Molekylfysik

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1. Quantum theory introduction and principles
1.1 The failures of classical physics 1.2
Wave-particle duality 1.3 The Schrödinger
equation 1.4 The Born interpretation of the
wavefunction 1.5 Operators and theorems of the
quantum theory 1.6 The Uncertainty Principle
?? c
?
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1.1 The failures of classical physics
A. Black-body radiation
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Tentative explanation via the classical mechanics
Equipartition of the energy Per degree of
freedom average energy kT (26 meV at 25C).
The total energy is equally partitioned over
all the available modes of motion. Rayleigh and
Jeans used the equipartition principle and
consider that the electromagnetic radiation is
emitted from a collection of excited oscillators,
which can have any given energy by controlling
the applied forces (related to T). It led to the
Rayleigh-Jeans law for the energy density ? as a
function of the wavelength ?.
It does not fit to the experiment. From this law,
every objects should emit IR, Vis, UV, X-ray
radiation. There should be no darkness!! This is
called the Ultraviolet catastrophe.
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Introduction of the quantization of energy to
solve the black-body problem
Max Planck quantization of energy.
Cross-check of the theory from the Planck
distribution, one can easily find the
experimental Wien displacement and the
Stefan-Boltzmann law. ? the quantization of
energy exists!
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C. Atomic and molecular spectra
Excitation energy
Photon absorption
Photon emission h?hc/?
Fe
The emission and absorption of radiation always
occurs at specific frequencies another proof of
the energy quantization.
NB wavenumber
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1.2 Wave-particle duality

• The particle character of
• electromagnetic radiation
• ? The photoelectric effect

• The photon h? ? particle-like projectile
• conservation of energy ½mv2 h? - ?
• metal workfunction, the minimum energy required
  to remove an electron from the metal to the
  infinity.
• Threshold does not depend on intensity of
  incident radiation.
• NB The photoelectron spectroscopy (UPS, XPS) is
  based on this photoelectric effect.

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B. The wave character of the particles ?
Electron diffraction
Diffraction is a characteristic property of
waves. With X-ray, Bragg showed that a
constructive interference occurs when ?2d sin?.
Davidsson and Germer showed also interference
phenomenon but with electrons!
?Particles are characterized by a wavefunction
? A link between the particle (pmv) and the wave
(?) natures
? An appropriate potential difference creates
electrons that can diffract with the lattice of
the nickel
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1.3 The Schrödinger Equation
From the wave-particle duality, the concepts of
classical physics (CP) have to be abandoned to
describe microscopic systems. The dynamics of
microscopic systems will be described in a new
theory the quantum theory (QT). ? A wave,
called wavefunction ?(r,t), is associated to each
object. The well-defined trajectory of an object
in CP (the location, r, and momenta, p m.v, are
precisely known at each instant t) is replaced by
?(r,t) indicating that the particle is
distributed through space like a wave. In QT, the
location, r, and momenta, p, are not precisely
known at each instant t (see Uncertainty
Principle). ? In CP, all modes of motions (rot,
trans, vib) can have any given energy by
controlling the applied forces. In the QT, all
modes of motion cannot have any given energy, but
can only be excited at specified energy levels
(see quantization of energy). ?The Planck
constant h can be a criterion to know if a
problem has to be addressed in CP or in QT. h can
be seen has a quantum of an action that has the
dimension of ML2T-1 (E h? where E is in ML2T-2
and ? is in T-1). With the specific parameters of
a problem, we built a quantity having the
dimension of an action (ML2T-1). If this quantity
has the order of magnitude of h (?10-34 Js), the
problem has to be treated within the QT.
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• Hamiltonian function H T V.
• T is the kinetic energy and V is the
  potential energy.
• correspondence principles are proposed to
  pass from the classical mechanics
• to the quantum mechanics

Quantum mechanics
Classical mechanics
Schrödinger Equation
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? The Schrödinger Equation (SE) shows that the
operator H and ih?/?t give the same results when
they act on the wavefunction. Both are equivalent
operators corresponding to the total energy E.
? In the case of stationary systems, the
potential V(x,y,z) is time independent. The
wavefunction can be written as a stationary wave
?(x,y,z,t) ?(x,y,z) e-i?t (with Eh?). This
solution of the SE leads to a density of
probability ?(x,y,z,t)2 ?(x,y,z)2, which is
independent of time. The Time Independent
Schrödinger Equation is
NB In the following, we only envisage the time
independent version of the SE.
or
? The Schrödinger equation is an eigenvalue
equation, which has the typical
form (operator)(function)(constant)(same
function) ? The eigenvalue is the energy E. The
set of eigenvalues are the only values that the
energy can have (quantization). ? The
eigenfunctions of the Hamiltonian operator H are
the wavefunctions ? of the system. ? To each
eigenvalue corresponds a set of eigenfunctions.
Among those, only the eigenfunctions that fulfill
specific conditions have a physical meaning.
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1.4 The Born interpretation of the wavefunction
? Physical meaning of the wavefunction probabilit
y of finding the particle in an infinitesimal
volume d?dxdydz at some point r is proportional
to ?(r)2d?

• ?(r)2 ?(r)?(r) is a probability density.
• It is always positive!
• wavefunction may have negative or complex values
  

Node
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A. Normalization Condition

• The solution of the differential equation of
  Schrödinger is defined within a constant N.
• If ?? is a known solution of H??E??, then
  ?N?? is a also solution for the same E.
• H?E? ? H(N??) E(N??) ? N(H??)N(E??)
  ? H??E??
• The sum of the probability of finding the
  particle over all infinitesimal volumes d? of the
  space is 1 Normalization condition.
• We have to determine the constant N, such that
  the solution ?N?? of the SE is normalized.

B. Other mathematical conditions
? ?(r)? ? ?r ? if not no physical meaning for
the normalization condition ? ?(r) should be
single-valued ?r ? if not 2 probability for the
same point!! ? The SE is a second-order
differential equation ?(r) and d?(r)/dr should
be continuous
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C. The kinetic energy and the wavefunction
We can expect a particle to have a high kinetic
energy if the average curvature of its
wavefunction is high.
The kinetic energy is then a kind of average over
the curvature of the wavefunction we get a large
contribution to the observed value from the
regions where the wavefunction is sharply curved
(?2? /? x2 is large) and the wavefunction itself
is large (? is large too).
Example the wave function in a periodic system
electrons in a metal
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1.5 Operators and principles of quantum mechanics
A. Operators in the quantum theory (QT)
An eigenvalue equation, ?f ?f, can be
associated to each operator ?. In the QT, the
operators are linear and hermitian. ? Linearity
? is linear if ?(c f) c ? f (cconstant)
and ?(f?) ? f ?? ? NB c can be defined to
fulfill the normalization condition ?
Hermiticity A linear operator is hermitian
if where f and ? are finite, uniform, continuous
and the integral for the normalization
converge. ? The eigenvalues of an hermitian
operator are real numbers (? ?) ? When the
operator of an eigenvalue equation is hermitian,
2 eigenfunctions (fj, fk) corresponding to 2
different eigenvalues (?j, ? k) are orthogonal.
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B. Principles of Quantum mechanics
? 1. To each observable or measurable property
lt?gt of the system corresponds a linear and
hermitian operator ?, such that the only
measurable values of this observable are the
eigenvalues ?j of the corresponding operator. ?f
?f ? 2. Each hermitian operator ? representing
a physical property is complete. Def An
operator ? is complete if any function (finite,
uniform and continuous) ?(x,y,z) can be developed
as a series of eigenfunctions fj of this operator.
? 3. If ?(x,y,z) is a solution of the Schrödinger
equation for a particle, and if we want to
measure the value of the observable related to
the complete and hermitian operator ? (that is
not the Hamiltonian), then the probability to
measure the eigenvalue ?k is equal to the square
of the modulus of fks coefficient, that is
Ck2, for an othornomal set of eigenfunctions
fj. Def The eigenfunctions are orthonormal if
NB In this case
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? 4. The average value of a large number of
observations is given by the expectation value
lt?gt of the operator ? corresponding to the
observable of interest. The expectation value of
an operator ? is defined as ? 5. If the
wavefunction ?f1 is the eigenfunction of the
operator ? (?f ?f), then the expectation value
of ? is the eigenvalue ?1.
For normalized wavefunction
See p305 in the book
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1.6 The Uncertainty Principle
?1. When two operators are commutable (and with
the Hamiltonian operator), their eigenfunctions
are common and the corresponding observables can
be determined simultaneously and accurately.
?2. Reciprocally, if two operators do not
commute, the corresponding observable cannot be
determined simultaneously and accurately. If
(?1?2- ?2?1) c, where c is a constant, then
an uncertainty relation takes place for the
measurement of these two observables where
Uncertainty Principle
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Example of the Uncertainty Principle
?1. For a free atom and without taking into
account the spin-orbit coupling, the angular
orbital moment L2 and the total spin S2 commute
with the Hamiltonian H. Hence, an exact value of
the eigenvalues L of L2 and S of S2 can be
measured simultaneously. L and S are good quantum
numbers to characterize the wavefunction of a
free atom ? see Chap 13 Atomic structure and
atomic spectra.
?2. The position x and the momentum px (along the
x axis). According to the correspondence
principles, the quantum operators are x? and
h/i(? /? x). The commutator can be calculated to
be
The consequence is a breakdown of the classical
mechanics laws if there is a complete certainty
about the position of the particle (?x0), then
there is a complete uncertainty about the
momentum (?px?).
?3. The time and the energy If a system stays
in a state during a time ?t, the energy of this
system cannot be determined more accurately than
with an error ?E. This incertitude is of major
importance for all spectroscopies ? see Chap 16,
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