1) NMR spectroscopy provides information about atomic nuclei by analyzing their magnetic properties when placed in a strong magnetic field.
2) For nuclei with spin 1/2 like 1H and 13C, NMR spectra show peaks representing different spin states that are split by spin-spin coupling with neighboring nuclei.
3) Nuclei with higher spin like 11B display more complex splitting patterns due to interactions between their multiple spin states and coupled nuclei.
1. The document provides an overview of concepts relevant to nanochemistry including the periodic table, atomic structure, size of atoms, molecules and phases, types of chemical bonds, quantum mechanics principles, and solid-state band theory.
2. Key topics covered include the periodic arrangement of elements, subatomic particles that make up atoms, sizes of atoms ranging from 0.1-0.5 nanometers, different states of matter, various types of chemical bonds between atoms, the four quantum numbers that describe electrons, Heisenberg's uncertainty principle, and how materials behave as semiconductors, conductors or insulators depending on their band structure.
3. The document also defines nano as a prefix meaning one billion
The document discusses the noble gases, which are unreactive elements in Group 18 of the periodic table. They rarely combine with other elements and are found in nature as uncombined atoms due to their low reactivity. Examples of noble gases discussed include helium, neon, argon, krypton, and radon. Common uses of the noble gases mentioned are balloons, lightbulbs, neon signs, and strobe lights. Radon gas requires special attention due to its radioactivity and potential to cause lung cancer if inhaled over long periods of time.
1. DFT+U is a method that adds Hubbard corrections to DFT to better account for localized electrons and electronic correlations in transition metal oxides that LDA/GGA cannot describe accurately.
2. It introduces an on-site Coulomb repulsion term U to the energy functional that favors electron localization and integer orbital occupations.
3. The U parameter can be computed using linear response theory by perturbing occupation matrices and evaluating screened response matrices in a supercell calculation.
Quantum chemistry is the application of quantum mechanics to solve problems in chemistry. It has been widely used in different branches of chemistry including physical chemistry, organic chemistry, analytical chemistry, and inorganic chemistry. The time-independent Schrödinger equation is central to quantum chemistry and can be used to model chemical systems like the particle in a box, harmonic oscillator, and hydrogen atom. Molecular orbital theory is also important in quantum chemistry for describing chemical bonding in molecules.
1) De Broglie hypothesized that particles like electrons can behave as waves, with a wavelength given by λ = h/mv, where h is Planck's constant, m is the particle's mass, and v is its velocity.
2) This hypothesis provided an explanation for the quantization of angular momentum and energy levels in Bohr's model of the hydrogen atom.
3) Experiments have verified that electrons and other particles do exhibit wave-like properties such as interference and diffraction, confirming the wave-particle duality predicted by De Broglie's hypothesis.
1) NMR spectroscopy provides information about atomic nuclei by analyzing their magnetic properties when placed in a strong magnetic field.
2) For nuclei with spin 1/2 like 1H and 13C, NMR spectra show peaks representing different spin states that are split by spin-spin coupling with neighboring nuclei.
3) Nuclei with higher spin like 11B display more complex splitting patterns due to interactions between their multiple spin states and coupled nuclei.
1. The document provides an overview of concepts relevant to nanochemistry including the periodic table, atomic structure, size of atoms, molecules and phases, types of chemical bonds, quantum mechanics principles, and solid-state band theory.
2. Key topics covered include the periodic arrangement of elements, subatomic particles that make up atoms, sizes of atoms ranging from 0.1-0.5 nanometers, different states of matter, various types of chemical bonds between atoms, the four quantum numbers that describe electrons, Heisenberg's uncertainty principle, and how materials behave as semiconductors, conductors or insulators depending on their band structure.
3. The document also defines nano as a prefix meaning one billion
The document discusses the noble gases, which are unreactive elements in Group 18 of the periodic table. They rarely combine with other elements and are found in nature as uncombined atoms due to their low reactivity. Examples of noble gases discussed include helium, neon, argon, krypton, and radon. Common uses of the noble gases mentioned are balloons, lightbulbs, neon signs, and strobe lights. Radon gas requires special attention due to its radioactivity and potential to cause lung cancer if inhaled over long periods of time.
1. DFT+U is a method that adds Hubbard corrections to DFT to better account for localized electrons and electronic correlations in transition metal oxides that LDA/GGA cannot describe accurately.
2. It introduces an on-site Coulomb repulsion term U to the energy functional that favors electron localization and integer orbital occupations.
3. The U parameter can be computed using linear response theory by perturbing occupation matrices and evaluating screened response matrices in a supercell calculation.
Quantum chemistry is the application of quantum mechanics to solve problems in chemistry. It has been widely used in different branches of chemistry including physical chemistry, organic chemistry, analytical chemistry, and inorganic chemistry. The time-independent Schrödinger equation is central to quantum chemistry and can be used to model chemical systems like the particle in a box, harmonic oscillator, and hydrogen atom. Molecular orbital theory is also important in quantum chemistry for describing chemical bonding in molecules.
1) De Broglie hypothesized that particles like electrons can behave as waves, with a wavelength given by λ = h/mv, where h is Planck's constant, m is the particle's mass, and v is its velocity.
2) This hypothesis provided an explanation for the quantization of angular momentum and energy levels in Bohr's model of the hydrogen atom.
3) Experiments have verified that electrons and other particles do exhibit wave-like properties such as interference and diffraction, confirming the wave-particle duality predicted by De Broglie's hypothesis.
Metal ions are used in medicine for both diagnostic and treatment purposes. Some metals like technetium, gadolinium, and iron are used in medical imaging techniques like MRI. Platinum compounds like cisplatin are commonly used to treat cancers by binding to DNA and interfering with cell division. While effective, metals can cause side effects if levels become unbalanced or toxicity occurs from treatment. Precise application of metal ions exploits their properties for medical benefit.
1) The document describes the canonical partition function for an ideal monatomic gas. It shows that the partition function can be broken into translational, electronic, and nuclear contributions which are independent.
2) Key relationships are developed between the partition function and thermodynamic properties like internal energy, entropy, heat capacity, and more. Knowing the partition function provides expressions for properties like the Helmholtz free energy.
3) For an ideal monatomic gas, the translational partition function is derived and shown to be proportional to volume to the power of 3/2. The electronic and nuclear contributions are accounted for but not explicitly derived.
Reaction mechanism in complex compoundsSPCGC AJMER
The document discusses reaction mechanisms in complex compounds. It begins by defining reaction mechanisms and factors studied like stereochemistry and equilibrium. It then differentiates between electrophilic and nucleophilic reagents. It describes substitution reactions of metal complexes, including ligand substitution reactions that can occur through dissociative or associative mechanisms. Specific ligand substitution reactions like ligand exchange, solvent exchange, and acid/base hydrolysis are mentioned. Trans effect and substitution reactions of square planar complexes are also summarized.
This document discusses solving the harmonic oscillator equation to model different types of vibrations. It covers undamped free vibrations which exhibit simple harmonic motion. Damped free vibrations are also examined, where damping causes the amplitude to decay over time. Forced vibrations, including cases of beats and resonance, are explored. The document suggests the beam vibration can be modeled as a harmonic oscillator. It shows how to write the second order differential equation as a first order system for numerical solution in Matlab. Finally, it notes that the solution depends on ratios of m, c, and k, not their individual values, which is important for solving the inverse problem.
This document contains lecture notes on quantum mechanics. It introduces key concepts like the Schrodinger equation, ket vectors, operators, and Hamiltonians. The notes are divided into multiple chapters that will cover topics such as the harmonic oscillator, angular momentum, perturbation theory, and other quantum systems. References are provided to textbooks where more of the material in the notes is based on. The notes are intended to review physical and mathematical concepts needed to formulate the theory of quantum mechanics.
(If visualization is slow, please try downloading the file.)
Part 2 of a tutorial given in the Brazilian Physical Society meeting, ENFMC. Abstract: Density-functional theory (DFT) was developed 50 years ago, connecting fundamental quantum methods from early days of quantum mechanics to our days of computer-powered science. Today DFT is the most widely used method in electronic structure calculations. It helps moving forward materials sciences from a single atom to nanoclusters and biomolecules, connecting solid-state, quantum chemistry, atomic and molecular physics, biophysics and beyond. In this tutorial, I will try to clarify this pathway under a historical view, presenting the DFT pillars and its building blocks, namely, the Hohenberg-Kohn theorem, the Kohn-Sham scheme, the local density approximation (LDA) and generalized gradient approximation (GGA). I would like to open the black box misconception of the method, and present a more pedagogical and solid perspective on DFT.
Charge-Transfer-Spectra. metal to metal, metal to ligandNafeesAli12
The document discusses charge transfer spectra in metal complexes. There are four main types of charge transfer transitions: ligand to metal (LMCT), metal to ligand (MLCT), intermetal or metal to metal (MMCT), and interligand (LLCT). LMCT involves electron transfer from ligand orbitals to metal orbitals, while MLCT is the reverse with electron transfer from metal to ligand orbitals. MMCT occurs between different oxidation states of the same metal. LLCT takes place between different ligands, one acting as an electron donor and the other as an acceptor. Examples are provided of each type of charge transfer and how they influence the color of complexes.
This document provides an overview of particle physics and cosmology concepts from a physics textbook. It discusses fundamental force particles, positrons and other antiparticles, positron emission tomography, the beginnings of particle physics with mesons, Feynman diagrams, pion exchange between protons and neutrons, particle classification, baryon number, conservation laws including baryon number, lepton number, and strangeness. Examples are given to check conservation of baryon number and lepton number. The document also discusses strange particles and strangeness conservation.
The document discusses the concept of effective nuclear charge. It explains that the actual charge experienced by valence electrons is less than the true nuclear charge due to shielding by inner electrons. This decreased charge is called the effective nuclear charge (Zeff). Slater's rules provide a method to calculate the screening constant σ and thus determine Zeff. The concept of Zeff is applied to explain trends in ionization energy, filling of electron shells, and properties of cations, anions, and across the periodic table.
CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I
5.1 X-Ray Scattering (review and some more material)
5.2 De Broglie Waves
5.3 Electron Scattering / Transmission electron microscopy
5.4 Wave Motion
5.5 Waves or Particles?
5.6 Uncertainty Principle
5.7 Probability, Wave Functions, and the Copenhagen Interpretation
5.8 Particle in a Box
This document provides an overview of quantum mechanics. It begins by explaining that quantum mechanics describes the motion of subatomic particles and is needed to understand the properties of atoms and molecules. It then discusses some key developments in quantum mechanics, including Planck's quantum theory of radiation, Einstein's explanation of the photoelectric effect, de Broglie's hypothesis of matter waves, Heisenberg's uncertainty principle, and Schrodinger's wave equation. The document also compares classical and quantum mechanics and provides examples of quantum mechanical applications like atomic orbitals and black body radiation.
IB Chemistry on Crystal Field Theory and Splitting of 3d orbitalLawrence kok
The document discusses the properties and behaviors of transition metals. Transition metals are d-block elements that have partially filled d orbitals. They can exist in multiple oxidation states and form colored complexes due to their variable electron configurations. Transition metals are also good catalysts as their partially filled d orbitals allow them to easily gain or lose electrons and form weak bonds with reactants to lower the activation energy of chemical reactions.
Density functional theory (DFT) and the concepts of the augmented-plane-wave ...ABDERRAHMANE REGGAD
Density functional theory (DFT) is a quantum mechanical method used to investigate the electronic structure of materials. The document discusses DFT and the linearized augmented plane wave plus local orbital (LAPW+lo) method implemented in the Wien2k software. Wien2k is widely used to study the properties of solids and surfaces using an all-electron, relativistic, and full-potential DFT approach. The document provides an overview of the theoretical foundations of DFT and LAPW methods as well as examples of applications studied with Wien2k.
This document provides an overview of metal carbonyls. It discusses how metal carbonyls are formed from transition metals and carbon monoxide, and examples like nickel tetracarbonyl and iron pentacarbonyl. The molecular orbital diagram of carbon monoxide is shown, explaining why it can participate in pi-backbonding. Infrared spectroscopy is described as a useful technique for analyzing metal carbonyls, as it can distinguish terminal from bridging carbonyl ligands based on the infrared absorption frequency. Factors like metal charge and other ligands that affect the carbonyl stretching frequency are also outlined. Finally, some applications of infrared spectra of metal carbonyls are mentioned.
Hidden Symmetries and Their Consequences in the Hubbard Model of t2g ElectronsABDERRAHMANE REGGAD
(1) The Hubbard model for t2g electrons in transition metal oxides possesses novel hidden symmetries that have significant consequences.
(2) These symmetries prevent long-range spin order at non-zero temperatures and lead to an extraordinary simplification in exact diagonalization studies.
(3) Even with spin-orbit interactions included, the excitation spectrum remains gapless due to a continuous symmetry arising from the hidden symmetries.
IB Chemistry on Standard Reduction Potential, Standard Hydrogen Electrode and...Lawrence kok
The document discusses standard electrode potentials and how they are measured. It explains that the standard hydrogen electrode is used as a reference with a potential of 0 V. Other half-cell potentials are measured against this to determine their standard electrode potential. Common half-cells include metal/metal ion, gas/ion, and ion/ion systems. Standard conditions of 1 M concentrations, 1 atm pressure, and 298K temperature must be used. The potentials of zinc/zinc ion, iron III/iron II, and chlorine/chloride ion half-cells are given as examples.
This document discusses the Schrodinger wave equation for hydrogen atoms. It begins by presenting the time-independent 3D Schrodinger wave equation and explains how it is converted to polar coordinates due to the radial symmetry of hydrogen atoms. The wave function is assumed to separate into three parts, leading to three equations involving the principal, azimuthal, and magnetic quantum numbers. Quantum numbers and their relationships to orbital shapes are also described. Finally, atomic orbitals are defined as regions of high probability of finding electrons based on the Schrodinger wave equation solution.
1. The document discusses moles, molar mass, and calculations involving moles. It provides examples of calculating the number of particles, ions, or molecules in 1 mole of various substances like iron, carbon dioxide, sodium chloride, and magnesium chloride.
2. It also explains how to calculate molar mass using relative atomic mass and gives examples for iron, carbon dioxide, sodium chloride, and magnesium chloride.
3. Key concepts discussed include the definition of 1 mole as 6.02 x 1023 particles and its relationship to the Avogadro constant, and how molar mass is used to express the mass of 1 mole of a substance.
Metal ions are used in medicine for both diagnostic and treatment purposes. Some metals like technetium, gadolinium, and iron are used in medical imaging techniques like MRI. Platinum compounds like cisplatin are commonly used to treat cancers by binding to DNA and interfering with cell division. While effective, metals can cause side effects if levels become unbalanced or toxicity occurs from treatment. Precise application of metal ions exploits their properties for medical benefit.
1) The document describes the canonical partition function for an ideal monatomic gas. It shows that the partition function can be broken into translational, electronic, and nuclear contributions which are independent.
2) Key relationships are developed between the partition function and thermodynamic properties like internal energy, entropy, heat capacity, and more. Knowing the partition function provides expressions for properties like the Helmholtz free energy.
3) For an ideal monatomic gas, the translational partition function is derived and shown to be proportional to volume to the power of 3/2. The electronic and nuclear contributions are accounted for but not explicitly derived.
Reaction mechanism in complex compoundsSPCGC AJMER
The document discusses reaction mechanisms in complex compounds. It begins by defining reaction mechanisms and factors studied like stereochemistry and equilibrium. It then differentiates between electrophilic and nucleophilic reagents. It describes substitution reactions of metal complexes, including ligand substitution reactions that can occur through dissociative or associative mechanisms. Specific ligand substitution reactions like ligand exchange, solvent exchange, and acid/base hydrolysis are mentioned. Trans effect and substitution reactions of square planar complexes are also summarized.
This document discusses solving the harmonic oscillator equation to model different types of vibrations. It covers undamped free vibrations which exhibit simple harmonic motion. Damped free vibrations are also examined, where damping causes the amplitude to decay over time. Forced vibrations, including cases of beats and resonance, are explored. The document suggests the beam vibration can be modeled as a harmonic oscillator. It shows how to write the second order differential equation as a first order system for numerical solution in Matlab. Finally, it notes that the solution depends on ratios of m, c, and k, not their individual values, which is important for solving the inverse problem.
This document contains lecture notes on quantum mechanics. It introduces key concepts like the Schrodinger equation, ket vectors, operators, and Hamiltonians. The notes are divided into multiple chapters that will cover topics such as the harmonic oscillator, angular momentum, perturbation theory, and other quantum systems. References are provided to textbooks where more of the material in the notes is based on. The notes are intended to review physical and mathematical concepts needed to formulate the theory of quantum mechanics.
(If visualization is slow, please try downloading the file.)
Part 2 of a tutorial given in the Brazilian Physical Society meeting, ENFMC. Abstract: Density-functional theory (DFT) was developed 50 years ago, connecting fundamental quantum methods from early days of quantum mechanics to our days of computer-powered science. Today DFT is the most widely used method in electronic structure calculations. It helps moving forward materials sciences from a single atom to nanoclusters and biomolecules, connecting solid-state, quantum chemistry, atomic and molecular physics, biophysics and beyond. In this tutorial, I will try to clarify this pathway under a historical view, presenting the DFT pillars and its building blocks, namely, the Hohenberg-Kohn theorem, the Kohn-Sham scheme, the local density approximation (LDA) and generalized gradient approximation (GGA). I would like to open the black box misconception of the method, and present a more pedagogical and solid perspective on DFT.
Charge-Transfer-Spectra. metal to metal, metal to ligandNafeesAli12
The document discusses charge transfer spectra in metal complexes. There are four main types of charge transfer transitions: ligand to metal (LMCT), metal to ligand (MLCT), intermetal or metal to metal (MMCT), and interligand (LLCT). LMCT involves electron transfer from ligand orbitals to metal orbitals, while MLCT is the reverse with electron transfer from metal to ligand orbitals. MMCT occurs between different oxidation states of the same metal. LLCT takes place between different ligands, one acting as an electron donor and the other as an acceptor. Examples are provided of each type of charge transfer and how they influence the color of complexes.
This document provides an overview of particle physics and cosmology concepts from a physics textbook. It discusses fundamental force particles, positrons and other antiparticles, positron emission tomography, the beginnings of particle physics with mesons, Feynman diagrams, pion exchange between protons and neutrons, particle classification, baryon number, conservation laws including baryon number, lepton number, and strangeness. Examples are given to check conservation of baryon number and lepton number. The document also discusses strange particles and strangeness conservation.
The document discusses the concept of effective nuclear charge. It explains that the actual charge experienced by valence electrons is less than the true nuclear charge due to shielding by inner electrons. This decreased charge is called the effective nuclear charge (Zeff). Slater's rules provide a method to calculate the screening constant σ and thus determine Zeff. The concept of Zeff is applied to explain trends in ionization energy, filling of electron shells, and properties of cations, anions, and across the periodic table.
CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I
5.1 X-Ray Scattering (review and some more material)
5.2 De Broglie Waves
5.3 Electron Scattering / Transmission electron microscopy
5.4 Wave Motion
5.5 Waves or Particles?
5.6 Uncertainty Principle
5.7 Probability, Wave Functions, and the Copenhagen Interpretation
5.8 Particle in a Box
This document provides an overview of quantum mechanics. It begins by explaining that quantum mechanics describes the motion of subatomic particles and is needed to understand the properties of atoms and molecules. It then discusses some key developments in quantum mechanics, including Planck's quantum theory of radiation, Einstein's explanation of the photoelectric effect, de Broglie's hypothesis of matter waves, Heisenberg's uncertainty principle, and Schrodinger's wave equation. The document also compares classical and quantum mechanics and provides examples of quantum mechanical applications like atomic orbitals and black body radiation.
IB Chemistry on Crystal Field Theory and Splitting of 3d orbitalLawrence kok
The document discusses the properties and behaviors of transition metals. Transition metals are d-block elements that have partially filled d orbitals. They can exist in multiple oxidation states and form colored complexes due to their variable electron configurations. Transition metals are also good catalysts as their partially filled d orbitals allow them to easily gain or lose electrons and form weak bonds with reactants to lower the activation energy of chemical reactions.
Density functional theory (DFT) and the concepts of the augmented-plane-wave ...ABDERRAHMANE REGGAD
Density functional theory (DFT) is a quantum mechanical method used to investigate the electronic structure of materials. The document discusses DFT and the linearized augmented plane wave plus local orbital (LAPW+lo) method implemented in the Wien2k software. Wien2k is widely used to study the properties of solids and surfaces using an all-electron, relativistic, and full-potential DFT approach. The document provides an overview of the theoretical foundations of DFT and LAPW methods as well as examples of applications studied with Wien2k.
This document provides an overview of metal carbonyls. It discusses how metal carbonyls are formed from transition metals and carbon monoxide, and examples like nickel tetracarbonyl and iron pentacarbonyl. The molecular orbital diagram of carbon monoxide is shown, explaining why it can participate in pi-backbonding. Infrared spectroscopy is described as a useful technique for analyzing metal carbonyls, as it can distinguish terminal from bridging carbonyl ligands based on the infrared absorption frequency. Factors like metal charge and other ligands that affect the carbonyl stretching frequency are also outlined. Finally, some applications of infrared spectra of metal carbonyls are mentioned.
Hidden Symmetries and Their Consequences in the Hubbard Model of t2g ElectronsABDERRAHMANE REGGAD
(1) The Hubbard model for t2g electrons in transition metal oxides possesses novel hidden symmetries that have significant consequences.
(2) These symmetries prevent long-range spin order at non-zero temperatures and lead to an extraordinary simplification in exact diagonalization studies.
(3) Even with spin-orbit interactions included, the excitation spectrum remains gapless due to a continuous symmetry arising from the hidden symmetries.
IB Chemistry on Standard Reduction Potential, Standard Hydrogen Electrode and...Lawrence kok
The document discusses standard electrode potentials and how they are measured. It explains that the standard hydrogen electrode is used as a reference with a potential of 0 V. Other half-cell potentials are measured against this to determine their standard electrode potential. Common half-cells include metal/metal ion, gas/ion, and ion/ion systems. Standard conditions of 1 M concentrations, 1 atm pressure, and 298K temperature must be used. The potentials of zinc/zinc ion, iron III/iron II, and chlorine/chloride ion half-cells are given as examples.
This document discusses the Schrodinger wave equation for hydrogen atoms. It begins by presenting the time-independent 3D Schrodinger wave equation and explains how it is converted to polar coordinates due to the radial symmetry of hydrogen atoms. The wave function is assumed to separate into three parts, leading to three equations involving the principal, azimuthal, and magnetic quantum numbers. Quantum numbers and their relationships to orbital shapes are also described. Finally, atomic orbitals are defined as regions of high probability of finding electrons based on the Schrodinger wave equation solution.
1. The document discusses moles, molar mass, and calculations involving moles. It provides examples of calculating the number of particles, ions, or molecules in 1 mole of various substances like iron, carbon dioxide, sodium chloride, and magnesium chloride.
2. It also explains how to calculate molar mass using relative atomic mass and gives examples for iron, carbon dioxide, sodium chloride, and magnesium chloride.
3. Key concepts discussed include the definition of 1 mole as 6.02 x 1023 particles and its relationship to the Avogadro constant, and how molar mass is used to express the mass of 1 mole of a substance.
2. Mechanika kwantowa
Równanie Schrödingera
ˆ
Hψ = Eψ
zasada
zachowania
energii
ˆ
H− operator różniczkowy Hamiltona
E− energia
ψ− funkcja falowa
operator energii kinetyczna operator energii potencjalnej
ˆ ˆ ˆ
H =T +V
h2 d 2 d2 d2 Z ⋅ e2
− 2 + 2 + 2 −
2m dx dy dz
4πε 0 r
przyciąganie Coulombowskie
energia kinetyczna elektronu jądro-elektron
m – masa cząstki Z – ładunek jądra ε0 – stała dielektryczna2próżni
h – stała Plancka E – ładunek elektronu r – promień
3. Atom wodoru
Równanie Schrödingera dla atomu wodoru
Z=1 (1 proton)
postać
h 2 ∂ 2ψ ∂ 2ψ ∂ 2ψ Ze 2
− 2 + 2 + 2 −
r ψ = Eψ
2πm ∂x
∂y ∂z
energia przyciąganie
kinetyczna Coulombowskie
elektronu jądro-elektron
rozwiązania
energia E
funkcja falowa Ψ
3
5. Atom wodoru
Równanie Schrödingera
rozwiązania
h
moment pędu M = l (l + 1)
2π
l = 0,1,2,3.....n-1
poboczna/orbitalna
liczba kwantowa
składowa momentu pędu
wzdłuż kierunku „z” M =
h
z m
2π
m = -l, ... ,0,…, +l
energia nie jest jedyną magnetyczna liczba
kwantowaną wielkością kwantowa
fizyczną
5
6. Atom wodoru
Równanie Schrödingera
rozwiązania przestrzenne kwantowanie
momentu pędu elektronu M w
atomie wodoru (l=2)
Obliczmy M dla l=2:
h h
M = 2⋅3 = 6
2π 2π
Składowe Mz wynoszą:
2h h h 2h
, , 0, − , −
2π 2π 2π 2π 6
7. Atom wodoru
Ostatnia liczba kwantowa
spin
nie wynika z r. Schroedingera
Ruch obrotowy elektronu nosi nazwę spinu. Elektron ma dwa stany
spinowe, oznaczane strzałkami ↑ i ↓. Możemy sobie wyobrazić, że
elektron obraca się z pewną prędkością w kierunku wskazówek
zegara przeciwnym (stan ↓, +1/2) lub z identyczna prędkością w
kierunku przeciwnym (stan ↑, -1/2). Ponieważ wirujący ładunek
elektryczny wytwarza pole magnetyczne, elektrony znajdujące się
w tych dwóch stanach spinowych można rozróżnić na podstawie
ich zachowania się w polu magnetycznym.
7
8. Atom wodoru
Liczby kwantowe
określa wzór przyjmuje określa funkcje
wielkość wartości falowe Ψ
fizyczną
energię 1 Z 2 2π 2 me 4 główna: rozmiar orbitalu
E=− 2
n h2 n=1,2,3,…
moment pędu h poboczna: kształt orbitalu
M = l (l + 1)
2π l=0,1,2,…n-1
składową magnetyczna: kierunek orbitalu
h
momentu pędu Mz = m m=-l, (-l+1),…(l-1), l
2π
składową spinu magnetyczna znak orbitalu
h spinowa:
σ z = ms
2π ms=-½ lub ½ 8
9. Atom wodoru
Liczby kwantowe n
rozwiązania gęstość prawdopodobieństwa
Ψ 1,0,0 1s l
Ψ 2,0,0
Ψ 2,1,-1
2s l= 0 1 2 3 4
Ψ 2,1,0 2p s p d f g
Ψ 2,1,1
Ψ 3,0,0 3s
Ψ 3,1,-1
Ψ 3,1,0 3p
ψ 3,1,1
Ψ 3,2,-2
Ψ 3,2,-1
Ψ 3,2,0 3d
Ψ 3,2,1 9
Ψ 3,2,2
10. Atom wodoru
Równanie Schrödingera
funkcja falowa
Funkcja falowa w interpretacji
interpretacja Funkcja falowa w interpretacji
Borna. Prawdopodobieństwo
Borna. Prawdopodobieństwo
znalezienia elektronu w danym
znalezienia elektronu w danym
punkcie jest proporcjonalne do
punkcie jest proporcjonalne do
kwadratu funkcji falowej (Ψ2):
kwadratu funkcji falowej (Ψ2):
prawdopodobieństwo to jest
prawdopodobieństwo to jest
wyrażone przez stopień
wyrażone przez stopień
zaczernienia paska u dołu.
zaczernienia paska u dołu.
Gęstość prawdopodobieństwa
Gęstość prawdopodobieństwa
w węźle wynosi 0. Węzeł jest
w węźle wynosi 0. Węzeł jest
punktem, w którym funkcja
punktem, w którym funkcja
falowa przechodzi przez 0.
falowa przechodzi przez 0.
10
11. Atom wodoru
Równanie Schrödingera
funkcja falowa
interpretacja
Zgodnie z postulatem Bohrna, prawdopodobieństwo
znalezienia elektronu w danym punkcie przestrzeni jest
proporcjonalne do kwadratu funkcji falowej w tym punkcie.
Tam gdzie funkcja falowa ma dużą ampitudę, istnieje duże
prawdopodobieństwo znalezienia opisanego przez nią
elektronu.
Tam gdzie funkcja falowa jest mała, znalezienie elektronu jest
mało prawdopodobne. Tam gdzie funkcja falowa jest równa 0,
znalezienie elektronu jest niemożliwe.
W mechanice kwantowej można przewidywać tylko
prawdopodobieństwo znalezienia cząsteczki w danym
miejscu.
11
12. Atom wodoru
Równanie Schrödingera
funkcja falowa
interpretacja
Funkcja falowa elektronu w atomie ma tak istotne
znaczenie, iż nadano jej specjalną nazwę – orbital
atomowy. Orbital można poglądowo przedstawić
jako chmurę otaczająca jądro atomu; gęstość
chmury reprezentuje prawdopodobieństwo
znalezienia elektronu w każdym punkcie.
12
13. Atom wodoru
Liczby kwantowe n
rozwiązania gęstość prawdopodobieństwa
Ψ 1,0,0 1s l
Ψ 2,0,0
Ψ 2,1,-1
2s l= 0 1 2 3 4
Ψ 2,1,0 2p s p d f g
Ψ 2,1,1
Ψ 3,0,0 3s
Ψ 3,1,-1
Ψ 3,1,0 3p
ψ 3,1,1
Ψ 3,2,-2
Ψ 3,2,-1
Ψ 3,2,0 3d
Ψ 3,2,1 13
Ψ 3,2,2
14. Atom wodoru
Równanie Schrödingera
rozwiązania
funkcja falowa
y
ϕ
r
ψ n ,l , m = R ( r ) ⋅ φ ( ϕ ) ⋅ θ ( γ ) γ y
x
część radialna
część kątowa
14
15. Atom wodoru
Równanie Schrödingera
rozwiązania
funkcja falowa ψ n ,l , m = R ( r ) ⋅ φ ( ϕ ) ⋅ θ ( γ )
n l m symbol Ψn,l,m
3
1 0 0 1s 1 2 − a0
r
2 e
a
0
2 0 0 2s
3
1 r
1 1 r − 2 a0
2
3
2 − e
2 2 a 0 a0
2 1 0 2pz
3
1 r
1 1 2 r − 2 a0
2
e cosγ
4( 6) 2 a0 a0
1
3
1 r
2 1 ±1 2px 1 1 2 r − 2 a0
2
e sinγ sinϕ h2
2 a a
4( 6) 0 0
1
a0 =
2py 3
1 r 4π 2 me e 2
1 1 2 r − 2 a0
2
e sinγ cosϕ
4( 6) 2 a0 a0
1
15
16. Atom wodoru
Równanie Schrödingera
rozwiązania
funkcja falowa
gęstość prawdopodobieństwa
ψ n ,l ,m = [ R( r )φ ( ϕ )θ ( γ ) ]
2 2
1s
P=Ψ2dV
odległość od jądra, r
16
24. Atom wodoru
Wizualizacja orbitali Ψ 3,2,-2
07_108B z z z Ψ 3,2,-1
Ψ 3,2,0 3d
y y Ψ 3,2,1y
Ψ 3,2,2
x x x
dxz dyz dxy
z z
y y
x x
24
(b) dx2 - y2 d z2
25. Atom wodoru
Wizualizacja orbitali
07_109
z z z Ψ 4,3,-3
Ψ 4,3,-2
Ψ 4,3,-1 4f
x x x Ψ 4,3,0
Ψ 4,3,1
y y y Ψ 4,3,2
fz3 - 3 zr
5
2 fx 3 - 3 xr 2
5
fy 3 - 3 yr 2
5
Ψ 4,3,3
z z z z
x x x x
y y y y
fxyz fy(x2 - z2) fx(z2 - y2) fz(x2 - y2)
25
27. Atom wodoru
Interpretacja widma
Oblicz długość fali fotonu emitowanego przez atom wodoru w
wyniku przejścia elektronu z poziomu n = 3 na poziom n = 2.
Zidentyfikuj na rysunku (widmo wodoru) linię spektralną
odpowiadającą temu przejściu.
Zmiana energii elektronu w czasie przejścia ze stanu nj do stanu ni (j>i)
2π 2 me 4 2π 2 me 4 2π 2 me 4 1
1
∆E = En j − Eni = − 2 2 − − 2 2 =
hn − 2
h nj i h 2 ni 2 n j
2π 2 me 4 1 1 1 1
∆E = E3 − E2 = − 2 = 2.179 ⋅10 −18 2 − 2 = 3.026 ⋅10 −19 J
h 2 22 3 2 3
h = 6.626 ⋅10 −34 J ⋅ s
m = 9.109 ⋅10 −31 kg
27
e = 1.602 ⋅10 −19 C
28. Atom wodoru
Interpretacja widma
c 1
E = hν = h = hc
λ λ
hc 6.626 ⋅10 −34 J ⋅ s × 3.00 ⋅108 m / s
λ= = = 6.57 ⋅10 −7 m
E 3.026 ⋅10 −19 J
= 657 nm
h = 6.626 ⋅10 −34 J ⋅ s
c = 3.00 ⋅108 m / s
28
30. Atom wodoru
Przykład 1 Identyfikacja linii w widmie wodoru
różnica energii między dwoma stanami:
∆E = (1/22 – 1/32) ⋅ h ⋅ (3.29⋅1015 Hz)
częstość emitowanego światła wynosi:
ν = ∆E/h = (1/22 – 1/32)⋅(3.29⋅1015 Hz)
długość fali promieniowania jest równa:
λ= c/ν = (3,00⋅108 m/s)/(1/22 – 1/32)⋅(3.29⋅1015 Hz)
λ = 6.57⋅10-7 m
30
35. Jon berylu w pułapce
W ostatnim "Nature" teoretyczne prace o dekoherencji zostały potwierdzone we
wspaniałym eksperymencie. Zespół Davida Winelanda z National Institute of
Standards and Technology w Boulder w USA złapał w pułapkę magnetyczną jon
berylu. Manipulacjami promieni laserowych fizycy sprawili, by jon znalazł się w takim
kwantowym stanie, iż był jednocześnie w dwóch różnych miejscach (oddalonych o
kilkadziesiąt nanometrów). Potem uczeni starali się zakłócić jego "podwójny" stan i
wyprowadzić go z tej "schizofrenii". By tego dokonać, symulowali różne wpływy
otoczenia. W końcu udało im się. Następowała dekoherencja - tzn. jon odnajdywał się
tylko w jednym miejscu. Co więcej, fizycy potwierdzili teoretyczne wyliczenia, że
dekoherencja następuje tym szybciej, im bardziej pozycje jonu były od siebie
oddalone. Słowem, szybkość dekoherencji zależy od rozmiaru obiektu. Dla
ogromnego (w porównaniu ze skalą atomową) kota kwantowe "być albo nie być"
rozstrzyga się niemal natychmiastowo. Kot w pudełku staje się albo martwy, albo
żywy. Nie ma szans tkwić tam w "dwuznacznej" sytuacji.
35
36. Być może, jak na razie cieszą się entuzjaści
możliwości podróży w czasie. Istnienie
wszechświatów równoległych rozwiązuje
paradoks dziadka: podróżnik cofa się do
przeszłości i zabija swojego dziadka, zanim ten
począł jego ojca - mamy problem: jak
podróżujący może w ogóle istnieć skoro jego
ojciec się nie narodzi? Odpowiedź: cofając się w
czasie, przenosi się do równoległego
wszechświata, w którym się nie narodzi.
36
