Introduction to Quantum Mechanic

Сентябрь 6, 2022

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Introduction to Quantum Mechanic

Содержание

2. When you find this image, you may skip this part This is less important
3. The idea of duality is rooted in a debate over the nature of light and matter
4. Radiations, terminology
5. Interferences in
6. Phase speed or velocity
7. Introducing new variables At the moment, let consider this just a formal change, introducing and we
8. Introducing new variables At the moment, h is a simple constant Later on, h will have
9. 2 different velocities, v and vϕ
10. If h is the Planck constant J.s Then Louis de BROGLIE French (1892-1987) Max Planck (1901)
11. Robert Millikan (1910) showed that it was quantified. Rutherford (1911) showed that the negative part was
12. Gustav Kirchhoff (1860). The light emitted by a black body is called black-body radiation] black-body radiation
13. black-body radiation Classical Theory Fragmentation of the surface. One large area (Small λ Large ν) smaller
14. Kirchhoff black-body radiation RED WHITE Small ν Large ν Shift of ν Radiation is emitted when
15. black-body radiation Max Planck (1901) Göttingen Why a decrease for small λ ? Quantification Numbering rungs
16. Quantum numbers In mathematics, a natural number (also called counting number) has two main purposes: they
17. black-body radiation Max Planck (1901) Göttingen Why a decrease for small λ ? Quantification
18. black-body radiation, quantification Max Planck Steps too hard to climb Easy slope, ramp Pyramid nowadays Pyramid
19. Max Planck
20. Johannes Rydberg 1888 Swedish Atomic Spectroscopy Absorption or Emission
21. Johannes Rydberg 1888 Swedish IR VISIBLE UV Atomic Spectroscopy Absorption or Emission Emission -R/12 -R/22 -R/32
22. Photoelectric Effect (1887-1905) discovered by Hertz in 1887 and explained in 1905 by Einstein. Heinrich HERTZ
23. Kinetic energy
24. Compton effect 1923 playing billiards assuming λ=h/p Arthur Holly Compton American 1892-1962
25. Davisson and Germer 1925 Clinton Davisson Lester Germer In 1927 Diffraction is similarly observed using a
26. Wave-particle Equivalence. Compton Effect (1923). Electron Diffraction Davisson and Germer (1925) Young's Double Slit Experiment In
27. Thomas Young 1773 – 1829 English, was born into a family of Quakers. At age 2,
28. Young's Double Slit Experiment Screen Mask with 2 slits
29. Young's Double Slit Experiment This is a typical experiment showing the wave nature of light and
30. Young's Double Slit Experiment Assuming a single electron each time What means interference with itself ?
31. Young's Double Slit Experiment There is no possibility of knowing through which split the photon went!
32. Macroscopic world: A basket of cherries Many of them (identical) We can see them and taste
33. Slot machine “one-arm bandit” After introducing a coin, you have 0 coin or X coins. A
34. de Broglie relation from relativity Popular expressions of relativity: m0 is the mass at rest, m
35. de Broglie relation from relativity Application to a photon (m0=0) To remember To remember
36. Max Planck Useful to remember to relate energy and wavelength
37. A New mathematical tool: Wave functions and Operators Each particle may be described by a wave
38. Wave functions describing one particle To represent a single particle Ψ(x,y,z) that does not evolve in
39. Operators associated to physical quantities We cannot use functions (otherwise we would end with classical mechanics)
40. Slot machine (one-arm bandit) Introducing a coin, you have 0 coin or X coins. A measure
41. Examples of operators in mathematics : P parity Even function : no change after x →
42. Examples of operators in mathematics : A y is an eigenvector; the eigenvalue is -1
43. Linearity The operators are linear: O (aΨ1+ bΨ1) = O (aΨ1 ) + O( bΨ1)
44. Normalization An eigenfunction remains an eigenfunction when multiplied by a constant O(λΨ)= o(λΨ) thus it is
45. Mean value If Ψ1 and Ψ2 are associated with the same eigenvalue o: O(aΨ1 +bΨ2)=o(aΨ1 +bΨ2)
46. Sum, product and commutation of operators (A+B)Ψ=AΨ+BΨ (AB)Ψ=A(BΨ) operators wavefunctions eigenvalues
47. Sum, product and commutation of operators not compatible operators [A,C]=AC-CA≠0 [A,B]=AB-BA=0 [B,C]=BC-CB=0
48. Compatibility, incompatibility of operators not compatible operators [A,C]=AC-CA≠0 [A,B]=AB-BA=0 [B,C]=BC-CB=0 When operators commute, the physical quantities
49. x and d/dx do not commute, are incompatible Translation and inversion do not commute, are incompatible
50. Introducing new variables Now it is time to give a physical meaning. p is the momentum,
51. Plane waves This represents a (monochromatic) beam, a continuous flow of particles with the same velocity
52. Niels Henrik David Bohr Danish 1885-1962 Correspondence principle 1913/1920 For every physical quantity one can define
53. Operators p and H We use the expression of the plane wave which allows defining exactly
54. Momentum and Energy Operators Remember during this chapter
55. Stationary state E=constant Remember for 3 slides after
56. Kinetic energy Classical quantum operator In 3D : Calling the laplacian Pierre Simon, Marquis de Laplace
57. Correspondence principle angular momentum Classical expression Quantum expression lZ= xpy-ypx
62. Erwin Rudolf Josef Alexander Schrödinger Austrian 1887 –1961 Without potential E = T With potential E
63. Schrödinger Equation for stationary states Kinetic energy Total energy Potential energy
64. Schrödinger Equation for stationary states H is the hamiltonian Sir William Rowan Hamilton Irish 1805-1865 Half
65. Chemistry is nothing but an application of Schrödinger Equation (Dirac) Paul Adrien Dirac 1902 – 1984
66. Uncertainty principle the Heisenberg uncertainty principle states that locating a particle in a small region of
67. It is not surprising to find that quantum mechanics does not predict the position of an
68. p and x do not commute and are incompatible For a plane wave, p is known
69. Superposition of two waves Δx/2 Δx/(2x(√2π)) Factor 1/2π a more realistic localization
71. Скачать презентацию

