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Introductory Quantum Mechanics/Chemistry Historical Bohr's Atom Wave vs. Particle De Brogile's Hypothesis Quantum Tools Heisenberg Operator Algebra Postulates Applications Spring 2014 Translational Vibrational Rotational Spectroscopy (NMR) Genealogy of Quantum Mechanics Classical Mechanics (Newton) High Low Velocity Mass Relativity Wave Theory of Light (Huygens) Maxwell’s EM Theory Quantum Theory Quantum Electrodyamics Electricity and Magnetism (Faraday, Ampere, et al.) Energy and Matter Size of Matter Particle Property Wave Property Large – macroscopic Mainly Unobservable Intermediate – electron Some Some Small – photon Mainly Few E = m c2 The Wave Nature of Light c E h The speed of light is constant (2:30 into video) Continuous Spectrum Line Spectra and the Bohr Model Line Spectra Bohr Model • Colors from excited gases arise because electrons move between energy states in the atom. (Electronic Transition) Class Data Line Spectra and the Bohr Model Line Spectra Line Spectra - Bohr Model En 2.178 10 18 Z2 J n2 Z charge ( H 1 ) Ei f E f Ei h Ei f 1 1 hc 18 h 2.178 10 J 2 2 n f ni Line Spectra Mathcad Calns Fall 12 Bohr Model Experiment 1. Calculate the expected electronic transitions ( λ in nm ) in the Balmer Series. [from higher n’s to n=2] 2. Calculate the series limit transition in the Balmer Series. 3. Using the Spectroscope, record observable lines (color and λ) for the Hydrogen discharge tube and compare them to your Bohr Model calculations. 4. Record observable lines for Oxygen and Water. [Calculate possible lines in the visible region for the Oxygen atom using an appropriate Z.] 5. Compare the observed lines for the three discharge tubes. 6. Using the following sample table (spreadsheet) as a guide, discuss your results. Tube Color Observed λ Bohr Model λ ( ni →nf ) Hydrogen ….. Class Data Bohr Model Calculations for the H-atom The Balmer Series: higher-n's down to nf=2 8 c 3.00 10 m s 1 n i 3 34 h 6.626 10 n f 2 19 E 3.025 10 h c E J s J 18 E 2.178 10 1 nf2 2 ni 1 9 nm 10 J 7 6.571 10 m m 657.1 nm Class Data Line Spectra and the Bohr Model • • • • Limitations of the Bohr Model Can only explain the line spectrum of hydrogen adequately. Can only work for (at least) one electron atoms. Cannot explain multi-lines with each color. Cannot explain relative intensities. • Electrons are not completely described as small particles. • Electrons can have both wave and particle properties. The Wave Behavior of Matter The Uncertainty Principle • Wavelength of Matter: h mv • Heisenberg’s Uncertainty Principle: on the mass scale of atomic particles, we cannot determine exactly the position, direction of motion, and speed simultaneously. • If x is the uncertainty in position and mv is the uncertainty in momentum, then h x· (mv) 4 Mathcad Calns The Heisenberg Uncertainty Principle Comparison between microscopic (electron) and macroscopic (SR) BBT QM Micro: Determine the uncertainty in finding the el ectron in an atom with a 1% uncertainty in determination of its speed. Macro: Determine the uncertainty in finding student along a 100 m track with a 1% uncertainty in determination of his speed. h x ( m v ) 4 h 6.626 10 4 h x 34 h x m ( v ) v 4 m v v 1% 34 xe 6.626 10 kg 1 107 4 9.1 10 31 10 xe 5.8 10 %xe xe de 100 J s 1 100 m s 1 m xe %xe 11 5 10 100 3 %xe 1.2 10 m 34 6.626 10 xSR J s 4 [ ( 90) kg ] ( 0.1) m s 36 xSR 5.9 10 m %xSR 1 xSR 100m 100 36 %xSR 5.9 10 J s Energy and Matter Size of Matter Particle Property Wave Property Large – macroscopic Mainly Unobservable Intermediate – electron Some Some Small – photon Mainly Few E = m c2 Quantum Mechanics and Atomic Orbitals • Schrödinger proposed an equation that contains both wave and particle terms. ^ H E • Solving the equation leads to wave functions. Quantum Chemistry • Bohr orbits replaced by Wavefunctions • Three Formulations – Differential Equation Approach by Schrödinger – Matrix Approach by Heisenberg – Operator/Linear Vector-Space approach by Dirac (Use Scaled-down version here) Operator Algebra • Operator Equation • Algebraic rules: – – – – • • • • Equality Addition/Subtraction (linear operators) Multiplication (order of operation) Division (reverse operation) Commutators Eigenvalue Equation Compound Operators Ladder Operators Operator Equation ^ f g Algebraic Rules (i) Equality (ii) Addition(/Subtraction) [Linear Operators follow Distributive Law] Algebraic Rules (iii) Multiplication [Order of Operation] (iv) Division Commutators ^ ^ ^ ^ ^ ^ ^ C [ , ] BBT-V Commutators – S10 ^ ^ ^ ^ ^ ^ ^ C [ , ] Eigenvalue Equation ^ f ( x) a f ( x) Eigenvalue Equation – S10 ^ f ( x) a f ( x) Compound Operators ^ ^2 h x d 2 2 2 2 (cartesian ) 2 2 2 x y z 2 1 2 1 1 2 2 r 2 sin r r r r sin r 2 sin 2 2 2 Ladder Operators For commutator s of the form : [ ˆ , ˆ ] k ˆ where : ˆ is a " generator" for the eigenfunct ions of ˆ . ( that is : ˆ is a Ladder Operator. ) Sequential examples : 1) Define : ĥ x̂ 2 dˆ 2 &  x̂ dˆ Then :  raising operator for ĥ B̂ lowering operator for ĥ & B̂ x̂ dˆ Ladder Operators: Seq. Eg. Cont/… 2) Show that : [ ĥ ,  ] 2  3) Show that : [ ĥ , B̂ ] - 2 B̂ 4) (a) (b) (c) Is f(x) e - x2 2 an eigfcn. of ĥ ? [Ans : Yes, e.v. 1] Show that  f(x) (x - d̂) e x2 2 2x e x2 2 Show that the new function, g(x) 2x e x2 2 , is an eignfcn. of ĥ with e.v. 2 1 3 . (d) - x2 2 Show that B̂ g(x) B̂ (2x e ) reproduces the original function, f(x) (within a constant); that is, the operators  & B̂ are raising & lowering operators for the eigfcn' s of ĥ . Ladder Operators Ladder Operators (S11) Postulates of Quantum Theory • The state of a system is defined by a function (usually denoted and called the wavefunction or state function) that contains all the information that can be known about the system. • Every physical observable is represented by a linear operator called the “Hermitian” operator. • Measurement of a physical observable will give a result that is one of the eigenvalues of the corresponding operator for that observable. TBBT: QM-joke Postulate I (x, y, z) stationary state Often, is complex : Z(a, b) where : a Real(Z) & b Imag(Z) For example : Z a ib in which Complex Conjugate Z a ib i complex number unit vecto r * 1 i 1 ; i i ; i 1 ; i . i 2 3 4 Z Z Z a 2 b2 * 2 Postulate I …cont… Probabilit y Distributi on Function * Criteria of must be single - valued over all space must be finite & continuous over all space must have a finite integral over all space 2 Normalizat ion of * d 1 all space (1D - cartesian) : d dx (3D - cartesian) : d dx dy dz (3D - spherical) : d r 2 dr sin d d Postulate I …cont… Probabilit y of finding particle within a certain region of space (1D - x only) : Prob( x1 xx ) 2 x2 x1 dx 2 dx 2 x2 dx if normalized x1 2 Postulate II Variable CM QM Position x,y,z xˆ , yˆ , zˆ Momentum Potential Energy mv V pˆ x i x i x pˆ y i y i y pˆ z i z i z Vˆ ( x, y, z ) V ( x, y, z ) Postulate II…cont… Variable Kinetic Energy Total Energy CM QM 1 2 p2 T mv 2 2m T+V ˆ Hamiltonia n 2 2 2 2 2 Tˆ 2 2 2 2 2m x y z 2m 2 ˆ 2 V ( x, y , z ) 2m Starting Point in all Quantum Mechanical Problems. Heisenberg’s Uncertainty Principle Variable A : ˆ i ai i Variable B : ˆ i bi i [ˆ , ˆ ] 0 A fundamental incompatibility exists in the measurement of physical variables that are represented by non-commuting operators: “A measurement of one causes an uncertainty in the other.” 1 A B [ˆ , ˆ ] 2 [ˆ , ˆ ] * (ˆˆ ˆˆ ) d Heisenberg’s Uncertainty Principle 1 A B [ˆ , ˆ ] 2 1 p x x [ pˆ x , xˆ ] 2 ˆ [ pˆ x , xˆ ] , x [d , xˆ ] i i x i 1 p x x 2 h 2 Particle in a Box (1D) – 1 – S14 ∞ ∞ V V=0 V=∞ V=∞ 0 x a Figure 11.6 Potential energy for the particle in a box. The potential (V) is zero for some finite region (0<x<a) and infinite elsewhere. Particle in a Box (1D) – 2 – S14 Particle in a Box (1D) – 3 – S14 Particle in a Box (1D) – 4 – S14 Wolfram Integrator Particle in a Box (1D) – 5 – S14 Particle in a Box (1D) – 1 – S13 ∞ ∞ V V=0 V=∞ V=∞ 0 x a Figure 11.6 Potential energy for the particle in