies of the two states,
Lower Temperature
.
If kT is small compared to the spacing between allowed energies, the distribution of states of energy will be
very different from those of classical mechanics where a continuous energy spectrum is present.
It kT is much larger than the energy spacing, classical and quantum mechanics will give the same result for
the relative numbers of atoms or molecules of different energy.
3
kT
2
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gi
gj
The degeneracy of an energy value is the number of ways that an atom can have an energy ǫi
within the interval ǫ − ∆ǫ < ǫi < ǫ + ∆ǫ
Kinetic Energy
• The root mean square energy is related to the absolute temperature by Ē =
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2. the temperature with an exponential dependence.
Higher Temperature
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The Boltzmann Distribution: Quantum vs. Classical Behavior
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The Boltzmann Distribution: Quantum vs. Classical Behavior
A quantum mechanical system has a discrete
energy spectrum with allowed values of energy called energy levels.
For ∆E ≈ kT , each discrete energy level is
sufficiently broadened by energy fluctuations
that adjacent energy levels can no longer be
distinguished.
For a molecule containing vibrational energy
levels, the allowed energy levels are equally
space with interval, ∆E, and these energy
levels are numbered with integers.
In this limit, any energy that is chosen lies
within the yellow area: the energy difference
corresponds to an allowed value, and the discrete energy spectrum appears continuous.
In a gas at equilibrium, the total energy of
an individual molecule will fluctuate within
a range of ∆E = kT through collisions of
molecules with one another.
Classical behavior will be observed under
these conditions.
Therefore, the energy of a molecule with
a particular vibrational quantum number will
fluctuate within a range of width kT .
A plot of the relative number of molecules
having a vibrational energy E as a function
of E is shown in the adjacent figure for sharp
energy levels.
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The Boltzmann Distribution: Quantum vs. Classical Behavior
The Heisenberg Uncertainty Principle
Classical mechanics does not involve any limitations on the accuracy
with which observables can be measured.
If ∆E >> kT , an arbitrarily chosen energy
withing this range will lie within the blue area
with high probability, because the yellow bars
are widely separated.
For example, the position and momentum of a classical particle may be
simultaneously measured to an any desired accuracy.
The blue area corresponds to forbidden energies, and the discontinuous nature of the
energy spectrum will be observed.
In 1927 Heisenberg formulated the principle that values of particular
pairs of observables cannot be determined simultaneously with
arbitrarily high precision in quantum mechanics.
Quantum mechanical behavior will be observed under these conditions.
Examples of pairs of observables that are restricted in this way are
• momentum and position
• energy and time
Such pairs are referred to as complementary.
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The Heisenberg Uncertainty Principle
The Heisenberg Uncertainty Principle
de Broglie Relation
The de Broglie wave for a particle is made up of a superposition of an infinitely large number of waves of the form
Quantitative expressions for the Heisenberg uncertainty principle can be derived by
combining the de Broglie relation (p = h/λ) and the Einstein relation (E = hν).
ψ(x, t) = A sin 2π
x
λ
− νt
Substituting the de Broglie relation into the previous expression for the uncertainty
principle gives the following for motion in the x-direction:
where A is the amplitude, and λ is the wavelength.
The waves that are added together have infinitesimally different wavelengths.
The superposition of waves produces a wave packet as
shown in the adjacent figure.
It is possible to show that for wave motion of any type
∆x∆
then
p 1
x
≥
h
4π
~
h
∆x∆px ≥
~=
2
2π
To determine the position of an electron, at least one photon would have to strike the
electron, and the momentum would inevitably be altered by this process.
∆x∆
1
4π
where ∆x is the extent of the wave packet in space, λ the
range of wavelengths, ∆ν the range of frequencies, and ∆t
is the measure of the time required for the packet to pass a
given point.
∆ is the above equations actually represents standard deviations.
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1
px
=
λ
h
1
1
≥
λ
4π
∆t∆ν ≥
30
Superposition of waves to give
But, this process would limit our ability to measure the momentum of the electron.
(a) a weakly localized and
(b) a strongly
packet.
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Photons of shorter wavelength would disturb the momentum to an even greater extent
localized
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The Heisenberg Uncertainty Principle
Experimental Error
The Heisenberg Uncertainty Principle
Lifetimes of Excited states
Another form of the Heisenberg uncertainty principle may be derived by
1
substituting E = hν into the equation ∆t∆ν ≥ 4π
1
E
≥
∆t∆
h
4π
~
2
Suppose that excited atoms emit electromagnetic radiation in going to a
lower state:
∆t∆E ≥
It is important to realize that the uncertainties in the equations for
the Heisenberg uncertainty principle are NOT experimental errors
that are dependent on the quality of the measuring apparatus but
are inherent in quantum mechanics.
