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FIZYKA 3 MICHAŁ MARZANTOWICZ
„Old” and „new” quantum theory
Faults of the Bohr model:
- gives only position of the lines and not the intensity- does not explain the number of electrons on each orbit- gives innacurate results for atoms with many electrons
Correspondence law:The quantum description becomes a classic one for large quantum numbers.
„Old” and „new” quantum theory
gives only position of the lines and not the intensitydoes not explain the number of electrons on each orbitgives innacurate results for atoms with many electrons
The quantum description becomes a classic one for large quantum numbers.
FIZYKA 3 MICHAŁ MARZANTOWICZ
The momentum of moving object corresponds to a wave
2hnrumL nnen nl=2r
The circumference of allowed orbit contains an integral number of de Broglie wavelengths
De Broglie hypothesis
The momentum of moving object corresponds to a wave
The circumference of allowed orbit contains an
FIZYKA 3 MICHAŁ MARZANTOWICZ
Wave properties of matter
Davisson-Germer experiment:wave properties of electrons
Thomson experiment: diffraction of electrons on thin polycrystalline foil
Stern experiment: diffraction of hydrogen and helium atoms on crystals of lithium fluoride and sodium fluoride.
Wave properties of matter
Thomson experiment: diffraction of electrons on thin polycrystalline foil
Stern experiment: diffraction of hydrogen and helium atoms on crystals of lithium fluoride and sodium fluoride.
FIZYKA 3 MICHAŁ MARZANTOWICZ
Heisenberg uncertainty principle
Momentu and position cannot be precisely determined at the same moment. The particle can occupy level with well defined energy for a long time. The „lifetime” of massive particles is limited
4hpx
Heisenberg uncertainty principle
Momentu and position cannot be precisely determined at the same
The particle can occupy level with well defined energy for a long time. The „lifetime” of massive particles is limited
4htE
FIZYKA 3 MICHAŁ MARZANTOWICZ
Heisenberg uncertainty principleHeisenberg uncertainty principle
FIZYKA 3 MICHAŁ MARZANTOWICZ
FIZYKA 3 MICHAŁ MARZANTOWICZ
Description of a wave
Wave vector (wavenumber)
Frequency and angular frequency
Wave equation(propagation along x axis, positive direction)
vxtf
2
22
2
2
xv
t
Differential equation
FIZYKA 3 MICHAŁ MARZANTOWICZ
The electron wavefunction
The wavefunction determines probability of finding the particle in a certain space (for a plane wave between x and x+dx) if the measurement was performed at time t
dxdxtxP *),(
The electron wavefunction
The wavefunction determines probability of finding the particle in a certain space (for a plane wave between x and x+dx) if the measurement was performed at time t
dx2 Density of probability
FIZYKA 3 MICHAŁ MARZANTOWICZ
Schrödinger equation
The wavefunctions are obtained by solving
Stationary (time independent ) equation:
Gradient
The potential in which the electron is
The wavefunctions – so called eigenfunctions of S.eq.
-finite-unequivocal-continuous The values of the functions and their first
derivatives are the same at the borders of areas with different potentials between these areas must be „smooth”
The wavefunctions are obtained by solving Schrödinger equation
The potential in which the electron is
Electron energy
so called eigenfunctions of S.eq. - have to be:
The electron exists
The values of the functions and their first derivatives are the same at the borders of areas with different potentials – the transition between these areas must be „smooth”
FIZYKA 3 MICHAŁ MARZANTOWICZ
Schrödinger equation –
E>V0
Reflection
I0
R
T Transmition
R+T=1
– potential step
III
V0
0 221
221
*1
*1
kkkk
AAvBBv
221
21*
1
*2 4
kkkk
AAvCCv
R+T=1
FIZYKA 3 MICHAŁ MARZANTOWICZ
Schrödinger equation –
E<V0 I
0
Area I
Area II
Classical:
Quantum
mEv 2
1 Area I
Electron penetrates area II, but the probability decreases exponentially
–potential step
I IIV0
V
0The electron does not penetrate area II
Electron penetrates area II, but the probability decreases exponentially
C=0, Exponential decrease
FIZYKA 3 MICHAŁ MARZANTOWICZ
Finite barrier
Electron can pass through, despite lower energy.The probability decreases with barrier width.
T2
exp
Electron can pass through, despite lower energy.The probability decreases with barrier width.
l
EVm
022
FIZYKA 3 MICHAŁ MARZANTOWICZ
Model of atom: infinite potential well
Inside the well:
Infinite well has „rigid” walls. The electron wavefunction must be a standing wave – the waves afre reflected on the walls.
Model of atom: infinite potential well
Inside the well:
The electron wavefunction must be a standing the waves afre reflected on the walls.