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When you find this image, you may skip this part

This is less important

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The idea of duality is rooted in a debate over the

nature of light and matter dating back to the 1600s, when competing theories of light were proposed by Huygens and Newton.

Christiaan Huygens
Dutch 1629-1695
light consists of waves

Sir Isaac Newton
1643 1727
light consists of particles

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Radiations, terminology

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Interferences
in

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Phase speed or velocity

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Introducing new variables
At the moment, let consider this just a formal

change, introducing
and
we obtain

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Introducing new variables
At the moment, h is a simple constant
Later on,

h will have a dimension and the p and E will be physical quantities
Then

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2 different velocities, v and vϕ

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If h is the Planck constant J.s
Then
Louis de BROGLIE
French
(1892-1987)
Max Planck

(1901)
Göttingen

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Robert Millikan (1910) showed that it was quantified.
Rutherford (1911) showed

that the negative part was diffuse while the positive part was concentrated.

Soon after the electron discovery in 1887

- J. J. Thomson (1887) Some negative part could be extracted from the atoms

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Gustav Kirchhoff (1860). The light emitted by a black body is

called black-body radiation]

black-body radiation

At room temperature, black bodies emit IR light, but as the temperature increases past a few hundred degrees Celsius, black bodies start to emit at visible wavelengths, from red, through orange, yellow, and white before ending up at blue, beyond which the emission includes increasing amounts of UV

RED WHITE
Small ν Large ν

Shift of ν

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black-body radiation
Classical Theory
Fragmentation of the surface. One large area (Small λ Large

ν) smaller pieces (Large λ Small ν)
Vibrations associated to the size, N2 or N3

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Kirchhoff
black-body radiation
RED WHITE
Small ν Large ν
Shift of ν
Radiation is

emitted when a solid after receiving energy goes back to the most stable state (ground state). The energy associated with the radiation is the difference in energy between these 2 states. When T increases, the average E*Mean is higher and intensity increases.
E*Mean- E = kT.
k is Boltzmann constant (k= 1.38 10-23 Joules K-1).

