Transcript n-1 - KAIST
1 H E or K V E h 2 2 2 2 2 2 2 2 V ( x, y, z ) ( x, y, z ) E ( x, y, z ) z 8 m x y Solving Schrodinger Eqn. for H atom Prob(r, , ) (r, , ) n, l, m Quantum numbers n, l, m Characteristics of the Orbital: 3D Waves 2 Born’s Interpretaion of Orbitals Orbitals and Orbits: Heisenberg’s Uncertainty Principle An Analogy Eu E E h Electronic Transitions “In Chapter 3, 3.6 and 3.7 will NOT be delivered in the classes, and will be excluded from midterm exam.” Chapter 3. Wave Mechanics and the Hydrogen Atom: Quantum Numbers, Energy Levels, and Orbitals 3.1 Solving the Schrödinger Equation for the H Atom (appendix B) An operator can be defined in mathematics as follows: df ( x) d ^ f ( x) D f ( x) dx dx differential operator Operator is an entity operating automatically on the factor to its right The Schrodinger Equation can be simplified by defining Hamiltonian operator. _ h2 2 2 2 2 2 2 2 V ( x, y, z ) ( x, y, z ) E ( x, y, z ) y z 8 m x Hamiltonian operator, Ĥ ^ H E or K^ V E ^ K kinetic energy operator Solving the Schrödinger Eq: ĤΨ=EΨ To find the wave function Ψ and E you need to solve the differential eq The solutions depend on the potential energy V(x,y,z) Demonstration by a particle in a box The Coulomb potential for e – nucleus interaction in a H atom Spherical coordinate is more convenient than the Cartesian coordinate for the central or radial Coulomb potential 3 Schrödinger Equation: Wave Equation for Particles The classical wave eq for describing a standing wave function ψ in one dimension: d2ψ/dx2 + (4π2/λ2)ψ = 0 From de Broglie: λ2 = h2/2mK = h2/[2m(E-V)] d2ψ/dx2 + (4π2/λ2)ψ = d2ψ/dx2 + [8π2m(E-V)/h2]ψ = 0 or -(h2/ 8π2m)(d2ψ/dx2) + Vψ = Eψ ………….(Eq 1) Let’s define ‘momentum operator’, p^ = (h/2πi)(d/dx), based on p = mv = mdx/dt (see p739). And the kinetic energy operator: K^ = p^2/2m = (1/2m)(h/2πi)2(d/dx)2 = - (h2/ 8π2m)(d2/dx2)…(Eq 2) Eq 1 and Eq 2 K^ψ + Vψ = (K^ + V)ψ = Eψ or H^ψ = Eψ 4 A Particle in a Box (Appendix B) Solving the Schrödinger Eq: ĤΨ=EΨ Boundary condition: Boundary Condition yields quantum numbers Energy Wavefunction A Particle in a Box (Appendix B) A Particle in a Box (Appendix B) Wavefunction Normalization condition Schrödinger Eq for Hydrogen Atom V(x, y, z) – can be expressed in various coordinate systems. r cosΘ= z Spherical polar coordinates (r, θ, Ф) SPC vs. Cartesian coordinates (x, y, z) Coulomb potential, V → convenient to express in SPC (V = -e2/r) ; Cartesian, V = -e2/[x2 + y2 +z2]1/2 - reducing to that of an electron moving around a fixed proton. h 2 2 2 2 2 2 2 2 V ( x, y, z ) ( x, y, z ) E ( x, y, z ) z 8 m x y Cartesian Coordinate h 2 2 2 2 e 2 2 2 2 2 ( x, y, z ) E ( x, y, z ) 2 2 2 z ( x y z ) 8 m x y Spherical Polar Coordn. h 2 1 2 1 1 2 e 2 2 2 r ( x, y, z ) E ( x, y, z ) 2 sin 2 2 2 8 m r r r r sin r sin r Spherical Polar Coordinate h 2 1 2 1 r sin 2 2 2 8 m r r r r sin 1 2 e 2 ( x, y, z ) E ( x, y, z ) 2 2 2 r sin r h 2 1 2 e 2 1 r E ( x , y , z ) sin 2 2 r 2 sin 8 m r r r r 1 2 ( x, y , z ) 0 2 2 2 r sin h 2 2 e 2 r 2 1 1 2 2 Er ( x, y, z ) ( x, y , z ) 0 2 r sin 2 2 8 m r r r sin sin Depends only on r Depends only on , Schrödinger wave function → factored into radial (R) and angular (Y) functions (r , , ) R(r )Y ( , ) 10 Boundary Condition constraint that reality places on the solutions to a physically relevant