FIZYKA 3 MICHAŁ MARZANTOWICZ
Quantum atom model
Schrödinger equation is solved in spherical coordinates
The solution can be separated in respect to the coordinates.Each coordinate depends on integer number – so called quantum number.
dinger equation is solved in spherical coordinates
The solution can be separated in respect to
Each coordinate depends on integer
so called quantum number.
FIZYKA 3 MICHAŁ MARZANTOWICZ
- principal quantum number (n = 1,2,3...) corresponds to the number of orbit from Bohr’s model- azimuthal quantum number (l = 0,1,...,value of orbital angular momentum L (number of electron sub- magnetic quantum number (ml = − projection of orbital angular momentum onto given axis. - spin quantum number S corresponds to the electron spin and equals 1/2. For any elementary particle, this is a constant value.- magnetic spin q. n. (ms = − m,mof the spin
Wektory orbitalny i spinowy sumują się.
Quantum numbers
퐽⃗ = 퐿⃗ + 푆⃗
= 1,2,3...) corresponds to the
= 0,1,...,n − 1) corresponds to the value of orbital angular momentum L (number of electron sub-shell)
= − l,..., − 1,0,1,...,l) shows the projection of orbital angular momentum onto given axis.
corresponds to the electron spin and equals 1/2. For any elementary particle, this is a constant value.
= 1 / 2, − 1 / 2) shows the direction
Wektory orbitalny i spinowy sumują się.
FIZYKA 3 MICHAŁ MARZANTOWICZ
Spin and fine structure of emission lines
The movement of electron around the core generates a magnetic field. This field interacts with electron magnetic moment, causing increase or decrease of total electron energy (spinThe emission lines split into two lines (fine structure)
Hydrogen fine structure
Spin and fine structure of emission lines
The movement of electron around the core generates a magnetic field. This field interacts with electron magnetic moment, causing increase or decrease of total electron energy (spin-orbit interaction)The emission lines split into two lines (fine structure)
Sodium doublet
FIZYKA 3 MICHAŁ MARZANTOWICZ
How can we see the quantum numbers?
Electron energy depends mainly on theprincipal number n
The azimuthal q.n. is related to the angularmomentum of electrons
Einstein
How can we see the quantum numbers?
1 llLangular
2220
42 124
)(n
eZmnE e
Einstein – de Haas experiment
FIZYKA 3 MICHAŁ MARZANTOWICZ
How can we see quantum
The electron energy in magneticfield depends on magnetic q.n. m(Zeeman effect).A similar effect is observed instrong electric field – Stark effect
BmE lBp
Stern-Gerlach moment of moment of single cancelpossible
quantum numbers?
Gerlach experiment: the total magneticmoment of silver atom equals the spin magneticmoment of single electron µs (rest of the moments
each other). Two values of spin arepossible: +1/2 i –1/2
FIZYKA 3 MICHAŁ MARZANTOWICZ
How can we see quantum quantum numbers?
Stern-Gerlach experiment
FIZYKA 3 MICHAŁ MARZANTOWICZ
Electron shells
Electron shellsK,L,M,N,O,P,Q2n2 electrons
An electron shellrepresenting
shellsK,L,M,N,O,P,Q
electrons on a shell
shell is formed by all atomic orbitalsthe same principal number n
FIZYKA 3 MICHAŁ MARZANTOWICZ
Pauli exclusion: In an atom, two electronscharacterized by the same quantum
Principles of occupation
The occupation of energy levels minimizes
Sommerfeld effect: States with differentl are splitted. States with lower l are occupied
Electrons on elypticalwith low l can getatomic core, thancircular orbits withThe states withmay representthan those with
electrons cannot be numbers : n, l, ml, ms
occupation of energy levels
minimizes the potential energy.
different values of azimuthal q.noccupied first.
lyptical orbitsget closed to the
than those on with lower n.
with high n and low llower energies,
with low n and high l
FIZYKA 3 MICHAŁ MARZANTOWICZ
Space distribution of probability
Energy depends mainly on the principal The wave functions are dependent on all
Example of electron configuration
2 electrons with n=1,l=0
2 electrons with n=2,l=0
4 electrons with n=2,l=1
Degeneration: two or more wavefunctions
For each n, there is n different values of l. For each l there is 2l+1 values of m
probability: orbitals
principal number nall quantum numbers
configuration: 1s22s22p4
n=2,l=0
n=2,l=1
wavefunctions represent the same energy.
.
FIZYKA 3 MICHAŁ MARZANTOWICZ
Orbitals – shapes
FIZYKA 3 MICHAŁ MARZANTOWICZ
Orbitals - shapes
FIZYKA 3 MICHAŁ MARZANTOWICZ
FIZYKA 3 MICHAŁ MARZANTOWICZ
The principles of occupationoccupation of states
FIZYKA 3 MICHAŁ MARZANTOWICZ
The periodic table
FIZYKA 3 MICHAŁ MARZANTOWICZ
The periodic table
Energy required for electron ionization (separationseparation from atom).