„Old” and „new” quantum theoryadam.mech.pw.edu.pl/~marzan/L3_quantumphys.pdf · Wektory...

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FIZYKA 3 MICHAŁ MARZANTOWICZ „Old” and „new” quantu Faults of the Bohr model: - gives only position of the lines and no - does not explain the number of electr - gives innacurate results for atoms wi Correspondence law: The quantum description becomes a um theory ot the intensity rons on each orbit ith many electrons classic one for large quantum numbers.

Transcript of „Old” and „new” quantum theoryadam.mech.pw.edu.pl/~marzan/L3_quantumphys.pdf · Wektory...

Page 1: „Old” and „new” quantum theoryadam.mech.pw.edu.pl/~marzan/L3_quantumphys.pdf · Wektory orbitalny i spinowy sumują się. Quantum numbers ,⃗= . ,⃗+ 5 ...

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.

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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

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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.

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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

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FIZYKA 3 MICHAŁ MARZANTOWICZ

Heisenberg uncertainty principleHeisenberg uncertainty principle

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FIZYKA 3 MICHAŁ MARZANTOWICZ

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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

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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

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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”

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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

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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

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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

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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.

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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.

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- 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.

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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

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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

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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

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FIZYKA 3 MICHAŁ MARZANTOWICZ

How can we see quantum quantum numbers?

Stern-Gerlach experiment

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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

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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

Page 22: „Old” and „new” quantum theoryadam.mech.pw.edu.pl/~marzan/L3_quantumphys.pdf · Wektory orbitalny i spinowy sumują się. Quantum numbers ,⃗= . ,⃗+ 5 ...

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.

.

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Orbitals – shapes

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Orbitals - shapes

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The principles of occupationoccupation of states

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FIZYKA 3 MICHAŁ MARZANTOWICZ

The periodic table

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FIZYKA 3 MICHAŁ MARZANTOWICZ

The periodic table

Energy required for electron ionization (separationseparation from atom).