equation (e.g., a quadratic equation having + & - two solutions for concentration) * Ψ is just the embodiment of de Broglie’s hypothesis of matter wave) Ψ must be smooth, single-valued, and finite everywhere in space Ψ must become small at large distances r from the nucleus (proton) Boundary Condition yields quantum numbers 11 3.2 Quantum Numbers n, ℓ, and m Wave function Ψ as the standing-wave for motions Characterized by 3 integers, n, ℓ, m → quantum numbers Why 3 quantum numbers? -> Because we have three dimensions (x, y, z) or (r, , ) one quantum number for each dimension is required.) Principal quantum number, n Energy from Schrödinger equation depends only on n n = 1, 2, 3, ···· Azimuthal quantum no, ℓ (Angular momentum q. no.) Associated with variation of an angle, θ ℓ = 0, 1, 2, 3, ···· , n-1 Magnetic quantum no, m (Projection quantum no.) behavior of H-atom in a magnetic field m = - ℓ, - ℓ+1, -ℓ +2, ···,-1, 0, 1, 2, ··· ℓ-1, ℓ 12 13 Degeneracy “Normally” the energy should depend on all three quantum numbers. Hydrogen atom is special in that the energy depends only on principal quantum number n. Two or more sets of quantum numbers corresponding to the same energy: referred as “degenerate” for example) 200, 211, 210, 21-1 states have the same energy. Each n: given total energy → n2 possible combinations of quantum numbers → degeneracy 14 Specification of the Energy: Ionization Ionization energy, IE Energy levels of hydrogen atom Ionization continuum: Electron’s energy is no longer quantized ! IE in excited state n is -En 15 Bohr – de Broglie Model Specification of the Wave Functions: Orbitals Schrödinger’s Wave Function Ψ should be single valued, continuous as a function of Θ and Φ: the wave meets itself as Θ and Φ cycle around the origin ← with quantum numbers Ex: For n = 3 (compare the one orbit in the Bohr-de Broglie model) 32 (nine) different ways the electron can vibrate to form a standing wave → 9 degeneracy Orbitals Mulliken: Ψ = a 3D Schrödinger’s wave function. He suggested the term, ‘orbital’, to refer to Ψ (= a 3D Schrödinger’s wave function) as a replacement of Bohr’s orbit Radial part, Rnℓ(r) Laguerre polynomial of (n-1) degree (highest power = rn-1) multiplied by e-r/(na0 ) Here, * a0 ← Bohr’s radius Angular part, Yℓm(θ,Ф) Legendre function (sin θ & cos θ) x eimФ 16 Department of Chemistry, KAIST 17 Department of Chemistry, KAIST 18 3.3 Characteristics of the Orbitals: Three-Dimensional Waves H-atom (the only atom for which the Schrödinger Eq can be solved exactly a model for bigger and many-electron atoms) n = 1, ℓ = 0, m = 0 → R10(r) and Y00(θ,Ф) 1s orbital of E1 a function of r only * spherically symmetric * exponential decaying * no nodes a) 1D Ψ vs. r b) 2D mountain by rotating graph (a) c) 2D contour map of the amplitude d) 3D cloud, proportional to the amplitude e) 3D boundary surface diagram 19 20 Nodes: places where the wave passes through zero amplitude “You can start to see nodes in the excited states, which brings us to n = 2” 21 n = 2, nℓm sets = 200, 210, 211, 21-1 2s (n = 2, ℓ = 0) 2p (n = 2, ℓ = 1) Ψ200((r,θ,Ф) = (R20)(Y00) ← 2s orbital zero at r = 2a0 = 1.06Å nodal sphere or radial node [r<2ao: Ψ>0, positive] [r>2ao: Ψ<0, negative] <node: due to a + - oscillation in the wave> Spectroscopic symbols ℓ = 0 → s (sharp), ℓ = 1 → p (principal: most intense) ℓ = 2 → d (diffuse), ℓ = 3 → f (fundamental) ℓ=4 → g ℓ = 5 → h, etc n=2 orbitals for H-Atom (a) 1D radial wave fn. vs. r (b) 2D cut through 3D clouds (c) 3D boundary surface shade-negative (right) Department of Chemistry, KAIST 22 23 n = 2, ℓ =1, m = 0 → 2p0 orbital : R21 Y10 · Ф = 0 → cylindrical symmetry about the z-axis · R21(r) → r/a0 no radial nodes except at the origin · cos θ → angular node at θ = 90o, x-y nodal plane (positive/negative) · r cos θ → z-axis 2p0 → labeled as 2pz n = 2, ℓ = 1, m = ±1 → 2p+1 and 2p-1 (complex functions containing both real and imaginary parts: a problem in graphing!) Take their linear combinations to obtain two real orbitals → constructed 2px and 2py · Use Y11(θ,Ф) → e±iФ = cosФ ± i sinФ ← Euler’s formula was used 2px and 2py (real functions using Euler’s formula) Px and py differ from pz only in the angular factors (orientations) Department of Chemistry, KAIST 24 25 n = 3, ℓ = 0(3s), 1(3p), 2(3d) Generalization · ns orbital → n -1 radial nodes, the same as the degree of R(r) polynomial (n-1 real roots) · ℓ increases from 0 to n -1 → number of nodes stays same · but, ℓ of the radial nodes → are exchanged for angular nodes, in the form of nodal plane or nodal cone · Algebraically, R(r) → loses one root for an increase in ℓ by one and the degree of Y(θ,Ф) in cosθ sinθ → increases by one Fig. 3.5 n = 3 orbitals for H (a) Radial parts (b) 2D cuts through 3D amplitude cloud (c) Boundary surface Department of Chemistry, KAIST 26 Fig. 3.6 Boundary surface Department of Chemistry, KAIST 27 28 29 h2 2 2 2 2 2 2 2 V ( x, y, z ) ( x, y, z ) E ( x, y, z ) z 8 m x y The kinetic and potential energies are transformed into the Hamiltonian which acts upon the wavefunction to give the quantized energies of the system and the form of the wavefunction so that other properties may be calculated. The wave nature of the electron has been clearly shown in experiments like the Davisson-Germer experiment. This raises the question "What is the nature of the wave?“: The wave is the wavefunction for the electron. Then, what is the nature of the wavefunction?? 30 31 3.4 Born’s Interpretations of Orbitals Schrödinger wavefunction: Not much physical significance, just a tool for predicting atomic properties Mulliken: Orbitals as a replacement of the planet-like orbits of the Bohr model Bohr: Anything that could create a pulse in one of Geiger’s electron detector tubes couldn’t be a smooth, continuous wave. Is there anything more we can say about the orbitals of the hydrogen atom? Born: Quantum version of the orbit of the electron. In those regions of space where the wave function is large, the electron is more likely to be found in orbit. Intensity of light ∝ amplitude 2 32 3.4 Born’s Interpretations of Orbitals Probability of finding an electron Prob(r,θ,Ф) ∝ ┃Ψ(r,θ,Ф)┃2 probability probability amplitude (Ψ) (3.13) probability density Normalization constant → a proportionality const. in Eq.