1. If these excited atoms live a long time, the radiation will be nearly
monochromatic, and the spectral line will be sharp.
2. If the excited atoms have a very short life, the electromagnetic
radiation will have a broader range of frequencies.
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Quantum Mechanical Waves and the Schrödinger Equation
Quantum Mechanical Waves and the Schrödinger Equation
The time-independent Schrödinger equation can be justified by combining classical
standing waves with the de Broglie relation.
Quantum mechanics is introduced by using the de Broglie relation, λ = h/p for the
wavelength:
For classical standing waves, the wave function is a product of two functions: one
which depends on spatial coordinates; the other depends only on time:
The momentum is related to the total energy, E and the potential energy, V (x) by
p2
= E − V (x)
2m
Ψ(x, t) = ψ(x) cos ωt
p
2m(E − V (x))
d2 ψ(x) 8π 2 m
+
[E − V (x)] ψ(x) = 0
dx2
h2
Using the abbreviation ~ = h/2π and rewriting the above equation, we obtain
d2 ψ(x) 4π 2
+ 2 ψ(x) = 0
dx2
λ
−
~2 d2 ψ(x)
+ V (x)ψ(x) = Eψ(x)
2m dx2
which is the time-independent Schrödinger equation in one dimension.
All of the above pertains to a classical wave.
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p=
Introducing the above expression for momentum into the de Broglie relation and
substituting the expression for λ into the wave equation yields
If the above function is substituted into the classical wave equation, the following is
obtained
d2 ψ(x) ω 2
+ 2 ψ(x) = 0
dx2
v
Using the relations ω = 2πν and νλ = v, where v is the velocity, ν the frequency, and λ
the wavelength of the wave, the above equation then becomes
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or
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The Born Interpretation of the Wave Function
The Physical Meaning Associated with the Wave Function
The Born interpretation of the wave function focuses on the square of
the wave function (ψ 2 ) or the square of the modulus:
The association of the wave function with probability has an important consequence
called normalization.
• The probability that a particle is found in an interval of width dx centered at the
position x must lie between 0 and 1.
| ψ |2 = ψ ∗ ψ
if ψ is complex.
• The sum of the probabilities over all intervals accessible to the particle is 1,
because the particle is somewhere in this range.
This states that the value of | ψ |2 at a point is proportional to finding the
particle at that point.
Consider a particle that is confined to a one-dimensional space of infinite extent which
leads to the following normalization condition:
Z ∞
ψ ∗ (x)ψ(x)dx = 1
| ψ |2 is a probability density and must be multiplied by an infinitesimal
region dx to obtain the probability
| ψ |2 dx
−∞
Such a definition is meaningless if the integral did not exist. Therefore, ψ(x) must
satisfy important mathematical conditions to ensure that it represents a possible
physical state.
The wave function ψ is called the probability amplitude and does not
have any physical significance.
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The Physical Meaning Associated with the Wave Function
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Operators
Another basic quantum mechanical concept is an operator which is a mathematical
operation that is applied to a function.
1. The wave function must be a single valued function of the spacial
coordinates. If this were not the case a particle would have more
than one probability of being found in the same interval.
∂
• For example, ∂x
is the operator that indicates that the function is to be
differentiated with respect to x.
• x̂ is the operator that indicates the function is to be multiplied by x.
2. The first derivative of the wave function must be continuous so that
the second derivative exists and is well-behaved. If this were not
the case, we could not set up the Schrödinger equation.
• Operators are designated with a caret, as in  or Ĥ.
The three-dimensional Schrödinger equation can be written in operator form:
~2 2
−
∇ + V (x, y, z) ψ(x, y, z) = Eψ(x, y, z)
2m
{z
}
|
3. The wave function cannot have an infinite amplitude over a finite
interval. If this were the case, the wave function could not be
normalized.
Ĥ
where
For example, the function ψ(x) = tan 2πx
, 0 ≤ x ≤ a cannot be
a
normalized.
∇2 =
∂2
∂2
∂2
+ 2+ 2
∂x2
∂y
∂z
The quantity in square brackets is called the Hamiltonian operator.
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Ĥψ(x,
y, z) =June
Eψ(x,
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Operators, Eigenfunctions, and Eigenvalues
Classical Functions to Quantum Mechanical Operators
Suppose an operator, e.g. Â operating on a function, e.g. ψn , yields a
constant, an multiplied by that function
For a particle of mass m moving in one dimension subject to a potential
energy V (x), the classical Hamiltonian function is
Âψn = an ψn
H=
The appropriate statement is that ψn is an eigenfunction of operator Â
with an eigenvalue of an .
There is no caret over H because this is a classical function.
The above equation should be compared with the corresponding
quantum mechanical equation.
~2 ∂ 2
−
+
V
(x)
ψ(x) = Eψ(x)
2m ∂x2
Thus, for the one-dimensional Schrödinger equation, ψ(x) is an
eigenfunction of Ĥ with eigenvalue E.
Ĥψn (x) = En ψn (x)
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Classical Mechanical Observables and
Corresponding Quantum Mechanical Operators
Classical Functions to Quantum Mechanical Operators
The process of converting a function for a classical system to the corresponding
operator for the quantum mechanical system is formalized by the following rules:
One-Dimensional Systems
Classical Observables
QM Operators
Name
Symbol
Symbol Operation
Position
x
x̂
Multiply by x
Position Squared
x2
x̂2
Multiply by x2
∂
Momentum
px
p̂x
−i~ ∂x
2
2
2 ∂2
Momentum Squared px
p̂x
−~ ∂x2
p2x
~2 ∂ 2
Kinetic Energy
Tx = 2m
Tˆx
− 2m
∂x2
Potential Energy
V (x)
V̂ (x)
Multiply by V (x)
~2 ∂ 2
+ V (x)
Total Energy
Tx + V (x) Ĥ
− 2m
∂x2
1. Each Cartesian coordinate in the Hamiltonian function is replaced by that
coordinate and a multiplication operator.
q̂ → q×
2. Each Cartesian component of a linear momentum, pq , q = x, y, z, in the
Hamiltonian function is replaced by the operator
p̂q = −i~
p2
+ V (x)
2m
∂
∂q
3. The potential energy function is not changed because of the first statement
given above:
V̂ (q̂) =→ V (q)×
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Classical Mechanical Observables and
Corresponding Quantum Mechanical Operators
Operators and Observables
Each classical observable is associated with a quantum mechanical
operator.
Three-Dimensional Systems
Classical Observables
QM Operators
Name
Symbol
Symbol Operation(a)
Position
r
r̂r
Multiply
by r
∂
∂
∂
+ j ∂y
+ k ∂z
Momentum
p
p̂p
−i~ i ∂x
Kinetic Energy
Potential Energy
Total Energy
Angular
Momentum
T
V (x, y, z)
E =T +V
lx = ypz − zpy
ly = zpx − xpz
T̂
V̂ (x, y, z)
Ĥ
ˆlx
ˆlx
ˆlx
lz = zpy − ypx
2
∂
(a) ∇2 = ∂x
2 +
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∂2
∂y 2
+
∂2
∂z 2
Another postulate of quantum mechanics is that the only possible
measured values of an observable are the eigenvalues of the operator
representing that observable.
~
∇2
− 2m
Multiply by V (x, y, z)
~2
2
− 2m
∇ + V (x, y,
z)
2
It is a postulate of quantum mechanics that the average value < a > of
an observable corresponding to an operator  is given by
Z
< a >= ψ ∗ Âψdτ
∂
∂
−i~ y ∂z
− z ∂y
∂
∂
−i~ z ∂x
− x ∂z
where ψ ∗ is the complex conjugate of ψ.
∂
∂
− y ∂x
−i~ x ∂y
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A complex complex conjugate of a complex function is obtained by
replacing i with −i.
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Operators and Observables
Hermetian Operators, Eigenfunctions, and Eigenvalues
Taking the energy of a one-dimensional system as an example of an
observable, energy eigenvalues can be obtained from the Schrödinger
equation:
Ĥψn (x) = En ψn (x)
Operators and wave functions may be complex, but eigenvalues of quantum
mechanical operators must be real because they are the only positive measured
values.
Operators which yield real values are called Hermitian and have the following
property:
Z
Z
where n is an index that labels the different eigenfunctions and
eigenvalues.
Multiplying the above equation by the complex conjugate of the wave
function and integrating over all values of x yields
Z
Z
Z
∗
∗
ψn (x)Ĥψn (x)dτ = ψn (x)En ψn (x)dτ = En ψn∗ (x)ψn (x)dτ = En
ψi∗ Âψj dτ =
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ψj (Âψi )∗ dτ
for any two well-behaved functions ψi and ψj , where (Âψi )∗ is the complex conjugate
of Âψi
If ψi = ψj = ψ and an eigenfunction of Â, then Âψ = aψ and
Z
Z
ψ ∗ Âψdτ = a;
ψ(Âψ)∗ dτ = a∗
since the wave functions are normalized.
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46
But, the Hermetian property requires that a = a∗ , proving that the eigenvalues are real.
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48
Hermetian Operators, Eigenfunctions, and Eigenvalues
Commuting and Non-commuting Operators
Eigenfunctions of an Hermitian operator corresponding to different
eigenvalues are orthogonal. If
As discussed on a previous slide, classical physics predicts that there is
no limit to the amount of information (or observables) that can be known
about a system at a given instant of time.