a box. The potential (V) is zero for some finite region (0<x<a) and infinite elsewhere. Particle in a Box (1D) – 2 – S13 Particle in a Box (1D) – 3 – S13 Particle in a Box (1D) – 4 – S13 Particle in a Box (1D) – 5 – S13 Particle in a Box (1D) - Interpretations ● MC plots of Wavefunctions n Excel plots ● 2 n x sin a a Plots of Squares of Wavefunctions a ● Check Normalizations 2 dx 1 0 ● How fast is the particle moving? Comparison of macroscopic versus microscopic particles. Calculate v(min) of an electron in a 20-Angstrom box. Calculate v(min) of a 1 g mass in a 1 cm-box n2 h2 En 8 m a2 Particle in a Box (1D) - Applications Particle in a Box (3D) – 1 – S14 z Explained using S13 slides a 0 a y a x Figure 11.8 The cubic box. For the three-dimensional particle in a box, the potential is zero inside a cube and infinite elsewhere. This could represent the situation of a particle inside a container with perfectly rigid, impenetrable walls. Particle in a Box (3D) – 2 – S14 Particle in a Box (3D) – 3 – S14 Particle in a Box (3D) – 1 – S13 z a 0 a y a x Figure 11.8 The cubic box. For the three-dimensional particle in a box, the potential is zero inside a cube and infinite elsewhere. This could represent the situation of a particle inside a container with perfectly rigid, impenetrable walls. Particle in a Box (3D) – 2 – S13 Particle in a Box (3D) – 3 – S13 Particle in a Box (3D) - Solutions 3 2 ny y nx x 2 nz z x y z sin sin sin a a a a h2 2 2 2 E Ex E y Ez n n n x y z 2 8ma Particle in a Box (3D) -Degeneracies Energy* 3 6 9 11 12 14 17 38 54 g 1 3 3 3 1 6 3 9 12 States (1,1,1) (2,1,1) (1,2,1) (1,1,2) (2,2,1) (2,1,2) (1,2,2) (3,1,1) (1,3,1) (1,1,3) (2,2,2) (3,2,1) (3,1,2) (2,3,1) (2,1,3) (1,2,3) (1,3,2) (3,2,2) etc (5,3,2) etc; (6,1,1,) etc (5,5,2) etc; (6,3,3) etc; (7,2,1) etc *Energy given in units of h2/8ma2 Particle in a 2D (non-Symmetric) Plane - Solutions y L V0 : 0xL & 0 yW W x 2 D ny y 2 2 nx x x y sin sin L W L W 2 2 ny nx h Ex E y 2 2 8m L W 2 E2 D MCad Particle in a 2D (non-symmetric) Plane L 20 ( x y ) W 10 2 L 2 W n x 10 nx x n y y sin L W sin n y 2 The Harmonic Oscillator – Model for Vibrations Use: V=½kx2 ; reduced mass; Ladder Operators. Applications include: Heat Cap’s, Blackbody radiation. Eigenfunct ions : v Av [ H v ( y )] e y2 2 v quantum number H v Hermite Polynomial s 1 Eigenvalue s : Ev v h vo 2 where : vo 1 k 2 reduced mass : m1 m2 m1 m2 Mathcad The Rigid Rotor – Model for Rotations Use: m & ; spherical polar coord’s; Ang. Mom. Operators. App’s: Structural Spectr; Rotat. Spectr; NMR, ESR, Mössbauer. Eigenfunct ions : combinatio ns of sin n and cos n ( 1) h 2 Eigenvalue s : E 8 2 I where : integral quantum numbers Moment of Inertia : I R 2 R radius of rotation of system Mathcad Quantum Numbers of Wavefuntions Quantum # Symbol Values Description Principle n 1,2,3,4,… Size & Energy of orbital Azimuthal 0,1,2,…(n-1) for each n Shape of orbital Magnetic m -…,0,…+ for each Relative orientation of orbitals within same Spin ms +1/2 or –1/2 Spin up or Spin down Azimuthal Quantum # Name of Orbital(CD) 0 s (sharp) 1 p (principal) 2 d (diffuse) 3 f (fundamental) 4 g Quantum Mechanics and Atomic Orbitals Orbitals and Quantum Numbers Figure 6.27 MO-Mcad’s Spectroscopy: Quantum Interpretations Measurements of quantized E-levels involving discrete transitions. E hc Selection Rules – Depends on allowed QM changes in quantum number between pairs of stationary states ( Ψ’s ). Golden Rules of Transition s I x i x j d * I y i y j d * I z i z j d * If any of above integrals is non-zero, then transition allowed. Intensity Ix2 + Iy2 + Iz2 If all integrals are zero => Forbidden Transition Introductory Quantum Mechanics Ei f h 1 1 hc 2.178 10 18 J 2 2 n f ni Bohr's Atom 1 2 p x x Wave vs. Particle h 2 Heisenberg 2 ˆ 2 V ( x, y , z ) 2m E hc E = m c2 Historical De Brogile's Hypothesis ^ Quantum Tools Operator Algebra Applications H E Postulates 2 n x n sin a a Translational Vibrational Rotational Spectroscopy (NMR) a 2 dx 1 0 n2 h2 En 8 m a2