Download wave

Wave Phenomena
Reflexion
Refraction
Diffraction is the bending
of a wave around an
obstacle or through an
opening.
qi
qi = qr
Interference
Diffraction
n1
qt
Wavelenght
dependence
Diffraction at Slits
n2
n1 sin (qi) = n2 sin (qt) w
q
p=w sin(q)=ml
bright fringes
Diffraction at a lattice
q
p=w sin(q)=ml
d
The path difference
must be a multiple
of a wavelength to
insure constructive
interference.
q
p=d sin(q)=ml
bright fringes
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
The Wave Nature of Matter
All material particles are
associated with Waves
(„Matter waves“
De Broglie
A central concept of Quantics:
wave–particle duality is the
concept that all matter and
energy exhibits both wave like and particle -like
properties.
E = hn
mc2 = hn = hc/l
E = mc2
or: mc = h/l
A normal particle with nonzero rest
mass m travelling at velocity v
Then, every particle with nonzero
rest mass m travelling at velocity v
has an related wave l
mv = p = h/l
l = h/ mv
1. The particle property is caused by their mass.
2. The wave property is related with particles' electrical charges.
3. Particle-wave duality is the combination of classical mechanics and electromagnetic field
theory.
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
Schrödinger's cat
It is a „Gedanken“ (thought experiment) often described as a paradox
I don‘t like it and I regret
that I got involved in it….
Superposition of two States:
Broadly stated, a quantum superposition is
the combination of all the possible states of a
system.
yAlife+ yDead
Schrödinger
Miau
!
yAlife
yDead
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
Schrödinger wrote:
„One can even set up quite ridiculous cases. A cat is penned up in a steel
chamber, along with the following device (which must be secured against direct
interference by the cat): in a Geiger counter there is a tiny bit of radioactive
substance, so small, that perhaps in the course of the hour one of the atoms
decays, but also, with equal probability, perhaps none; if it happens, the counter
tube discharges and through a relay releases a hammer which shatters a small
flask of hydrocyanic acid. If one has left this entire system to itself for an hour,
one would say that the cat still lives if meanwhile no atom has decayed. The Yfunction of the entire system would express this by having in it the living and dead
cat (pardon the expression) mixed or smeared out in equal parts.
It is typical of these cases that an indeterminacy originally restricted to the atomic
domain becomes transformed into macroscopic indeterminacy, which can then be
resolved by direct observation. That prevents us from so naively accepting as
valid a "blurred model" for representing reality. In itself it would not embody
anything unclear or contradictory. There is a difference between a shaky or outof-focus photograph and a snapshot of clouds and fog banks.“
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
Schrödinger's famous tought experiment poses the question: when does a
quantum system stop existing as a mixture of states and become one or the
other? (More technically, when does the actual quantum state stop being a linear
combination of states, each of which resemble different classical states, and
instead begin to have a unique classical description?) If the cat survives, it
remembers only being alive. But explanations of experiments that are consistent
with standard microscopic quantum mechanics require that macroscopic objects,
such as cats and notebooks, do not always have unique classical descriptions.
The purpose of the thought experiment is to illustrate this apparent paradox: our
intuition says that no observer can be in a mixture of states, yet it seems cats, for
example, can be such a mixture. Are cats required to be observers, or does their
existence in a single well-defined classical state require another external
observer?
An interpretation of quantum mechanics. A key feature of quantum
mechanics is that the state of every particle is described by a wavefunction,
which is a mathematical representation used to calculate the probability for it to
be found in a location, or state of motion. In effect, the act of measurement
causes the calculated set of probabilities to "collapse" to the value defined by the
measurement. This feature of the mathematical representations is known as
wave function collapse.
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
The Waves and the Incertainty Principle of Heisenberger
„The measurement of particle position
leads to loss of knowledge about particle
momentum and visceversa.“
y
v
Dy
p
q
m
2p sin (q) =
Dpy=2pl/Dy
sin (q) = ±l/Dy
Dpy . Dy ≈
2pl = 2h
x
The momentum of the incoming beam is all in the x direction. But as a result of
diffraction at the slit, the diffracted beam has momentum p with components on
both x and y directions.
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
The Rydberg formula is used
in atomic physics to describe
the wavelengths of spectral
lines of many chemical
elements.
Rydberg
Continuous spectrum emission lines
absorptiom lines
Rydberg formula for hydrogen
Where
λvac is the wavelength of the light emitted in vacuum,
RH is the Rydberg constant for hydrogen,
n1 and n2 are integers such that n1 < n2
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
Bohr‘s Atom Model
Bohr
He suggested that electrons could only have certain classical motions:
1.The electrons can only travel in special orbits: at a certain discrete set of
distances from the nucleus with specific energies.
2.The electrons do not continuously lose energy as they travel. They can only