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black-body radiation
Max Planck (1901)
Göttingen
Why a decrease for small λ ? Quantification
Numbering

rungs of ladder introduces quantum numbers (here equally spaced)

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Quantum numbers
In mathematics, a natural number (also called counting number) has

two main purposes: they can be used for counting ("there are 6 apples on the table"), and they can be used for ordering ("this is the 3rd largest city in the country").

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black-body radiation
Max Planck (1901)
Göttingen
Why a decrease for small λ ? Quantification

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black-body radiation,
quantification
Max Planck
Steps too hard to climb Easy slope, ramp
Pyramid

nowadays Pyramid under construction

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Max Planck

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Johannes Rydberg 1888
Swedish
Atomic Spectroscopy
Absorption or Emission

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Johannes Rydberg 1888
Swedish
IR
VISIBLE
UV
Atomic Spectroscopy
Absorption or Emission
Emission
-R/12
-R/22
-R/32
-R/42
-R/52
-R/62
-R/72
Quantum numbers n, levels are

not equally spaced R = 13.6 eV

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Photoelectric Effect (1887-1905)
discovered by Hertz in 1887 and explained in 1905

by Einstein.

Heinrich HERTZ
(1857-1894)

Albert EINSTEIN
(1879-1955)

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Kinetic energy

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Compton effect 1923 playing billiards assuming λ=h/p
Arthur Holly Compton
American
1892-1962

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$Davisson and Germer 1925 Clinton Davisson Lester Germer In 1927 Diffraction$

Davisson and Germer 1925
Clinton Davisson
Lester Germer
In 1927
Diffraction is similarly observed using

a mono-energetic electron beam
Bragg law is verified assuming λ=h/p

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$Wave-particle Equivalence. Compton Effect (1923). Electron Diffraction Davisson and Germer (1925)$

Wave-particle Equivalence.
Compton Effect (1923).
Electron Diffraction Davisson and Germer (1925)
Young's Double Slit

Experiment

In physics and chemistry, wave–particle duality is the concept that all matter and energy exhibits both wave-like and particle-like properties. A central concept of quantum mechanics, duality, addresses the inadequacy of classical concepts like "particle" and "wave" in fully describing the behavior of small-scale objects. Various interpretations of quantum mechanics attempt to explain this apparent paradox.

Wave–particle duality

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Thomas Young 1773 – 1829
English, was born into a family

of Quakers.
At age 2, he could read.
At 7, he learned Latin, Greek and maths.
At 12, he spoke Hebrew, Persian and could handle optical instruments.
At 14, he spoke Arabic, French, Italian and Spanish, and soon the Chaldean Syriac. "…
He is a PhD to 20 years "gentleman, accomplished flute player and minstrel (troubadour). He is reported dancing above a rope."
He worked for an insurance company, continuing research into the structure of the retina, astigmatism ...
He is the rival Champollion to decipher hieroglyphics.
He is the first to read the names of Ptolemy and Cleopatra which led him to propose a first alphabet of hieroglyphic scriptures (12 characters).

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Young's Double Slit Experiment
Screen
Mask with 2 slits

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Young's Double Slit Experiment
This is a typical experiment showing the wave

nature of light and interferences.
What happens when we decrease the light intensity ?
If radiation = particles, individual photons reach one spot and there will be no interferences
If radiation ≠ particles there will be no spots on the screen
The result is ambiguous
There are spots
The superposition of all the impacts make interferences

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Young's Double Slit Experiment
Assuming a single electron each time
What means interference

with itself ?
What is its trajectory?
If it goes through F1, it should ignore the presence of F2

Screen

Mask with 2 slits

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Young's Double Slit Experiment
There is no possibility of knowing through which

split the photon went!
If we measure the crossing through F1, we have to place a screen behind.
Then it does not go to the final screen.
We know that it goes through F1 but we do not know where it would go after.
These two questions are not compatible

Screen

Mask with 2 slits

Two important differences with classical physics:
measurement is not independent from observer
trajectories are not defined; hν goes through F1 and F2 both! or through them with equal probabilities!

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Macroscopic world:
A basket of cherries
Many of them (identical)
We can see

them and taste others
Taking one has negligible effect
Cherries are both red and good

Microscopic world:
A single cherry
Either we look at it without eating
It is red
Or we eat it, it is good
You can not try both at the same time
The cherry could not be good and red at the same time

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Slot machine “one-arm bandit”
After introducing a coin, you have 0 coin

or X coins.
A measure of the profit has been made: profit = X

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de Broglie relation from relativity
Popular expressions of relativity:
m0 is the mass

at rest, m in motion

E like to express E(m) as E(p) with p=mv

Ei + T + Erelativistic + ….

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de Broglie relation from relativity
Application to a photon (m0=0)
To remember
To remember

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Max Planck
Useful to remember to relate energy
and wavelength

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A New mathematical tool:
Wave functions and Operators
Each particle may

be described by a wave function Ψ(x,y,z,t), real or complex,
having a single value when position (x,y,z) and time (t) are defined.
If it is not time-dependent, it is called stationary.
The expression Ψ=Aei(pr-Et) does not represent one molecule but a flow of particles: a plane wave

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Wave functions describing one particle
To represent a single particle Ψ(x,y,z) that

does not evolve in time, Ψ(x,y,z) must be finite (0 at ∞).
In QM, a particle is not localized but has a probability to be in a given volume:
dP= Ψ* Ψ dV is the probability of finding the particle in the volume dV.
Around one point in space, the density of probability is dP/dV= Ψ* Ψ
Ψ has the dimension of L-1/3
Integration in the whole space should give one
Ψ is said to be normalized.

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Operators associated to physical quantities
We cannot use functions (otherwise we would

end with classical mechanics)
Any physical quantity is associated with an operator. An operator O is “the recipe to transform Ψ into Ψ’ ”
We write: O Ψ = Ψ’
If O Ψ = oΨ (o is a number, meaning that O does not modify Ψ, just a scaling factor), we say that Ψ is an eigenfunction of O and o is the eigenvalue. We have solved the wave equation O Ψ = oΨ by finding simultaneously Ψ and o that satisfy the equation.
o is the measure of O for the particle in the state described by Ψ.

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Slot machine (one-arm bandit)
Introducing a coin, you have 0 coin or

X coins.
A measure of the profit has been made: profit = X

O is a Vending machine (cans)
Introducing a coin, you get one can.
No measure of the gain is made unless you sell the can (return to coins)

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Examples of operators in mathematics : P parity
Even function : no

change after x → -x
Odd function : f changes sign after x → -x
y=x2 is even
y=x3 is odd
y= x2 + x3 has no parity: P(x2 + x3) = x2 - x3

Pf(x) = f(-x)

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Examples of operators in mathematics : A
y is an eigenvector; the

eigenvalue is -1

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Linearity
The operators are linear:
O (aΨ1+ bΨ1) = O (aΨ1 ) +

O( bΨ1)

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Normalization
An eigenfunction remains an eigenfunction when multiplied by a constant
O(λΨ)= o(λΨ)

thus it is always possible to normalize a finite function

Dirac notations <ΨIΨ>

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Mean value
If Ψ1 and Ψ2 are associated with the same eigenvalue

o: O(aΨ1 +bΨ2)=o(aΨ1 +bΨ2)
If not O(aΨ1 +bΨ2)=o1(aΨ1 )+o2(bΨ2)
we define ō = (a2o1+b2o2)/(a2+b2)

Dirac notations

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Sum, product and commutation of operators
(A+B)Ψ=AΨ+BΨ (AB)Ψ=A(BΨ)
operators
wavefunctions
eigenvalues

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Sum, product and commutation of operators
not compatible
operators
[A,C]=AC-CA≠0
[A,B]=AB-BA=0
[B,C]=BC-CB=0

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Compatibility, incompatibility of operators
not compatible
operators
[A,C]=AC-CA≠0
[A,B]=AB-BA=0
[B,C]=BC-CB=0
When operators commute, the physical quantities may

be simultaneously defined (compatibility)
When operators do not commute, the physical quantities can not be simultaneously defined (incompatibility)

compatible
operators

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x and d/dx do not commute, are incompatible
Translation and inversion do

not commute, are incompatible

Translation vector

Inversion center

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Introducing new variables
Now it is time to give a physical meaning.
p

is the momentum, E is the Energy
H=6.62 10-34 J.s

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Plane waves
This represents a (monochromatic) beam, a continuous flow of particles

with the same velocity (monokinetic).
k, λ, ω, ν, p and E are perfectly defined
R (position) and t (time) are not defined.
ΨΨ*=A2=constant everywhere; there is no localization.
If E=constant, this is a stationary state, independent of t which is not defined.

