Download Introduction to quantum mechanics

Introduction to
quantum mechanics
Lecture 3
MTX9100
Nanomaterjalid
OUTLINE
-What is electron – particle or wave?
- How large is a potential well?
-What happens at nanoscale?
What is inside?
Matter
Molecule
Atom
Nucleus
Baryon
Quark
(Hadron)
u
10-14m
10-9m
10-10m
10-2m
Condensed matter/Nano-Science/Chemistry
Atomic Physics
Nuclear
Physics
10-15m
protons, neutrons,
mesons, etc.
π,Ω,Λ...
<10-19m
top, bottom,
charm, strange,
up, down
Electron
(Lepton)
<10-18m
High Energy Physics
Quantum mechanics milestones – the
Bohr atomic model
the ground energy for
the hydrogen atom is
−13.6 eV
Each energy level or shell is represented
by the principal quantum number n
Electron energy states for a hydrogen atom
Quantum mechanics basics
The simplified Bohr atomic model, in which
electrons are assumed to revolve around the atomic
nucleus in discrete orbitals, and the position of any
particular electron is more or less well
defined in terms of its orbital.
If an atom is in one of the excited states E1,
E2, and so on, it does not remain in that
state forever. Sooner or later it drops to a
lower state and radiates energy in the form
of light.
The frequency of light υ that is liberated in a
transition, for example, from energy E3 to
energy E1,
h is Planck’s constant
The energies of electrons are
quantized;
electrons are permitted to have
only specific
values of energy.
Quantum mechanics milestones
Between 1900 and 1925 Quantum Physics
was developed by a number of physicists,
including Planck, Einstein, Bohr and de
Broglie.
From 1925 onwards a more mathematical
approach was developed by Schrödinger
(wave mechanics), Heisenberg (matrix
mechanics) and Dirac (who developed a
Werner Heisenberg and Erwin Schrödinger, founders
of Quantum Mechanics - Wikipedia
more general formulation).
If you are not confused by Quantum Physics then you
haven't really understood
it.
N.Bohr
Quantum mechanics basics (2)
Quantum mechanics is the study of mechanical systems whose
dimensions are close to the atomic scale. Quantum mechanics is a
fundamental branch of physics with wide applications. Quantum
theory generalizes classical mechanics to provide accurate
descriptions for many previously unexplained phenomena such as
black body radiation and stable electron orbits.
The effects of quantum mechanics become evident at the atomic and
subatomic level, and they are typically not observable on macroscopic
scales. - Wikipedia
What is quantum mechanics good for?
Atoms are governed by the laws of quantum mechanics, and
quantum mechanics is essential for an understanding of atomic
physics.
The interactions between atoms are governed by quantum
mechanics, and so an understanding of quantum mechanics is a
prerequisite for understanding the science
This information enables us to calculate the
average value of the measurement of a physical
variable.
Quantum mechanics does not explain how a quantum
particle behaves.
Instead, it gives a recipe for determining the probability
of the measurement of the value of a physical variable
(e.g. energy, position or momentum).
Wave-particle duality
In physics and chemistry,
wave–particle duality
is the concept that all matter and
energy exhibits both wave-like
and particle-like properties.
Particles are waves,
waves are particles.
In quantum mechanics,
the motion of particles is described with
probabilities.
Probability distribution of the Bohr atom –
Wikipedia
8
Exclusion principle
The exclusion principle says that two electrons
cannot get into exactly the same energy state.
In other words, it is
not possible for two electrons to have the same momentum, be
at the same location, and spin in the same direction.
How many quantum numbers?
Using wave mechanics, every electron in an atom is characterized
by four parameters called quantum numbers.
The Bohr model was a one-dimensional model that used one
quantum number to describe the distribution of electrons in the atom.
The only information that was important was the size of the orbit,
which was described by the n quantum number.
Schrödinger's model allowed the electron to occupy
three-dimensional space.
It therefore required three coordinates, or three quantum
numbers, to describe the orbitals in which electrons can be found.
The three coordinates that come from Schrödinger's wave equations are the principal (n),
angular (l), and magnetic (m) quantum numbers.
These quantum numbers describe the size, shape, and orientation in space of the orbitals on an
atom.
10
Quantum numbers
The principal quantum number (n) describes the size (distance of an electron
from the nucleus) of the orbital.
Orbitals for which n = 2 are larger than those for which n = 1.
The principal quantum number therefore indirectly describes the energy of an
orbital.
The angular quantum number (l) describes the shape of the orbital.
Orbitals have shapes that are best described as spherical (l = 0), polar (l = 1), or
cloverleaf (l = 2). They can even take on more complex shapes as the value of the
angular quantum number becomes larger.
A third quantum number, known as the magnetic quantum number (m), describes