Âψn = an ψn
then
Z
and
Âψm = am ψm
This is not the case in quantum mechanics.
• Two observables can be known simultaneously only if the outcome
of the measurements is independent of the order in which they
were measured.
ψn∗ ψm dτ = 0 m 6= n
Since the ψm can always be normalized, they form an orthonormal set
of functions:
Z
ψn∗ ψm dτ = δmn
• An uncertainty relation limits the degree to which observables of
non-commuting operators can be known simultaneously.
where δmn is the Kronecker delta which is defined by
(
0 for m 6= n
δmn =
1 for m = n
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Commuting and Non-commuting Operators
Commuting and Non-commuting Operators
The values of two different observables α and β, which correspond to
two operators  and B̂, can be simultaneously determined only if the
measurement process does not change the system.
The only case in which the measurement does not change the state of
the system is if ψn (x) is simultaneously an eigenfunction of  and B̂.
B̂[Âψn (x)] = αn B̂ψn (x) = αn βn ψn (x)
Otherwise the system on which the two measurements are carried out
is not the same.
Â[B̂ψn (x)] = βn Âψn (x) = βn αn ψn (x)
Let ψn (x) be the wave function that characterizes the system, but
possibly not an eigenfunction of the operators  and B̂.
Because the eigenvalues αn and βn are constants,
βn αn ψn (x) = αn βn ψn (x)
Now perform a measurement of the observables corresponding to the
first operator  and subsequently to the operator B̂ which is equivalent
to
B̂[Âψn (x)] = αn B̂ψn (x)
and
[ÂB̂ − B̂ Â]ψn (x) = 0
We have just shown that the act of measurement changes the state of
the system unless the system wave function is simultaneously an
eigenfunction of two different operators.
and
Â[B̂ψn (x)] = βn Âψn (x)
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52
Commuting and Non-commuting Operators
Symmetry: Commuting and Non-commuting Operators
Two operators commute, if they have a common set of eigenfunctions
The Hamiltonian, Ĥ, for a molecule is unchanged under other symmetry operation of
a group, because the total energy remains the same.
Â[B̂f (x)] − B̂[Âf (x)]
Thus, Ĥ belongs to the totally symmetric representation.
If a molecule that contains a plane of symmetry undergoes the reflection symmetry
operation (σ̂), is indistinguishable from the original molecule.
which is abbreviated as
[Â, B̂]f (x)
Ĥ and σ̂ commute because the order of operating on the wave function is unimportant.
The expression in square brackets [ ] is called the commutator of
operators  and B̂.
[Ĥ, σ̂] = Ĥ σ̂ − σ̂ Ĥ
Thus, eigenfunctions of Ĥ can be found that are simultaneously eigenfunctions of
σ̂ and all other symmetry operarions of the point group.
If the value of the commutator is not zero for an arbitrary function f (x),
then the corresponding observables cannot be determined
simultaneously and exactly.
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Symmetry-adapted MOs which exhibit the symmetry of the molecule can be
obtained.
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Expectation Values and Superposition
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Expectation Values and Superposition
The average value of  is given by
The only possible measured values of an observable are the eigenvalues of the
operator representing that observable.
< a >=
The average value of an observable, Â, when the system is in the state ψ is
Z
< a >= ψ ∗ Âψdτ
Z
X
n
cn φ n
!∗
Â
X
cm φ m
m
Then
< a >=
X
n
!
dτ =
XX
n
c∗n cm
m
Z
φ∗n Âφm dτ
|cn |2 an
In most cases, ψ is not an eigenfunction of Â, but it can always be written as a linear
superposition of the eigenfunctions of Â:
X
cn φn
ψ=
Thus, the average value < a > is the sum of possible measured values (an ) multiplied by |cn |2 , a
non-negative number. |cn |2 is the probability of measuring the value an .
where the cn are constants.
The probability of measuring the eigenvalue am is given by
cm =
n
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m
n
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φ∗m ψdτ =
Z
φ∗m
X
cn φn dτ
n
2
Z
| cm |2 = φ∗m ψdτ Since the φn are orthonormal, ψ has to be normalized
!
!∗
Z
Z
Z X
X
X
X
|cn |2 = 1
c∗n cn φ∗n φn dτ =
cm φm dτ =
cn φn
ψ ∗ ψdτ =
n
Z
Note that the probability amplitude may be complex, but the probability | cm |2 is always real.
When ψ is an eigenfunction of Â, e.g. φk , then the probability of measuring ak is 1, and the probability of
measuring any other eigenvalue is 0.
n
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56
What day is it?
A test
<?php
if ($day == "monday")
{
$callInSick = true;
}
else
{
$callInSick = false;
}
?>
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