gain and lose energy by jumping from one allowed orbit to another, absorbing or
emitting electromagnetic radiation with a frequency n determined by the energy
difference ΔE = E2 − E1 = hn of the levels according to Bohr's formula where h is
Planck‘s Constant.
3.the frequency of the radiation emitted at an orbit with period T is as it would be
in classical mechanics--- it is the reciprocal of the classical orbit period:
n = 1/T = c/l
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
The angular momentum A is restricted to be an integer multiple of a fixed
unit:
A= n h / 2p = mvr
[1]
where n = 1,2,3,… and is called the principal quantum number. The lowest
value of n is 1. This gives a smallest possible orbital radius of 0.0529 nm.
This is known as the Bohr radius. Once an electron is in this lowest orbit, it
can get no closer to the proton. Starting from the angular momentum
quantum rule Bohr was able to calculate the energies of the allowed orbits of
the hydrogen atom and other hydrogenlyke atoms and ions.
Bohr's condition, that the angular momentum is an integer multiple of h/2p
was later reinterpreted by de Broglie as a standing wave condition:
n l =2p r = L [2]
the electron is described by a wave and a whole number of wavelengths
must fit along the circumference of the electron's orbit.
X=0
sin(px/L)
sin(2px/L)
sin(3px/L)
X=L
1,0
Y Axis Title
0,5
0,0
-0,5
-1,0
0
X Axis Title
L
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
To calculate the orbits requires two assumptions:
1. Classical mechanics
The electron is held in a circular orbit by electrostatic attraction. The
centripetal force is equal to the Coulom force:
mv2/r = ke2 /r2
[3]
where m is the mass and e is the charge of the electron. This determines the
speed at any radius:
v=√(ke2 /mr)
[4]
It also determines the total energy at any radius:
E=mv2/2 - ke2 /r = -ke2 /2r
[5]
The total energy is negative and inversely proportional to r. This means that
it takes energy to pull the orbiting electron away from the proton. For infinite
values of r, the energy is zero, corresponding to a motionless electron
infinitely far from the proton. For larger nuclei, the only change is that ke2 is
everywhere replaced by Z ke2 where Z is the number of protons.
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
2. . Quantum rule
Substituting the expression for the velocity [4] in the expression for the
angular momentum [1] gives an equation for r in terms of n:
√(ke2mr) = nh/2p
[6]
so that the allowed orbit radius at any n is:
r = n2 h2 / (4p2 ke2 m) [7]
The smallest possible value of r is 0.51 x 10-10m (n=1) is called the Bohr
radius The energy of the n-th level is determined by the radius (replacing
[7] in [5] :
E= (2pke2) 2 m/(2n2 h2 )
The combination of natural constants in the energy formula is called the Rydberg
energy RH (n=1):
RH = (2pke2) 2 m/(2h2 )
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
Einstein
Bohr
Heisenberg
Planck
Photons
electrons
de Broglie
Schrödinger
Energy
Waves
Uncertainty
diffraction
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
The book of nature you can only
understand, if you have
previously learnt its language
and the letters. It is written in
mathematical language and the
letters are geometrical figures,
and without these means it is
impossible for human beings to
understand even a word of it.
Galileo Galilei, 16th century
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
Dirac Notation, Hilbert Space
The mathematical concept of a Hilbert space,
generalizes the notion of Euclidean space. It
extends the methods of vector algebra from the
two-dimensional plane and three-dimensional
space to infinite-dimensional spaces.
Dirac
Hilbert
Mathematically, a pure quantum state is typically represented by a vector in a
Hilbert space.
Bra-ket notation is a standard notation for describing quantum states
composed of angle brackets and vertical bars. It can also be used to denote
abstract vectors. It is so called because the inner product of two states is
denoted by a bracket,
, consisting of a left part,
, called the bra, and a
right part,
, called the ket. The notation was invented by Paul Dirac and is
also known as Dirac notation.
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
Operators
An Operator transforms the original vector |u> in general by changing its
amount (magnitude) and its direction into a new vector
Eigenvalue: The Operator transforms the original vector |u> in particular by
changing its amount (magnitude) (elongation or compression) but mantaining
its direction into a new vector
Hermitian Operator: which fulfil the condition: (always real Eigenvalues)
Oˆ u u = u Oˆ u
If in addition an Hermitian Operator also satisfies the condition:
Oˆ = Oˆ *
Adjugate
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
Conclusions, Summary, Some of more important Postulates
•Duality Behaviour: Wavelike ↔ Particlelike: Associated with any particle is a
wavefunction having wavelength related to a particle momentum by: l = h/p = h/√(2m(E-V)) (de Broglie)
•Wave Function:y: its absolute square is proportional to the probability density for finding the
particle. It describes a state as completely as possible.
First Postulate
y ,
y
•Wave Function:y: it is an eigenfunction of Schrödinger‘s Equation, which can be
constructed from the classical wave equation requiring: l = h/p = h/√(2m(E-V))
•Wave Function:y: to be acceptable: it must be single-valued, continuous, nowhere infinite, with
a piecewise continuous first derivative, square-integrable.
•Wave Function:y - Normalization:

*
y
 y dV = 1 = y y
-
•Operator:Ô: For any Observable there is an operator which is constructed from the classical
expression according to a simple recipe. Second
Postulate
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
•Time-Dependent Schrödinger Equation: The
satisfy the equation:
Hˆ ψ ( x, y, z, t ) =
State functions or Wave functions
-h 
ψ ( x, y, z, t ) Hˆ y = E y
2pi t
•Both, Hamiltonian and Wave functions are time-dependent. Third
Postulate
• If the Hamiltonian Operator for a system is time.independent, stationary eingenfunctions exist of the
form:
y ( x, y, z,  ) exp( -2piEt / h)
The time-dependent exponential does not affect the measurable properties of a system in this state and is
almost always completely ignored in any time-independent problem.
•Operator:Ô: The Eigenvalues for such an Operator are the possible values we can measure for
that quantity. Fourth
Postulate
•Operator:Ô: The act of measuring the quantity forces the system into a state described by an
Eigenfunction of the Operator
•Expectation Value m of an Operator M:
Fifth Postulate

m = y * Mˆ y dV, = y Mˆ y
-
It is equivalent to the arithmetic average of all the possible measured values of a property times their
frequency of occurrence.
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
•Completeness of Eigenfunctions of an Operator M: Sixth Postulate
The Eigenfunctions for any quantum mechanical operator corresponding
to an observable variable constitute a complete set.
•Briefly, a series of functions {f } having certain restrictions is said to be complete if an arbitrary
function y having the same restrictions can be expressed in terms of the series:
•
y =  cifi , y =  ci fi
i
i
•Time-Dependent Schrödinger Equation:
Hˆ ψ ( x, y, z, t ) =
-h 
ψ ( x, y, z , t ) Hˆ y = E y
2pi t
•Time-independent Schrödinger Equation:
2



Hˆ ψ( x, y, z ) = 
D  Vˆ  ψ( x, y, z ) Hˆ y = E y
 2m

PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves