Atoms 2012 update -- start with single electron: H-atom
|
|
- Morgan Fisher
- 5 years ago
- Views:
Transcription
1 Atoms 2012 update -- start with single electron: H-atom x z φ θ e -1 y 3-D problem - free move in x, y, z - easier if change coord. systems: Cartesian Spherical Coordinate (x, y, z) (r, θ, φ) Reason: V(r) = -Ze 2 / r depend - separation not orientation electrostatic potential basis of chemistry Note: (if proton is fixed at 0,0,0 then r =[x 2 +y 2 +z 2 ] 1/2 ) r = (x e x p )i + (y e y p )j + (z e z p )k --vector, p e r = [(x e x p ) 2 + (y e y p ) 2 + (z e z p ) 2 ] 1/2 --length (scalar) Goal separate variables V(r) x,y,z mixed no problem for K.E. T already separated First step formal: reduce from 6-coord: x e,y e,z e & x p,y p,z p to 3-internal coordinates. Eliminate center of mass -- see notes (Center of Mass) on the Web site whole atom: R = X+Y+Z X = (m e x e +m p x p )/(m e +m p ).. etc. this normalizes the position correct for mass for H-atom these are almost equal to: x p, y p, z p but process is general move equal mass issue other 3 coord: relative x = x e x p etc. ideal for V(r) Problem separates, V = V(r) only depend on internal coord. Hψ = [-h 2 /2M R 2 + -h 2 /2μ r 2 + V(r)] Ψ(R,r) = EΨ H - sum independent coord. M = m e + m p μ = m e. m p /( m e + m p ) 33 VII 33
2 Ψ(R,r) = Ξ(R) ψ(r) product wave function separates summed Η like before (eg.3-d p.i.b.) i.e. Operate H on Ψ(R,r) and operators pass through R and r dependent terms, Ξ(R) and ψ(r), to give: First term: + second two terms: (= E [Ξ(R) ψ(r)]) (-h 2 /2M)ψ(r) 2 R Ξ(R) and Ξ(R)[(-h 2 /2μ) 2 r +V(r)]ψ(r) so divide through by Ψ(R,r) = Ξ(R) ψ(r) and results are independent (R and r) sum equal constant, E VII 34 R-dependent equation: (-h 2 /2M)(1/Ξ(R)) R 2 Ξ(R) = E T Motion (T) of whole atom free particle -not quantized -ignore r-dependent equation: (1/ψ(r))[(-h 2 /2μ) r 2 +V(r)]ψ(r) = E int relative (internal) coord. - formal: we let E = E T +E int internal equation simplified by conversion: x,y,z r,θ,φ result internal: H(r)ψ(r) = [(-h 2 /2μ) r 2 Ze 2 /r]ψ(r) = E int ψ(r) (idea potential only depends on r, so other two coordinates, φ,θ only contribute K.E.) r,θ,φ 2 = 1/r 2 { / r(r 2 / r)+[1/sinθ] / θ(sinθ / θ) + [1/sin 2 θ] 2 / φ 2 } Separation, one coord. at time. Easy to separate φ depend. to get φ only in 1 term, multiply r 2 sin 2 θ through Hψ = Eψ Aside, just for information, Step by step separation, start: [(-h 2 /2μ) r,θ,φ 2 Ze 2 /r] ψ(r,θ,φ) = (-h 2 /2μr 2 ){ / r(r 2 / r)+ [1/sinθ] / θ(sinθ / θ) + [1/sin 2 θ] 2 / φ 2 }ψ(r,θ,φ) Ze 2 /r ψ(r,θ,φ) = E int ψ(r,θ,φ) 34
3 multiply by r 2 sin 2 θ : 1. (-h 2 /2μ) sin 2 θ{ / r(r 2 / r)+ [1/sinθ] / θ(sinθ / θ) + 2 / φ 2 [r 2 sin 2 θ Ζe 2 /r]}ψ(r,θ,φ) = r 2 sin 2 θe int ψ(r,θ,φ) VII 35 isolate the φ dependent terms: 2. (-h 2 /2μ) {sin 2 θ / r(r 2 / r)+ sinθ / θ(sinθ / θ) (2μ/h 2 ) r 2 sin 2 θ(e int + Ze 2 /r)}ψ(r,θ,φ) = (h 2 /2μ) 2 / φ 2 ψ(r,θ,φ) see all φ operators one side Use ψ(r, θ, φ) = R(r) Θ(θ) Φ(φ) and cancel out h 2 /2μ 3. {sin 2 θ / r(r 2 / r)+ sinθ / θ(sinθ / θ)} R(r)Θ(θ)Φ(φ) r 2 sin 2 θ(2μ/h 2 ) (E int + Ze 2 /r) R(r)Θ(θ)Φ(φ) = - 2 / φ 2 R(r)Θ(θ)Φ(φ) Recall 2 / φ 2 only operate on Φ(φ) part, rest pass through Same the other side, Φ(φ) pass through / r and / θ 4. Φ(φ) {sin 2 θ / r(r 2 / r)+ sinθ / θ(sinθ / θ)} R(r)Θ(θ) Φ(φ)r 2 sin 2 θ(2μ/h 2 ) (E int + Ze 2 /r)r(r)θ(θ) = - R(r)Θ(θ) 2 / φ 2 Φ(φ) divide through as before by Ψ(r,θ,φ) 5. [R(r)Θ(θ)] 1 {sin 2 θ / r(r 2 / r)+sinθ / θ(sinθ / θ)} R(r)Θ(θ) r 2 sin 2 θ(2μ/h 2 )(E int +Ze 2 /r)r(r)θ(θ) = - [Φ(φ)] 1 2 / φ 2 Φ(φ) = const. right side (red) independent of left side, so must be constant a) Let Φ part equal constant, m 2 : 2 / φ 2 Φ(φ) = -m 2 Φ(φ) y x +m -m rotation about z-axis Φ(φ) = e imφ m = 0,±1,±2... note: equivalent to particle on a ring problem -m 2 important, requires complex exponential 35
4 b) Can similarly separate Θ(θ) but arithmetic messier 1 st divide through by sin 2 θ -- separate r and θ terms 1. [R(r)Θ(θ)] 1 { / r(r 2 / r)+(sinθ) 1 / θ(sinθ / θ)} R(r)Θ(θ) r 2 (2μ/h 2 ) (E int + Ze 2 /r) R(r)Θ(θ) = m 2 / sin 2 θ Note 1 st & 3 rd terms have r but middle term only θ, separate 2. [Θ(θ)] 1 { (sinθ) 1 / θ(sinθ / θ) - m 2 /sin 2 θ} Θ(θ) = [R(r) ] 1 { / r(r 2 / r) + (2μ/h 2 ) r 2 (E int + Ze 2 /r)} R(r) = const. 36 VII 36 Result- only has θ on left and r on right, let const. = -l(l+1): 3. { (sinθ) 1 / θ(sinθ / θ) - m 2 /sin 2 θ + l(l +1)} Θ(θ) = 0 LeGendre polynomial: Θ lm (θ) = P l m (cos θ) l = 0,1,2, l = 0 2 /2 l = 1 m = 0 (3/2) 1/2 cos θ l = 1 m = ±1 (3/4) 1/2 sin θ l = 2 m = 0 (5/8) 1/2 (3 cos 2 θ 1) l = 2 m = ±1 (15/4) 1/2 (sin θ cos θ) l = 2 m = ±1 (15/16) 1/2 (sin 2 θ).... polynom: P(x) but x = cosθ, l = 3 m = 0 ~ (5 cos 3 θ 3 cos θ) as m inc. cosθ --> sinθ c) Radial function messier, has V(r), yet but must fit B.C. r R nl (r) 0 (must be integrable) exponential decay, penetrate potential ~ e -αr damp (works, r always +) Must be orthogonal this works when fct. oscillate (wave-like) power series will do
5 Associated LaGuerre Polynomial solves radial equation { r -2 / r(r 2 / r) + (2μ/h 2 ) (E int + Ze 2 /r) - l(l +1)/r 2 } R(r) = 0 VII 37 divide by r 2, multiply by R(r) & move l(l+1) term to left side R nl = [const] (2σ/n) l 2l 1 L + n + l (2σ/n) e -σ/n σ = Zr/a 0 Quantum #: n = 1, 2, 3, l = 0, 1, 2, n 1, l m n = 1, l = 0 ~ e -σ n = 2 l = 0 ~ (2 σ)e -σ/2 n = 2 l = 1 ~ σe -σ/2 n = 3 l = 0 ~ (1 2σ/3 + 2σ 2 /27)e -σ/3 n = 3 l = 1 ~ (σ σ 2 /6)e -σ/3 n = 3 l = 2 ~ σ 2 e -σ/3.... Note: n restricts l ( n 1) and l restricts m ( l) Comparison of potentials when potential not infinite, levels collapse, when sides not infinite and vertical w/f penetrates potential wall Particle in box; Stubby box; infinite, steep potential, get expanding Energylevels, ψ(wall)=0 37 Finite potential get collapsing energy levels continuous at top dissociation ψ(x) penetrate wall
6 Harmonic oscillator; V(x) Anharmonic oscillator VII 38 sloped potential wavefct penetrate wall, finite pot. levels collapse, H-atom Solutions for l = 0 higher l, less nodes If rotate this around r = 0, get a symmetric well and shapes look like υ = even harmonic oscillator shapes finite potential bends over, V = 0 at r =, get collapse of E-levels as n Energy vary with nodes curvature as before Yellow (Engel) penetrate classic forbidden region 38
7 These functions (3 independent solutions) can be combined ψ(r,θ,φ) = R nl (r) Θ l m (θ) Φ m (φ) VII 39 Note: only R nl depend on r as does V(r) Energy will not depend on θ,φ for H-atom, differ for He etc. Often write: Y l m (θ,φ) = Θ m l (θ) Φ m (φ) spherical harmonics Angular parts give shapes but not size these are eigenfunctions of Angular Momentum recall : L = r x p, L 2 ~ -h 2 2 r, L z = xp y -yp x, L z = -ih / φ L 2 Y l m (θ,φ) = l(l + 1) h 2 Y l m [H,L 2 ] = 0 commute: L z Y l m (θ φ) = mhy l m [H,L z ] = 0 simul.sol n This is source of familiar orbitals, l = 0,1,2,3.. or s,p,d,f Solving R nl equation E n = -Z 2 e 4 μ/2h 2 n 2 = -Z 2 R/n 2 exactly Bohr solution, R Rydberg, (must be, since works) Familiar: n = 1 l = 0 m = 0 1s n = 2 l = 0 m = 0 2s l = 1 m = 0,±1 2p (2p 0 + 2p ±1 ) n = 3 l = 0 m = 0 3s l = 1 m = 0,±1 3p (3p 0 + 3p ±1 ) l = 2 m=0,±1,±2 3d (3d 0, 3d ±1, 3d ±2 ) (lt) compare 1s and 2s (rt) probability (ψ ψ) and radial distribution (4πr 2 ψ ψ) Note: 1s decays, 2s has node (2-σ) term, 2p starts at 0 39
8 Compare probability distributions, see expansion in size with n, l VII 40 Note: 2s node makes dip Note: # radial nodes (in R nl ) in probability (e- density) decrease with l, i.e. # = n-l-1 Similarly mix Θ ±2 to get d xy, d x 2-y 2 and Θ ±2 for d xz, d yz 40
9 Angular functions have no radial value --> just surface, combine with radial function to get magnitude or e - density, Best represented as a contour map or probability surface VII 41 Real orbitals, take linear combinations of ±m values, eliminate i dependent terms, get x,y,z functions Cartesian form: e imφ + e -imφ = cosφ + i sinφ + cosφ i sinφ = 2 cos φ ~ x i (e imφ - e -imφ ) = 2 sin φ ~ y recall: x = r cos φ and y = r sin φ 41
10 42 VII 42
11 H-atom solutions, complex orbitals, eigen values of angular momentum: VII 43 Linear combinations of H-atom solutions Real orbitals 43
12 Contour plots of orbitals VII 44 Radial function effect represented by contours, each line represents lower e- density sign change makes node between lobes or radially (e.g. see top, 3p vs. 2p) 44
13 Linear comb. of degenerate orbitals also solutions H-atom show effect of mixing s and p orbitals: VII 45 Hybrids linear comb. of s and p orient for bonding sp 2 orbitals / opposite direction, (s±p) are 180 sp 2 3 orbitals / in plane / 120 apart sp 3 4 orbitals / 4 vertices of tetrahedron (109 ) 45
14 Energy level diagram H-atom E n VII 46 Allowed any n change: Δn 0 l, m l as before: Δl = ±1, Δm l = 0, ±1 no energy dependence on l, but spectral transitions do depend on l Spectral transitions match Balmer series but also must account for Θ,Φ functions Allowed selection rules (see box) n n' = 1 Lyman must start p orbital --> end 1s n n' = 2 Balmer must start d or s orbital end 2p or start in p orbital end in 2s etc. Test with Zeeman effect m l βh = E' add E from field More complex than Bohr, but same energies and angular momenta, similar restriction on solution to line spectra 46
15 47 VII 47
Atoms 2010 update -- start with single electron: H-atom
VII 33 Atoms 2010 update -- start with single electron: H-atom x z φ θ e -1 y 3-D problem - free move in x, y, z - easier if change coord. systems: Cartesian Spherical Coordinate (x, y, z) (r, θ, φ) Reason:
More informationAtoms 09 update-- start with single electron: H-atom
Atoms 09 update-- start with single electron: H-atom VII 33 x z φ θ e -1 y 3-D problem - free move in x, y, z - handy to change systems: Cartesian Spherical Coordinate (x, y, z) (r, θ, φ) Reason: V(r)
More informationIV. Electronic Spectroscopy, Angular Momentum, and Magnetic Resonance
IV. Electronic Spectroscopy, Angular Momentum, and Magnetic Resonance The foundation of electronic spectroscopy is the exact solution of the time-independent Schrodinger equation for the hydrogen atom.
More informationQuantum Mechanics: The Hydrogen Atom
Quantum Mechanics: The Hydrogen Atom 4th April 9 I. The Hydrogen Atom In this next section, we will tie together the elements of the last several sections to arrive at a complete description of the hydrogen
More informationPhysics 228 Today: Ch 41: 1-3: 3D quantum mechanics, hydrogen atom
Physics 228 Today: Ch 41: 1-3: 3D quantum mechanics, hydrogen atom Website: Sakai 01:750:228 or www.physics.rutgers.edu/ugrad/228 Happy April Fools Day Example / Worked Problems What is the ratio of the
More informationCHEM-UA 127: Advanced General Chemistry I
1 CHEM-UA 127: Advanced General Chemistry I Notes for Lecture 11 Nowthatwehaveintroducedthebasicconceptsofquantummechanics, wecanstarttoapplythese conceptsto build up matter, starting from its most elementary
More informationA Quantum Mechanical Model for the Vibration and Rotation of Molecules. Rigid Rotor
A Quantum Mechanical Model for the Vibration and Rotation of Molecules Harmonic Oscillator Rigid Rotor Degrees of Freedom Translation: quantum mechanical model is particle in box or free particle. A molecule
More informationAngular Momentum. Classically the orbital angular momentum with respect to a fixed origin is. L = r p. = yp z. L x. zp y L y. = zp x. xpz L z.
Angular momentum is an important concept in quantum theory, necessary for analyzing motion in 3D as well as intrinsic properties such as spin Classically the orbital angular momentum with respect to a
More informationONE AND MANY ELECTRON ATOMS Chapter 15
See Week 8 lecture notes. This is exactly the same as the Hamiltonian for nonrigid rotation. In Week 8 lecture notes it was shown that this is the operator for Lˆ 2, the square of the angular momentum.
More informationThe Hydrogen Atom Chapter 20
4/4/17 Quantum mechanical treatment of the H atom: Model; The Hydrogen Atom Chapter 1 r -1 Electron moving aroundpositively charged nucleus in a Coulombic field from the nucleus. Potential energy term
More informationHarmonic Oscillator (9) use pib to think through 2012
Harmonic Oscillator (9) use pib to think through 01 VI 9 Particle in box; Stubby box; Properties of going to finite potential w/f penetrate walls, w/f oscillate, # nodes increase with n, E n -levels less
More informationThe Hydrogen atom. Chapter The Schrödinger Equation. 2.2 Angular momentum
Chapter 2 The Hydrogen atom In the previous chapter we gave a quick overview of the Bohr model, which is only really valid in the semiclassical limit. cf. section 1.7.) We now begin our task in earnest
More informationFun With Carbon Monoxide. p. 1/2
Fun With Carbon Monoxide p. 1/2 p. 1/2 Fun With Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results p. 1/2 Fun With Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results C V (J/K-mole) 35 30 25
More informationH atom solution. 1 Introduction 2. 2 Coordinate system 2. 3 Variable separation 4
H atom solution Contents 1 Introduction 2 2 Coordinate system 2 3 Variable separation 4 4 Wavefunction solutions 6 4.1 Solution for Φ........................... 6 4.2 Solution for Θ...........................
More informationQuantum Theory of Angular Momentum and Atomic Structure
Quantum Theory of Angular Momentum and Atomic Structure VBS/MRC Angular Momentum 0 Motivation...the questions Whence the periodic table? Concepts in Materials Science I VBS/MRC Angular Momentum 1 Motivation...the
More informationThe Hydrogen Atom. Chapter 18. P. J. Grandinetti. Nov 6, Chem P. J. Grandinetti (Chem. 4300) The Hydrogen Atom Nov 6, / 41
The Hydrogen Atom Chapter 18 P. J. Grandinetti Chem. 4300 Nov 6, 2017 P. J. Grandinetti (Chem. 4300) The Hydrogen Atom Nov 6, 2017 1 / 41 The Hydrogen Atom Hydrogen atom is simplest atomic system where
More informationSolving the Schrödinger Equation for the 1 Electron Atom (Hydrogen-Like)
Stockton Univeristy Chemistry Program, School of Natural Sciences an Mathematics 101 Vera King Farris Dr, Galloway, NJ CHEM 340: Physical Chemistry II Solving the Schröinger Equation for the 1 Electron
More information1.6. Quantum mechanical description of the hydrogen atom
29.6. Quantum mechanical description of the hydrogen atom.6.. Hamiltonian for the hydrogen atom Atomic units To avoid dealing with very small numbers, let us introduce the so called atomic units : Quantity
More informationOne-electron Atom. (in spherical coordinates), where Y lm. are spherical harmonics, we arrive at the following Schrödinger equation:
One-electron Atom The atomic orbitals of hydrogen-like atoms are solutions to the Schrödinger equation in a spherically symmetric potential. In this case, the potential term is the potential given by Coulomb's
More information20 The Hydrogen Atom. Ze2 r R (20.1) H( r, R) = h2 2m 2 r h2 2M 2 R
20 The Hydrogen Atom 1. We want to solve the time independent Schrödinger Equation for the hydrogen atom. 2. There are two particles in the system, an electron and a nucleus, and so we can write the Hamiltonian
More informationModel Problems update Particle in box - all texts plus Tunneling, barriers, free particle - Tinoco (pp455-63), House Ch 3
VI 15 Model Problems update 010 - Particle in box - all texts plus Tunneling, barriers, free particle - Tinoco (pp455-63), House Ch 3 Consider E-M wave 1st wave: E 0 e i(kx ωt) = E 0 [cos (kx - ωt) i sin
More informationPHYS 3313 Section 001 Lecture # 22
PHYS 3313 Section 001 Lecture # 22 Dr. Barry Spurlock Simple Harmonic Oscillator Barriers and Tunneling Alpha Particle Decay Schrodinger Equation on Hydrogen Atom Solutions for Schrodinger Equation for
More informationThe Central Force Problem: Hydrogen Atom
The Central Force Problem: Hydrogen Atom B. Ramachandran Separation of Variables The Schrödinger equation for an atomic system with Z protons in the nucleus and one electron outside is h µ Ze ψ = Eψ, r
More informationPhysics 2203, 2011: Equation sheet for second midterm. General properties of Schrödinger s Equation: Quantum Mechanics. Ψ + UΨ = i t.
General properties of Schrödinger s Equation: Quantum Mechanics Schrödinger Equation (time dependent) m Standing wave Ψ(x,t) = Ψ(x)e iωt Schrödinger Equation (time independent) Ψ x m Ψ x Ψ + UΨ = i t +UΨ
More informationPhysics 2203, Fall 2012 Modern Physics
Physics 03, Fall 01 Modern Physics. Monday, Oct. 8 th, 01. Finish up examples for Ch. 8 Computer Exercise. Announcements: Take home Exam #1: Average 84.1, Average both 63.0 Quiz on Friday on Ch. 8 or Ch.
More informationParticle in a 3 Dimensional Box just extending our model from 1D to 3D
CHEM 2060 Lecture 20: Particle in a 3D Box; H atom L20-1 Particle in a 3 Dimensional Box just extending our model from 1D to 3D A 3D model is a step closer to reality than a 1D model. Let s increase the
More informationA 2 sin 2 (n x/l) dx = 1 A 2 (L/2) = 1
VI 15 Model Problems 014 - Particle in box - all texts, plus Tunneling, barriers, free particle Atkins(p.89-300),ouse h.3 onsider E-M wave first (complex function, learn e ix form) E 0 e i(kx t) = E 0
More informationModel Problems 09 - Ch.14 - Engel/ Particle in box - all texts. Consider E-M wave 1st wave: E 0 e i(kx ωt) = E 0 [cos (kx - ωt) i sin (kx - ωt)]
VI 15 Model Problems 09 - Ch.14 - Engel/ Particle in box - all texts Consider E-M wave 1st wave: E 0 e i(kx ωt) = E 0 [cos (kx - ωt) i sin (kx - ωt)] magnitude: k = π/λ ω = πc/λ =πν ν = c/λ moves in space
More informationChemistry 795T. NC State University. Lecture 4. Vibrational and Rotational Spectroscopy
Chemistry 795T Lecture 4 Vibrational and Rotational Spectroscopy NC State University The Dipole Moment Expansion The permanent dipole moment of a molecule oscillates about an equilibrium value as the molecule
More informationQuantum Mechanics in 3-Dimensions
Quantum Mechanics in 3-Dimensions Pavithran S Iyer, 2nd yr BSc Physics, Chennai Mathematical Institute Email: pavithra@cmi.ac.in August 28 th, 2009 1 Schrodinger equation in Spherical Coordinates 1.1 Transforming
More informationModel Problems update Particle in box - all texts plus Tunneling, barriers, free particle Atkins (p ), House Ch 3
VI 15 Model Problems update 01 - Particle in box - all texts plus Tunneling, barriers, free particle Atkins (p.33-337), ouse h 3 onsider E-M wave first (complex function, learn e ix form) E 0 e i(kx ωt)
More informationSolved radial equation: Last time For two simple cases: infinite and finite spherical wells Spherical analogs of 1D wells We introduced auxiliary func
Quantum Mechanics and Atomic Physics Lecture 16: The Coulomb Potential http://www.physics.rutgers.edu/ugrad/361 h / d/361 Prof. Sean Oh Solved radial equation: Last time For two simple cases: infinite
More informationThe Hydrogen Atom. Dr. Sabry El-Taher 1. e 4. U U r
The Hydrogen Atom Atom is a 3D object, and the electron motion is three-dimensional. We ll start with the simplest case - The hydrogen atom. An electron and a proton (nucleus) are bound by the central-symmetric
More information1 Schroenger s Equation for the Hydrogen Atom
Schroenger s Equation for the Hydrogen Atom Here is the Schroedinger equation in D in spherical polar coordinates. Note that the definitions of θ and φ are the exact reverse of what they are in mathematics.
More information5.111 Lecture Summary #6
5.111 Lecture Summary #6 Readings for today: Section 1.9 (1.8 in 3 rd ed) Atomic Orbitals. Read for Lecture #7: Section 1.10 (1.9 in 3 rd ed) Electron Spin, Section 1.11 (1.10 in 3 rd ed) The Electronic
More informationApplied Statistical Mechanics Lecture Note - 3 Quantum Mechanics Applications and Atomic Structures
Applied Statistical Mechanics Lecture Note - 3 Quantum Mechanics Applications and Atomic Structures Jeong Won Kang Department of Chemical Engineering Korea University Subjects Three Basic Types of Motions
More information1. We want to solve the time independent Schrödinger Equation for the hydrogen atom.
16 The Hydrogen Atom 1. We want to solve the time independent Schrödinger Equation for the hydrogen atom.. There are two particles in the system, an electron and a nucleus, and so we can write the Hamiltonian
More informationIntroduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world,
Introduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world, x p h π If you try to specify/measure the exact position of a particle you
More information5.61 Lecture #17 Rigid Rotor I
561 Fall 2017 Lecture #17 Page 1 561 Lecture #17 Rigid Rotor I Read McQuarrie: Chapters E and 6 Rigid Rotors molecular rotation and the universal angular part of all central force problems another exactly
More informationSchrödinger equation for the nuclear potential
Schrödinger equation for the nuclear potential Introduction to Nuclear Science Simon Fraser University Spring 2011 NUCS 342 January 24, 2011 NUCS 342 (Lecture 4) January 24, 2011 1 / 32 Outline 1 One-dimensional
More information2m r2 (~r )+V (~r ) (~r )=E (~r )
Review of the Hydrogen Atom The Schrodinger equation (for 1D, 2D, or 3D) can be expressed as: ~ 2 2m r2 (~r, t )+V (~r ) (~r, t )=i~ @ @t The Laplacian is the divergence of the gradient: r 2 =r r The time-independent
More informationWelcome back to PHY 3305
Welcome back to PHY 3305 Today s Lecture: Hydrogen Atom Part I John von Neumann 1903-1957 One-Dimensional Atom To analyze the hydrogen atom, we must solve the Schrodinger equation for the Coulomb potential
More informationFinal Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall Duration: 2h 30m
Final Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall. ------------------- Duration: 2h 30m Chapter 39 Quantum Mechanics of Atoms Units of Chapter 39 39-1 Quantum-Mechanical View of Atoms 39-2
More information( ( ; R H = 109,677 cm -1
CHAPTER 9 Atomic Structure and Spectra I. The Hydrogenic Atoms (one electron species). H, He +1, Li 2+, A. Clues from Line Spectra. Reminder: fundamental equations of spectroscopy: ε Photon = hν relation
More informationApplications of Calculus II
Applications of Calculus II Applications of Polar Coordinates in Chemistry Dr. Christian Clausen III Department of Chemistry Spherical Polar Coordinates x = rcosθ ; y = rsinθ ; z = rcosφ 2 The Schrödinger
More informationPhysics 401: Quantum Mechanics I Chapter 4
Physics 401: Quantum Mechanics I Chapter 4 Are you here today? A. Yes B. No C. After than midterm? 3-D Schroedinger Equation The ground state energy of the particle in a 3D box is ( 1 2 +1 2 +1 2 ) π2
More informationQuantum Mechanics & Atomic Structure (Chapter 11)
Quantum Mechanics & Atomic Structure (Chapter 11) Quantum mechanics: Microscopic theory of light & matter at molecular scale and smaller. Atoms and radiation (light) have both wave-like and particlelike
More informationAngular momentum. Quantum mechanics. Orbital angular momentum
Angular momentum 1 Orbital angular momentum Consider a particle described by the Cartesian coordinates (x, y, z r and their conjugate momenta (p x, p y, p z p. The classical definition of the orbital angular
More informationif trap wave like violin string tied down at end standing wave
VI 15 Model Problems 9.5 Atkins / Particle in box all texts onsider E-M wave 1st wave: E 0 e i(kx ωt) = E 0 [cos (kx - ωt) i sin (kx - ωt)] magnitude: k = π/λ ω = πc/λ =πν ν = c/λ moves in space and time
More informationMore On Carbon Monoxide
More On Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results Jerry Gilfoyle The Configurations of CO 1 / 26 More On Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results Jerry Gilfoyle The Configurations
More informationLecture 4 Quantum mechanics in more than one-dimension
Lecture 4 Quantum mechanics in more than one-dimension Background Previously, we have addressed quantum mechanics of 1d systems and explored bound and unbound (scattering) states. Although general concepts
More informationAngular Momentum - set 1
Angular Momentum - set PH0 - QM II August 6, 07 First of all, let us practise evaluating commutators. Consider these as warm up problems. Problem : Show the following commutation relations ˆx, ˆL x = 0,
More informationThe 3 dimensional Schrödinger Equation
Chapter 6 The 3 dimensional Schrödinger Equation 6.1 Angular Momentum To study how angular momentum is represented in quantum mechanics we start by reviewing the classical vector of orbital angular momentum
More informationQuantum Numbers. principal quantum number: n. angular momentum quantum number: l (azimuthal) magnetic quantum number: m l
Quantum Numbers Quantum Numbers principal quantum number: n angular momentum quantum number: l (azimuthal) magnetic quantum number: m l Principal quantum number: n related to size and energy of orbital
More informationThe one and three-dimensional particle in a box are prototypes of bound systems. As we
6 Lecture 10 The one and three-dimensional particle in a box are prototypes of bound systems. As we move on in our study of quantum chemistry, we'll be considering bound systems that are more and more
More informationChapter 6: Quantum Theory of the Hydrogen Atom
Chapter 6: Quantum Theory of the Hydrogen Atom The first problem that Schrödinger tackled with his new wave equation was that of the hydrogen atom. The discovery of how naturally quantization occurs in
More informationLecture 4 Quantum mechanics in more than one-dimension
Lecture 4 Quantum mechanics in more than one-dimension Background Previously, we have addressed quantum mechanics of 1d systems and explored bound and unbound (scattering) states. Although general concepts
More informationChapter 7 The Quantum-Mechanical Model of the Atom
Chapter 7 The Quantum-Mechanical Model of the Atom Electron Energy electron energy and position are complimentary because KE = ½mv 2 for an electron with a given energy, the best we can do is describe
More informationSummary: angular momentum derivation
Summary: angular momentum derivation L = r p L x = yp z zp y, etc. [x, p y ] = 0, etc. (-) (-) (-3) Angular momentum commutation relations [L x, L y ] = i hl z (-4) [L i, L j ] = i hɛ ijk L k (-5) Levi-Civita
More information1 Reduced Mass Coordinates
Coulomb Potential Radial Wavefunctions R. M. Suter April 4, 205 Reduced Mass Coordinates In classical mechanics (and quantum) problems involving several particles, it is convenient to separate the motion
More informationAngular momentum & spin
Angular momentum & spin January 8, 2002 1 Angular momentum Angular momentum appears as a very important aspect of almost any quantum mechanical system, so we need to briefly review some basic properties
More informationChapter 2. Atomic Structure. Inorganic Chemistry1 CBNU T.-S.You
Chapter 2. Atomic Structure Chapter 2. Atomic Structure The theory of atomic and molecular structure depend on quantum mechanics to describe atoms and molecules in mathematical terms. Fortunately, it is
More informationCollection of formulae Quantum mechanics. Basic Formulas Division of Material Science Hans Weber. Operators
Basic Formulas 17-1-1 Division of Material Science Hans Weer The de Broglie wave length λ = h p The Schrödinger equation Hψr,t = i h t ψr,t Stationary states Hψr,t = Eψr,t Collection of formulae Quantum
More informationChapter 6. Quantum Theory of the Hydrogen Atom
Chapter 6 Quantum Theory of the Hydrogen Atom 1 6.1 Schrodinger s Equation for the Hydrogen Atom Symmetry suggests spherical polar coordinates Fig. 6.1 (a) Spherical polar coordinates. (b) A line of constant
More informationKey Developments Leading to Quantum Mechanical Model of the Atom
Key Developments Leading to Quantum Mechanical Model of the Atom 1900 Max Planck interprets black-body radiation on the basis of quantized oscillator model, leading to the fundamental equation for the
More informationECEN 5005 Crystals, Nanocrystals and Device Applications Class 20 Group Theory For Crystals
ECEN 5005 Crystals, Nanocrystals and Device Applications Class 20 Group Theory For Crystals Laporte Selection Rule Polarization Dependence Spin Selection Rule 1 Laporte Selection Rule We first apply this
More informationCh120 - Study Guide 10
Ch120 - Study Guide 10 Adam Griffith October 17, 2005 In this guide: Symmetry; Diatomic Term Symbols; Molecular Term Symbols Last updated October 27, 2005. 1 The Origin of m l States and Symmetry We are
More informationSchrödinger equation for central potentials
Chapter 2 Schrödinger equation for central potentials In this chapter we will extend the concepts and methods introduced in the previous chapter ifor a one-dimenional problem to a specific and very important
More informationQUANTUM MECHANICS AND ATOMIC STRUCTURE
5 CHAPTER QUANTUM MECHANICS AND ATOMIC STRUCTURE 5.1 The Hydrogen Atom 5.2 Shell Model for Many-Electron Atoms 5.3 Aufbau Principle and Electron Configurations 5.4 Shells and the Periodic Table: Photoelectron
More information(a) Determine the general solution for φ(ρ) near ρ = 0 for arbitary values E. (b) Show that the regular solution at ρ = 0 has the series expansion
Problem 1. Curious Wave Functions The eigenfunctions of a D7 brane in a curved geometry lead to the following eigenvalue equation of the Sturm Liouville type ρ ρ 3 ρ φ n (ρ) = E n w(ρ)φ n (ρ) w(ρ) = where
More informationExpansion of 1/r potential in Legendre polynomials
Expansion of 1/r potential in Legendre polynomials In electrostatics and gravitation, we see scalar potentials of the form V = K d Take d = R r = R 2 2Rr cos θ + r 2 = R 1 2 r R cos θ + r R )2 Use h =
More informationSchrödinger equation for central potentials
Chapter 2 Schrödinger equation for central potentials In this chapter we will extend the concepts and methods introduced in the previous chapter for a one-dimensional problem to a specific and very important
More information(Refer Slide Time: 1:20) (Refer Slide Time: 1:24 min)
Engineering Chemistry - 1 Prof. K. Mangala Sunder Department of Chemistry Indian Institute of Technology, Madras Lecture - 5 Module 1: Atoms and Molecules Harmonic Oscillator (Continued) (Refer Slide Time:
More information1 Commutators (10 pts)
Final Exam Solutions 37A Fall 0 I. Siddiqi / E. Dodds Commutators 0 pts) ) Consider the operator  = Ĵx Ĵ y + ĴyĴx where J i represents the total angular momentum in the ith direction. a) Express both
More informationOutlines of Quantum Physics
Duality S. Eq Hydrogen Outlines of 1 Wave-Particle Duality 2 The Schrödinger Equation 3 The Hydrogen Atom Schrödinger Eq. of the Hydrogen Atom Noninteracting Particles and Separation of Variables The One-Particle
More informationAngular Momentum - set 1
Angular Momentum - set PH0 - QM II August 6, 07 First of all, let us practise evaluating commutators. Consider these as warm up problems. Problem : Show the following commutation relations ˆx, ˆL x ] =
More informationeff (r) which contains the influence of angular momentum. On the left is
1 Fig. 13.1. The radial eigenfunctions R nl (r) of bound states in a square-well potential for three angular-momentum values, l = 0, 1, 2, are shown as continuous lines in the left column. The form V (r)
More informationWe now turn to our first quantum mechanical problems that represent real, as
84 Lectures 16-17 We now turn to our first quantum mechanical problems that represent real, as opposed to idealized, systems. These problems are the structures of atoms. We will begin first with hydrogen-like
More informationLegendre Polynomials and Angular Momentum
University of Connecticut DigitalCommons@UConn Chemistry Education Materials Department of Chemistry August 006 Legendre Polynomials and Angular Momentum Carl W. David University of Connecticut, Carl.David@uconn.edu
More information4/21/2010. Schrödinger Equation For Hydrogen Atom. Spherical Coordinates CHAPTER 8
CHAPTER 8 Hydrogen Atom 8.1 Spherical Coordinates 8.2 Schrödinger's Equation in Spherical Coordinate 8.3 Separation of Variables 8.4 Three Quantum Numbers 8.5 Hydrogen Atom Wave Function 8.6 Electron Spin
More informationQuantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours.
Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours. There are 10 problems, totalling 180 points. Do all problems. Answer all problems in the white books provided.
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 21 Quantum Mechanics in Three Dimensions Lecture 21 Physics 342 Quantum Mechanics I Monday, March 22nd, 21 We are used to the temporal separation that gives, for example, the timeindependent
More information= ( Prove the nonexistence of electron in the nucleus on the basis of uncertainty principle.
Worked out examples (Quantum mechanics). A microscope, using photons, is employed to locate an electron in an atom within a distance of. Å. What is the uncertainty in the momentum of the electron located
More informationOh, the humanity! David J. Starling Penn State Hazleton PHYS 214
Oh, the humanity! -Herbert Morrison, radio reporter of the Hindenburg disaster David J. Starling Penn State Hazleton PHYS 24 The hydrogen atom is composed of a proton and an electron with potential energy:
More informationLecture #21: Hydrogen Atom II
561 Fall, 217 Lecture #21 Page 1 Lecture #21: Hydrogen Atom II Last time: TISE For H atom: final exactly solved problem Ĥ in spherical polar coordinates Separation: ψ nlml ( r,θ,φ) = R nl (r)y m l (θ,φ)
More informationIntroduction to Quantum Physics and Models of Hydrogen Atom
Introduction to Quantum Physics and Models of Hydrogen Atom Tien-Tsan Shieh Department of Applied Math National Chiao-Tung University November 7, 2012 Physics and Models of Hydrogen November Atom 7, 2012
More informationTHE RIGID ROTOR. mrmr= + m K = I. r 2 2. I = m 1. m + m K = Diatomic molecule. m 1 r 1. r 2 m 2. I moment of inertia. (center of mass) COM K.E.
5.6 Fall 4 Lecture #7-9 page Diatoic olecule THE RIGID ROTOR r r r r rr= (center of ass) COM r K.E. K = r r K = I ( = r r ) I oent of inertia I = r r = µ r µ = (reduced ass) K = µ r = µv z Prob l e reduced
More informationSeparation of Variables in Polar and Spherical Coordinates
Separation of Variables in Polar and Spherical Coordinates Polar Coordinates Suppose we are given the potential on the inside surface of an infinitely long cylindrical cavity, and we want to find the potential
More information1/2. e ikx + 1 1/2. e i2kx. 1/2. e i2kx We wish to evaluate Eq. (12.9) using the w(k) function of Eq. (12.10): = e (k k 0 )2 /2(δk) 2 e ikx dk
CHAPTER 1 Quantum Mechanical Model Systems SECTION 1.1 1.1 Half the particles here have twice the momentum of the other half and therefore also have twice the wavevector of the others. The analog of the
More informationThe Electronic Structure of Atoms
The Electronic Structure of Atoms Classical Hydrogen-like atoms: Atomic Scale: 10-10 m or 1 Å + - Proton mass : Electron mass 1836 : 1 Problems with classical interpretation: - Should not be stable (electron
More informationPhysics 221A Fall 2017 Notes 15 Orbital Angular Momentum and Spherical Harmonics
Copyright c 2018 by Robert G. Littlejohn Physics 221A Fall 2017 Notes 15 Orbital Angular Momentum and Spherical Harmonics 1. Introduction In Notes 13, we worked out the general theory of the representations
More information6 Wave equation in spherical polar coordinates
6 Wave equation in spherical polar coorinates We now look at solving problems involving the Laplacian in spherical polar coorinates. The angular epenence of the solutions will be escribe by spherical harmonics.
More informationPHYS 404 Lecture 1: Legendre Functions
PHYS 404 Lecture 1: Legendre Functions Dr. Vasileios Lempesis PHYS 404 - LECTURE 1 DR. V. LEMPESIS 1 Legendre Functions physical justification Legendre functions or Legendre polynomials are the solutions
More informationLecture 37. Physics 2170 Fall
Lecture 37 Will do hydrogen atom today After Thanksgiving break we have only two weeks before finals. We will talk about multielectron atoms, Pauli Exclusion Principle, etc. up thru Chapter 10. A few interesting
More informationCrystal field effect on atomic states
Crystal field effect on atomic states Mehdi Amara, Université Joseph-Fourier et Institut Néel, C.N.R.S. BP 66X, F-3842 Grenoble, France References : Articles - H. Bethe, Annalen der Physik, 929, 3, p.
More informationModern Physics. Unit 6: Hydrogen Atom - Radiation Lecture 6.3: Vector Model of Angular Momentum
Modern Physics Unit 6: Hydrogen Atom - Radiation ecture 6.3: Vector Model of Angular Momentum Ron Reifenberger Professor of Physics Purdue University 1 Summary of Important Points from ast ecture The magnitude
More informationSCIENCE VISION INSTITUTE For CSIR NET/JRF, GATE, JEST, TIFR & IIT-JAM Web:
Test Series: CSIR NET/JRF Exam Physical Sciences Test Paper: Quantum Mechanics-I Instructions: 1. Attempt all Questions. Max Marks: 185 2. There is a negative marking of 1/4 for each wrong answer. 3. Each
More informationQuantization of the E-M field
April 6, 20 Lecture XXVI Quantization of the E-M field 2.0. Electric quadrupole transition If E transitions are forbidden by selection rules, then we consider the next term in the expansion of the spatial
More information