the orientation in space of a particular orbital and shows the energy states in sub-shells.
11
Shells and sub-shells of orbitals
Orbitals that have the same value of the
principal quantum number form a shell.
Orbitals within a shell are divided into
subshells that have the same value of the
angular quantum number.
Bohr and (b)
wavemechanical
atom models in
terms of electron
distribution.
12
Sub-shells
•s, p, d and f signify the
subshells which the
electrons occupy.
• Different types of subshells
have different
numbers of energy states
•Within each energy state
there are two
possible spin orientations
13
Schematic
representation of the relative
energies of the electrons for the
various shells and subshells
Electron states
14
Electron configurations
15
Chemical Bonding Between Atoms
• Electron states is controlling factor for atomic bonding
• Types of primary (strong) bonds: ionic, covalent, metallic
• Types of secondary (weak) bonds: van der Waals, hydrogen
• Properties that are controlled by interatomic potentials: melting point,
bond stiffness, thermal expansion coefficient
16
Bonding forces
Many properties of materials are determined by the interatomic forces that bind the atoms
together.
Equilibrium spacing
These forces are of two types, attractive and repulsive, and the magnitude of each is a function
of the separation or interatomic
distance.
17
Energy of bonding
Bonding
energy
18
Once in this position, the two atoms
will counteract
any attempt to separate them by an
attractive force, or to push them
together by
a repulsive action.
This typical curve has a
minimum at equilibrium
distance R0
R > R0 ;
the potential increases
gradually, approaching 0
as R∞
the force is attractive
R < R 0;
the potential increases
very rapidly, approaching
∞ at small separation.
the force is repulsive
V(R)
Repulsive
0
R0
Attractive
r
R
Force between the atoms is the negative of the slope of this curve. At equlibrium, repulsive force
becomes equals to the attractive part.
R
Heinserberg Uncertainty Principle
This idea claims that we are
We can only say that there
not allowed to know
is a probability that a
simultaneously the definite
particle will have a position
location and the definite speed
near some coordinate x.
of a particle.
The laws of motion for a
quantum particle have to be
framed in such a way that lets
us make predictions only for
the uncertainty in position, x, and
the uncertainty in momentum, p,
quantities that are the average
of many individual
measurements.
The probability that a particle is at a certain point in
space and time is given by the square of a complex number
called the probability amplitude, ψ - wavefunction
Wavefunction
(a) Si is in Group IV in the Periodic Table. An isolated Si atom has two
electrons in the 3s and two electrons in the 3p orbitals.
(b) When Si is about to bond, the one 3s orbital and the three 3p orbitals
become perturbed and mixed to form four hybridized orbitals, ψhyb, called
sp3 orbitals, which are directed toward the corners of a tetrahedron. The
ψhyb orbital has a large major lobe and a small back lobe. Each ψhyb orbital
takes one of the four valence electrons.
Probability
In quantum
mechanics the
probability of
finding a particle at
a certain position at
time t,
P(x,y,z,t),
is the square of the
wave function.
Probability densities for the electron
of a hydrogen atom in different
quantum states.-Wikipedia
Wavefunctions and Probabilities
Summary
The quantum state of a particle is characterized by a wave function,
which contains all the information about the system an observer can
possibly obtain.
The wave function is interpreted as a
probability amplitude of the particles presence.
|Ψ(r,t)|2 is the probability density.
For a single particle the total probability of finding it anywhere in
space at time t is equal to 1.
A proper wave function must be square-integrable.
The Schrödinger equation - Intro
The Schrodinger equation plays the role of Newton's
laws and conservation of energy in classical mechanics
- i.e., it predicts the future behavior
of a dynamic system.
It is a wave equation in terms of the wavefunction
which predicts analytically and precisely the
probability of events or outcome.
The detailed outcome is not strictly determined, but
given a large number of events, the Schrodinger
equation will predict the distribution of results.
The Schrödinger equation - basics
Schrodinger equation
The Schrodinger equation is a differential equation that
describes the time evolution of
Kinetic energy of a free particle
For a particle moving in a potential V(x,t)
Schrodinger equation
Planck’s constant
h
= 1 .05459 × 10 − 34 Js
h=
2π
 h2 2

∂
 −
∇ + V (r , t ) ψ (r , t ) = ih ψ (r , t )
∂t
 2m

wave function
Time-independent Schrödinger equation
values of E constitute the allowed energies
H is the Hamiltonian operator
The word “operator” means that it is a
mathematical set of operations to be
carried out on a function placed to its
right in this case
Solution of the time-independent
Schrodinger equation
This is the quantum number of the state.
Application of Schrodinger equation
Summary