SO FAR: We Solved some of the Problems with Classical Physics

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1 SO FAR: We Solved some of the Problems with Classical Physics Blackbody Radiation? Plank Quantized Energy of Atoms: En nhf Photoelectric Effect? Einstein Quantized energy states of light photons: E hf Compton Effect? Light is a particle with momentum! h Matter Waves? debroglie waves! e p Discrete Spectra? Atoms?

3 Continuous vs Discrete This is a continuous spectrum of colors: all colors are present. This is a discrete spectrum of colors: only a few are present. Copyright 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

4 Continuous Discrete

5 Review: Dispersion: Diffraction Gratings How does dispersion with a grating compare with a prism? Longer wavelength light is bent more with a grating. Shorter wavelength light is bent more with a prism. d sin m Copyright 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

11 The Evidence.

Kirkoff s Rules for Spectra: 1859 German physicist who developed the spectroscope and the

Kirkoff Bunsen * Rule 1 : A hot and opaque solid, liquid or highly compressed gas emits a

* Rule 2 : A hot, transparent gas produces an emission spectrum with bright lines.

12 Kirkoff s Rules for Spectra: 1859 German physicist who developed the spectroscope and the science of emission spectroscopy with Bunsen. Kirkoff Bunsen * Rule 1 : A hot and opaque solid, liquid or highly compressed gas emits a continuous spectrum. * Rule 2 : A hot, transparent gas produces an emission spectrum with bright lines. * Rule 3 : If a continuous spectrum passes through a gas at a lower temperature, the transparent cooler gas generates dark absorption lines. Copyright 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

13 Compare absorption lines in a source with emission lines found in the laboratory! Kirchhoff deduced that elements were present in the atmosphere of the Sun and were absorbing their characteristic wavelengths, producing the absorption lines in the solar spectrum. He published in 1861 the first atlas of the solar spectrum, obtained with a prism ; however, these wavelengths were not very precise : the dispersion of the prism was not linear at all. Copyright 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

$spectrum, obtained with a diffraction grating, whose dispersion is linear, and replaced Kirchhoff's$ arbitrary scale by the wavelengths, expressed in the metric system, using a small unit (10-10 m) with which

arbitrary scale by the wavelengths, expressed in the metric system, using a small unit (10-10 m) with which

15 Anders Jonas Ångström 1869 Ångström measured the wavelengths on the four visible lines of the hydrogen spectrum, obtained with a diffraction grating, whose dispersion is linear, and replaced Kirchhoff's arbitrary scale by the wavelengths, expressed in the metric system, using a small unit (10-10 m) with which his name was to be associated. Line color red blue-green violet violet Wavelength Å Å Å Å

Balmer Series: 1885 Johann Balmer found an empirical equation that correctly predicted the four visible emission lines of hydrogen Johannes Robert Rydberg generalized it in 1888 for all transitions:

16 Balmer Series: 1885 Johann Balmer found an empirical equation that correctly predicted the four visible emission lines of hydrogen Johannes Robert Rydberg generalized it in 1888 for all transitions: 1 R 1 1 λ 2 n H 2 2 R H is the Rydberg constant R H = x 10 7 m -1 n is an integer, n = 3, 4, 5, The spectral lines correspond to different values of n H α is red, λ = nm H β is green, λ = nm H γ is blue, λ = nm H δ is violet, λ = nm

17 Theories.

19 Let no one unversed in geometry enter here. The Universe is made of pure mathematical ideas the Platonic Solids. Plato believed that the stars, planets, Sun and Moon move round the Earth in crystalline spheres.

Earth and the universe were seen as constructed out of

20 Earth and the universe were seen as constructed out of five basic elements: earth, water, air, fire, and ether. The natural place of the motionless Earth was at the centre of that universe. The stars in the heavens were made up of an indestructible substance called ether (aether) and were considered as eternal and unchanging.

the deflection of cathode rays in an electric field His model of the

21 Joseph John Thomson Plum Pudding Model 1904 Received Nobel Prize in 1906 Usually considered the discoverer of the electron Worked with the deflection of cathode rays in an electric field His model of the atom A volume of positive charge Electrons embedded throughout the volume

22 1911: Rutherford s Planetary Model of the Atom A beam of positively charged alpha particles hit and are scattered from a thin foil target. Large deflections could not be explained by Thomson s pudding model. (Couldn t explain the stability or spectra of atoms.)

23 1911: Rutherford s Planetary Model of the Atom A beam of positively charged alpha particles hit and are scattered from a thin foil target. Large deflections could not be explained by Thomson s model. (Couldn t explain the stability or spectra of atoms.)

24 Classical Physics at the Limit WHY IS MATTER (ATOMS) STABLE?

25 Electrons exist in quantized orbitals with energies given by multiples of Planck s constant. Light is emitted or absorbed when an electron makes a transition between energy levels. The energy of the photon is equal to the difference in the energy levels: E nhf, n= 0,1,2,3, h 6.626x10 Js E E E hf i f

26 Light Absorption & Emission E E E hf i f

27 Bohr s Model Electrons in an atom can occupy only certain discrete quantized states or orbits. E nhf, n= 0,1,2,3, Electrons are in stationary states: they don t accelerate and they don t radiate. 3. Electrons radiate only when making a transition from one orbital to another, either emitting or absorbing a photon. Postulate: E E E hf i f The angular momentum of an electron is always quantized and cannot be zero: h L n ( n 1,2,3,...) 2

28 Bohr s Derivation of the Energy for Hydrogen: Conservation of E: E K U F is centripetal: Sub back into E: From Angular Momentum: (2) h From : L n = mvr ( n 1,2,3,...) 2 (1) Sub r back into (1): Sub into (2): v kq 2 q q e e e nh 4 0hn 2 0hn Why is it negative?

29 Bohr Orbital Binding Energy for The Hydrogen Atom En eV n 2 Ground State: n = 1 First Excited: n = 2 2 nd Excited: n = eV is the energy of the H ground state. 2. Negative because it is the Binding Energy and work must be done on the atom (by a photon) to ionize it. 3. A 13.6eV photon must be absorbed to ionize the ground state 4. Bohr model only works for single electron atoms since it doesn t take into account electron-electron interaction forces.

30 The frequency of the photon emitted when the electron makes a transition from an outer orbit to an inner orbit is E E k e i f e ƒ 2 2 h 2aoh nf ni It is convenient to look at the wavelength instead The wavelengths are found by 2 1 ƒ kee R 2 2 H 2 2 λ c 2a ohc n f n i n f n i

31 Bohr Line Spectra of Hydrogen Bohr s Theory derived the spectra equations that Balmer, Lyman and Paschen had previously found experimentally! ( ) 2 RZ n 2 2 f n i 7 1 R x10 m Balmer: Visible Lyman: UV Paschen: IR

32 Bohr: Allowed Orbital Radii r n 2 2 n m k e e e n 1, 2, 3, 2 r (5.9x10 m) n n 11 2 rn a n 0 2 n = 1, 2, 3 Bohr Radius (ground state): ao = 5.9x10-11 m

33 Problem A hydrogen atom is in the third (n = 4) excited state. It can make a jump to a different state by transition A, B, or C as shown, and a photon is either emitted or absorbed. a) Which transition, A, B, or C, emits a photon with the greatest energy? Calculate the wavelength in m and the energy in ev. b) What is the energy of a photon needed to ionize the atom when the electron is initially in the third excited state? Calculate in ev. c) What is the orbital radius of the n=4 state? ( ) 2 RZ n 2 2 f n i m (1.097 x10 m )1 ( ) x10 8 m E 13.6eV Z eV eV 2 r n a n 0 ao = 5.9x10-11 m

34 Spontaneous Emission: Random Transition Probabilities Fermi s Golden Rule Transition probabilities correspond to the intensity of light emission.

35 IMPORTANT NOTE!! Free electrons have continuous energy states! Only bound electrons have quantized energy states! This is because they are not forced to fit in a confined space like a wave in a box!!! Continuous or Discrete? Bound-Bound Transitions: Atoms, boxes D Bound-Free Transitions: Ionization C Free-Bound Transitions: Ion captures an electron C Free-Free Transitions: Collisions, electron absorption C

36 1. Bohr model does not explain angular momentum postulate. 2. Bohr model does not explain splitting of spectral lines. 3. Bohr model does not explain multi-electron atoms. 4. Bohr model does not explain ionization energies of elements. E Z ev

37 1. Bohr model does not explain angular momentum postulate: The angular momentum of an electron is always quantized and cannot be zero*: h L n = nh 2 ( n 1,2,3,...) *If L=0 then the electron travels linearly and does not orbit. But if it is orbiting then it should radiate and the atom would be unstable eek gads! WHAT A MESS!

38 If photons can be particles, then why can t electrons be waves? p E hf h c c debroglie Wavelength: e h p Electrons are STANDING WAVES in atomic orbitals. 34 h 6.626x10 J s

39 2 r n n 1924: de Broglie Waves Explains Bohr s postulate of angular h p momentum quantization: 2 r n n h h 2 r n n n p e m v m vr e n h n 2 L m vr nh e n ( n 1,2,3,...)

40 1926:Schrodinger s Equation Rewrite the SE in spherical coordinates. Solutions to it give the possible states of the electron. Solution is sepearable:

41 Quantum Numbers and their quantization are derived purely from Mathematical Solutions of Schrodinger s Equation (boundary conditions) However, they have a phenomenological basis!! To be discussed later Solution:

42 Wave Functions given by Quantum Numbers!! The probability of finding an electron within a shell of radius r and thickness δr around a proton is where the first three radial wave functions of the electron in a neutral hydrogen atom are

43 Quantum H Atom 1s Wave Function: Radial Probability Density: 1 1 s () r e a 3 0 r/ a P(r) = 4πr 2 ψ 2 0 Most Probable value: solve dp 0 dr Average value: rave r rp() r dr 0 Probability in (0-r): P r P() r dr 0

44 Stationary States of Hydrogen Solutions to the Schrödinger equation for the hydrogen atom potential energy exist only if three conditions are satisfied: 1. The atom s energy must be one of the values where a B is the Bohr radius. The integer n is called the principal quantum number. These energies are the same as those in the Bohr hydrogen atom.

45 Stationary States of Hydrogen 2. The angular momentum L of the electron s orbit must be one of the values The integer l is called the orbital quantum number. 3. The z-component of the angular momentum must be one of the values The integer m is called the magnetic quantum number. Each stationary state of the hydrogen atom is identified by a triplet of quantum numbers (n, l, m).

46 Interpretation: Orbital Angular Momentum & its Z-component are Quantized Angular momentum magnitude is quantized: L l( l 1) h ( l 0,1,2,... n 1) Angular momentum direction is quantized: L m h ( m l,.. 1,0,1,2,... l) Z l l For each l, there are (2l+1) possible m l states. cos L z L Note: for n = 1, l = 0. This means that the ground state angular momentum in hydrogen is zero, not h/2, as Bohr assumed. What does it mean for L = 0 in this model?? Standing Wave

47 Zeeman Effect Explained The Zeeman Effect is the splitting of spectral lines when a magnetic field is applied. This is due to the interaction between the external field and the B field produced by the orbital motion of the electron. Only certain angles are allowed between the orbital angular momentum and the external magnetic field resulting in the quantization of space! L m ( m l,.. 1,0,1,2,... l) h L l( l 1) h ( l 0,1,2,... n 1) Z l l

48 Stern-Gerlach 1921 A beam of silver atoms sent through a non-uniform magnetic field was split into two discrete components. Classically, it should be spread out because the magnetic moment of the atom can have any orientation. QM says if it is due to orbital angular momentum, there should be an odd number of components, (2l+1), but only two components are ever seen. This required an overhaul of QM to include Special Relativity. Then the electron magnetic spin falls out naturally from the mathematics. The spin magnetic moment of the electron with two spin magnetic quantum number values: ms 1 2

50 Intrinsic Spin is Quantized Fine Line Splitting: B field due to intrinsic spin interacts with B field due to orbital motion and produces additional energy states. Spin magnitude is quantized and fixed: S h 3 h s( s 1) Spin direction is quantized: h 1 1 S m ( m, ) Z s 2 s 2 2 spin e m S

51 Spin is the Spin Quantum Entanglement Quantum Computing The Qubit At the heart of the realm of quantum computation is the qubit. The quantum bit, by analogy with the binary digit, the bit, used by everyday computers, the qubit is the quantum computer's unit of currency. Instead of being in a 1 or zero state, a qubit can be in a superposition of both states.

53 Quantum Numbers Once the principal quantum number is known, the values of the other quantum numbers can be found. The possible states of a system are given by combinations of quantum numbers! Quantum Rules n 1,2,3,4... l 0,1,2... n 1 m l, l 1,...,0,1,... l 1, l m l s 1/ 2, 1/ 2

54 Shells and Subshells Shells are determined by the principle quantum number, n. Subshells are named after the type of line they produce in the emission spectrum and are determined by the l quantum number. s: Sharp line (l = 0) p: Very bright principle line( l = 1) d: Diffuse line (l = 2) f: Fundamental (hydrogen like) ( l =3) 1s 2s 2p

55 Wave Functions given by Quantum Numbers!! The probability of finding an electron within a shell of radius r and thickness δr around a proton is where the first three radial wave functions of the electron in a neutral hydrogen atom are

56 Wave Functions for Hydrogen The simplest wave function for hydrogen is the one that describes the 1s (ground) state and is designated ψ 1s (r) 1 rao ψ1 s() r e 3 πa As ψ 1s (r) approaches zero, r approaches and is normalized as presented ψ 1s (r) is also spherically symmetric This symmetry exists for all s states o

Radial Probability Density The radial probability density function P(r) is the probability per unit radial length of finding the electron in a spherical shell of radius r and thickness dr 1 A

57 Radial Probability Density The radial probability density function P(r) is the probability per unit radial length of finding the electron in a spherical shell of radius r and thickness dr 1 A spherical shell of radius r and thickness dr has a volume of 4πr 2 dr The radial probability density function is P(r) = 4πr 2 ψ 2 rao ψ1 s() r e πa 3 o where ψ 1 o 2 2ra 1s e 3 πao P r 4r e 2 2rao 1s () 3 ao

P(r) for 1s State of Hydrogen The radial probability density function for the hydrogen atom in its ground state is P r 4r e 2 2rao 1s () 3 ao The peak indicates the most

58 P(r) for 1s State of Hydrogen The radial probability density function for the hydrogen atom in its ground state is P r 4r e 2 2rao 1s () 3 ao The peak indicates the most probable location, the Bohr radius. The average value of r for the ground state of hydrogen is 3/2 a o The graph shows asymmetry, with much more area to the right of the peak

59 Electron Clouds The charge of the electron is extended throughout a diffuse region of space, commonly called an electron cloud This shows the probability density as a function of position in the xy plane The darkest area, r = a o, corresponds to the most probable region

62 Hydrogen 1s Radial Probability

63 Sample Problem

64 Sample Problem

65 Wave Function of the 2s state The next-simplest wave function for the hydrogen atom is for the 2s state n = 2; l = 0 The wave function is 3 2 r r 2a o 1 1 ψ2s( r ) 2 e 4 2π ao ao ψ 2s depends only on r and is spherically symmetric

66 Hydrogen 2s Radial Probability The next-simplest wave function for the hydrogen atom is for the 2s state: n = 2; l = r r 2a o 1 1 ψ2s( r ) 2 e 4 2π ao ao

67 Comparison of 1s and 2s States The plot of the radial probability density for the 2s state has two peaks The highest value of P corresponds to the most probable value In this case, r 5a o

68 Hydrogen 1s Radial Probability

69 Hydrogen 2s Radial Probability

70 Hydrogen 2p Radial Probability

71 Hydrogen 3s Radial Probability

72 Hydrogen 3p Radial Probability

73 Hydrogen 3d Radial Probability

76 1926:Schrodinger s Equation Rewrite the SE in spherical coordinates. Solutions to it give the possible states of the electron. Solution is sepearable:

77 Quantum Numbers and their quantization are derived purely from Mathematical Solutions of Schrodinger s Equation (boundary conditions) However, they have a phenomenological basis!! To be discussed later Solution:

78 Wave Functions given by Quantum Numbers!! The probability of finding an electron within a shell of radius r and thickness δr around a proton is where the first three radial wave functions of the electron in a neutral hydrogen atom are

80 Stationary States of Hydrogen Solutions to the Schrödinger equation for the hydrogen atom potential energy exist only if three conditions are satisfied: 1. The atom s energy must be one of the values where a B is the Bohr radius. The integer n is called the principal quantum number. These energies are the same as those in the Bohr hydrogen atom.

81 Stationary States of Hydrogen 2. The angular momentum L of the electron s orbit must be one of the values The integer l is called the orbital quantum number. 3. The z-component of the angular momentum must be one of the values The integer m is called the magnetic quantum number. Each stationary state of the hydrogen atom is identified by a triplet of quantum numbers (n, l, m).

82 Interpretation: Orbital Angular Momentum & its Z-component are Quantized Angular momentum magnitude is quantized: L l( l 1) h ( l 0,1,2,... n 1) Angular momentum direction is quantized: L m h ( m l,.. 1,0,1,2,... l) Z l l For each l, there are (2l+1) possible m l states. cos L z L Note: for n = 1, l = 0. This means that the ground state angular momentum in hydrogen is zero, not h/2, as Bohr assumed. What does it mean for L = 0 in this model?? Standing Wave

83 Intrinsic Spin is Quantized Fine Line Splitting: B field due to intrinsic spin interacts with B field due to orbital motion and produces additional energy states. Spin magnitude is quantized and fixed: S h 3 h s( s 1) Spin direction is quantized: h 1 1 S m ( m, ) Z s 2 s 2 2 spin e m S

84 Spin is the Spin Quantum Entanglement Quantum Computing The Qubit At the heart of the realm of quantum computation is the qubit. The quantum bit, by analogy with the binary digit, the bit, used by everyday computers, the qubit is the quantum computer's unit of currency. Instead of being in a 1 or zero state, a qubit can be in a superposition of both states.

86 Quantum Numbers Once the principal quantum number is known, the values of the other quantum numbers can be found. The possible states of a system are given by combinations of quantum numbers! Quantum Rules n 1,2,3,4... l 0,1,2... n 1 m l, l 1,...,0,1,... l 1, l m l s 1/ 2, 1/ 2

87 Shells and Subshells Shells are determined by the principle quantum number, n. Subshells are named after the type of line they produce in the emission spectrum and are determined by the l quantum number. s: Sharp line (l = 0) p: Very bright principle line( l = 1) d: Diffuse line (l = 2) f: Fundamental (hydrogen like) ( l =3) 1s 2s 2p

90 Pauli Exclusion Principle No two electrons can have the same set of quantum numbers. That is, no two electrons can be in the same quantum state. From the exclusion principle, it can be seen that only two electrons can be present in any orbital: One electron will have spin up and one spin down. Maximum number of electrons in a shell is: # = 2n 2 Maximum number of electrons in a subshell is: # = 2(2l+1) You must obey Pauli to figure out how to fill orbitals, shells and subshells. (This is not true of photons or other force carrying particles such as gravitons or gluons (bosons))

91 Hund s Rule Hund s Rule states that when an atom has orbitals of equal energy, the order in which they are filled by electrons is such that a maximum number of electrons have unpaired spins Some exceptions to the rule occur in elements having subshells that are close to being filled or half-filled

92 Hund s Rule Hund s Rule states that when an atom has orbitals of equal energy, the order in which they are filled by electrons is such that a maximum number of electrons have unpaired spins Some exceptions to the rule occur in elements having subshells that are close to being filled or half-filled The filling of electronic states must obey both the Pauli Exclusion Principle and Hund s Rule.

94 How many electrons in n=3 Shell? n 3 l 0,1,2 m m l s Quantum Rules n 1,2,3,4... l 0,1,2... n 1 m l, l 1,...,0,1,... l 1, l m l s 1/ 2, 1/ 2 2, 1,0,1,2 1/ 2, 1/ 2 2(2l 1) 2e 6e 10e =18e = 2n 2

95 Bohr model does not explain ionization energies of elements but QM does. How?

96 Spontaneous Emission: Random Transition Probabilities Fermi s Golden Rule Transition probabilities correspond to the intensity of light emission.

97 Excited States and Spectra An atom can jump from one stationary state, of energy E 1, to a higher-energy state E 2 by absorbing a photon of frequency In terms of the wavelength: Note that a transition from a state in which the valence electron has orbital quantum number l 1 to another with orbital quantum number l 2 is allowed only if

98 Hydrogen Energy Level Diagram Selection Rules The selection rules for allowed transitions are Δl = ±1 Δm l = 0, ±1 Transitions in which l does not change are very unlikely to occur and are called forbidden transitions Such transitions actually can occur, but their probability is very low compared to allowed transitions

99 X-Ray Production The discrete lines are called characteristic x-rays. These are created when A bombarding electron collides with a target atom The electron removes an inner-shell electron from orbit An electron from a higher orbit drops down to fill the vacancy Typically, the energy is greater than 1000 ev The continuous spectrum is called bremsstrahlung, the German word for braking radiation

100 Synchrotron Radiation X-Ray and Radio

101

102 X Ray Imaging when X-ray light shines on us, it goes through our skin, but allows shadows of our bones to be projected onto and captured by film. When X-ray light shines on us, it goes through our skin, but allows shadows of our bones to be projected onto and captured by film.

103 Lasers Light Amplification by Stimulated Emission of Radiation "A splendid light has dawned on me about the absorption and emission of radiation..." Albert Einstein, 1916

104 Stimulated Emission 1 photon in, 2 out "When you come right down to it, there is really no such thing as truly spontaneous emission; its all stimulated emission. The only distinction to be made is whether the field that does the stimulating is one that you put there or one that God put there..." David Griffths, Introduction to Quantum Mechanics

105

106

107 Stimulated Emission

108 Sample Problem What average wavelength of visible light can pump neodymium into levels above its metastable state? From the figure it takes 2.1eV to pump neodymium into levels above its metastable state. Thus, hc E ev m 2.10 ev m 590 nm

109 Laser Applications

110 Phaser?

111 UV in, vibrant color out. Fluorescence

112 Phosphorescence Time Delayed Fluorescence Glow in the Dark: Day Glow

113 BTW: Iridescence: Diffraction

We now realize that the phenomena of chemical interactions, and, ultimately life itself, are to be understood in terms of electromagnetism".

We now realize that the phenomena of chemical interactions, and, ultimately life itself, are to be understood in terms of electromagnetism". -Richard Feynman Quantum H Atom Review Radia Wave Function (1s):