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Niels Henrik David Bohr
Danish
1885-1962
Correspondence principle 1913/1920
For every physical quantity

one can define an operator. The definition uses formulae from classical physics replacing quantities involved by the corresponding operators

QM is then built from classical physics in spite of demonstrating its limits

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Operators p and H
We use the expression of the plane wave

which allows defining exactly p and E.

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Momentum and Energy Operators
Remember during this chapter

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Stationary state E=constant
Remember for 3 slides after

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Kinetic energy
Classical quantum operator
In 3D :
Calling the laplacian
Pierre Simon,

Marquis de Laplace
(1749 -1827)

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Correspondence principle angular momentum
Classical expression Quantum expression
lZ= xpy-ypx

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Erwin Rudolf Josef Alexander Schrödinger
Austrian
1887 –1961
Without potential E =

T
With potential E = T + V

Time-dependent Schrödinger Equation

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Schrödinger Equation for stationary states
Kinetic energy
Total energy
Potential energy

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Schrödinger Equation for stationary states
H is the hamiltonian
Sir William Rowan Hamilton

Irish 1805-1865

Half penny bridge in Dublin

Remember

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Chemistry is nothing but an application of Schrödinger Equation (Dirac)
Paul Adrien

Dirac 1902 – 1984
Dirac’s mother was British and his father was Swiss.

< ΨI Ψ> <Ψ IOI Ψ >
Dirac notations

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Uncertainty principle
the Heisenberg uncertainty principle states that locating a particle

in a small region of space makes the momentum of the particle uncertain; and conversely, that measuring the momentum of a particle precisely makes the position uncertain
We already have seen incompatible operators

Werner Heisenberg
German
1901-1976

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It is not surprising to find that quantum mechanics does not

predict the position of an electron exactly. Rather, it provides only a probability as to where the electron will be found.
We shall illustrate the probability aspect in terms of the system of an electron confined to motion along a line of length L. Quantum mechanical probabilities are expressed in terms of a distribution function.
For a plane wave, p is defined and the position is not.
With a superposition of plane waves, we introduce an uncertainty on p and we localize. Since, the sum of 2 wavefucntions is neither an eigenfunction for p nor x, we have average values.
With a Gaussian function, the localization below is 1/2π

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p and x do not commute and are incompatible
For a plane

wave, p is known and x is not (Ψ*Ψ=A2 everywhere)
Let’s superpose two waves…
this introduces a delocalization for p and may be localize x
At the origin x=0 and at t=0 we want to increase the total amplitude,
so the two waves Ψ1 and Ψ2 are taken in phase
At ± Δx/2 we want to impose them out of phase
The position is therefore known for x ± Δx/2
the waves will have wavelengths

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Introduction to Quantum Mechanic

Содержание

When you find this image, you may skip this part

The idea of duality is rooted in a debate over the

Radiations, terminology

Interferencesin

Phase speed or velocity

Introducing new variablesAt the moment, let consider this just a formal

Introducing new variablesAt the moment, h is a simple constantLater on,

2 different velocities, v and vϕ

If h is the Planck constant J.sThen Louis de BROGLIEFrench (1892-1987) Max Planck

Robert Millikan (1910) showed that it was quantified.Rutherford (1911) showed

Gustav Kirchhoff (1860). The light emitted by a black body is

black-body radiationClassical TheoryFragmentation of the surface. One large area (Small λ Large

Kirchhoff black-body radiationRED WHITESmall ν Large ν Shift of νRadiation is

black-body radiationMax Planck (1901)GöttingenWhy a decrease for small λ ? QuantificationNumbering

Quantum numbersIn mathematics, a natural number (also called counting number) has

black-body radiationMax Planck (1901)GöttingenWhy a decrease for small λ ? Quantification

black-body radiation, quantificationMax PlanckSteps too hard to climb Easy slope, rampPyramid

Max Planck

Johannes Rydberg 1888Swedish Atomic SpectroscopyAbsorption or Emission

Johannes Rydberg 1888Swedish IRVISIBLEUVAtomic SpectroscopyAbsorption or EmissionEmission-R/12-R/22-R/32-R/42-R/52-R/62-R/72Quantum numbers n, levels are

Photoelectric Effect (1887-1905) discovered by Hertz in 1887 and explained in 1905

Kinetic energy

Compton effect 1923 playing billiards assuming λ=h/pArthur Holly ComptonAmerican 1892-1962

Davisson and Germer 1925Clinton DavissonLester GermerIn 1927Diffraction is similarly observed using

Wave-particle Equivalence.Compton Effect (1923).Electron Diffraction Davisson and Germer (1925)Young's Double Slit

Thomas Young 1773 – 1829 English, was born into a family

Young's Double Slit ExperimentScreenMask with 2 slits

Young's Double Slit ExperimentThis is a typical experiment showing the wave

Young's Double Slit ExperimentAssuming a single electron each timeWhat means interference

Young's Double Slit ExperimentThere is no possibility of knowing through which

Macroscopic world:A basket of cherriesMany of them (identical) We can see

Slot machine “one-arm bandit”After introducing a coin, you have 0 coin

de Broglie relation from relativityPopular expressions of relativity:m0 is the mass

de Broglie relation from relativityApplication to a photon (m0=0)To rememberTo remember

Max PlanckUseful to remember to relate energy and wavelength

A New mathematical tool: Wave functions and OperatorsEach particle may

Wave functions describing one particleTo represent a single particle Ψ(x,y,z) that

Operators associated to physical quantitiesWe cannot use functions (otherwise we would

Slot machine (one-arm bandit)Introducing a coin, you have 0 coin or

Examples of operators in mathematics : P parityEven function : no

Examples of operators in mathematics : Ay is an eigenvector; the

LinearityThe operators are linear:O (aΨ1+ bΨ1) = O (aΨ1 ) +

NormalizationAn eigenfunction remains an eigenfunction when multiplied by a constantO(λΨ)= o(λΨ)

Mean valueIf Ψ1 and Ψ2 are associated with the same eigenvalue

Sum, product and commutation of operators(A+B)Ψ=AΨ+BΨ (AB)Ψ=A(BΨ)operatorswavefunctionseigenvalues

Sum, product and commutation of operatorsnot compatibleoperators[A,C]=AC-CA≠0[A,B]=AB-BA=0[B,C]=BC-CB=0

Compatibility, incompatibility of operatorsnot compatibleoperators[A,C]=AC-CA≠0[A,B]=AB-BA=0[B,C]=BC-CB=0When operators commute, the physical quantities may

x and d/dx do not commute, are incompatibleTranslation and inversion do

Introducing new variablesNow it is time to give a physical meaning.p

Plane wavesThis represents a (monochromatic) beam, a continuous flow of particles

Niels Henrik David Bohr Danish1885-1962 Correspondence principle 1913/1920For every physical quantity

Operators p and HWe use the expression of the plane wave

Momentum and Energy OperatorsRemember during this chapter

Stationary state E=constantRemember for 3 slides after

Kinetic energyClassical quantum operatorIn 3D : Calling the laplacian Pierre Simon,

Correspondence principle angular momentumClassical expression Quantum expressionlZ= xpy-ypx

Erwin Rudolf Josef Alexander SchrödingerAustrian 1887 –1961 Without potential E =

Schrödinger Equation for stationary statesKinetic energyTotal energyPotential energy

Schrödinger Equation for stationary statesH is the hamiltonianSir William Rowan Hamilton

Chemistry is nothing but an application of Schrödinger Equation (Dirac)Paul Adrien

Uncertainty principle the Heisenberg uncertainty principle states that locating a particle