(3.13) to meet the total probability = 1. ┃Ψ┃2 x r2 sinθ dr dθ dФ → the probability of finding an electron within that tiny volume volume element ┃Ψ┃2 → electron density -e┃Ψ┃2 → charge density : “wave function must be the quantum version of the orbit of the electrons” : “in space where the wave function is large, the electron is more likely to be found in orbit” 33 In 1926 he collaborated with his student Werner Heisenberg to develop the mathematical formulation of the wave function in the new quantum theory. He later showed, in the work for which he is perhaps best known, that the solution of the Schrödinger equation has a statistical meaning of physical significance. Max Born (1882-1970, Germany) 1926, Probability density function 1954 Nobel Prize in Physics Einstein: “Probability cloud” is a strange sort of mist made up of a single electron ! “The theory yields a lot, but it hardly brings us any closer to the secret of the Old One. In any case I am convinced that He does not throw dice.” 34 Radial Distribution Functions: RDFs Prob(r) = r2[R(r)]2 2 0 r 2 sin d d 2 0 4 r 2 4 r 2 R(r )Y ( , ) 2 4 r 2 R(r ) Y ( , ) 2 4 r R(r ) 1/ 2 2 2 2 2 r 2 R(r ) 2 2 ┃Ψ┃2 x 4πr2 dr 35 Schrödinger’s waves vs Bohr’s orbits Bohr’s orbits: Each orbit has a fixed radius, rn = n2 ao ; the electron is always found at r = rn ; Bohr’s RDF is a sharp spike at rn. Schrödinger’s RDF is a smooth curve with one or more peaks Probability amplitude and Probability density ┃Ψ┃2 : electron density -e┃Ψ┃2 : charge density Ψ vs ┃Ψ┃2 1) nodal surfaces are not changed 2) negative parts of orbitals become positive 3) boundary surfaces are same 37 RDF for H RDF for H Smooth with one or more peaks Nodes appear at radii of zero probability Falls to zero smoothly at large r Department of Chemistry, KAIST 38 39 Bohr’s Model for Hydrogen Atom 1s electron density ao r 40 2s RDF: the electrons spend most of its time in the range of outer peak, and much less around the inner peak. How ?? How the electron gets from the outer to inner region without trapping into the node at the nodal space ?? Key is in the Wave Property of Electrons: the standing wave has amplitude everywhere simultaneously Outer turning point – In Newton’s mechanics, a bound particle cannot stray beyond a fixed radius. Quantum tunneling – An electron in the orbital has a small but finite probability beyond the outer turning point. 3.5 Heisenberg’s Uncertainty Principle e- passing through x slit becomes uncertain in v, by Δx, due to the wave nature. λ ~ 0.1 Å or 105 eV, short enough to resolve the position of e-, can distroy or at least disturb the atom. IE (H) = 13.6 eV Figure 3.10.Two views of Heisenberg’s uncertainty principle 3.5 Heisenberg’s Uncertainty Principle So, “you cannot accurately measure Δx and Δpx at the same time.” Position uncertainty x momentum uncertainty ≥ h/4π, at the same time ( = 5.27 x 10-35J s ) (This principle is valid between any pair of the conjugate variables) ΔE Δ t ≥ h / 4 (In effect, the electron’s energy fluctuates within narrow bounds, and what is supposed as the electron’s energy is, in fact, an average value over the very narrow time parameter) (This fluctuating electron energy might suggest a violation of the conservation of energy, but not if the electron is exchanging energy at the Planck level with other electrons or particles) The basis for the initial realization of fundamental uncertainties in the ability of an experimenter to measure more than one quantum variable at a time. 43 Why does a hydrogen atom have a ground state, and what determines the energy of that state? To require the electron to be roughly a distance ao from the nucleus implies an uncertainty in momentum ∆p ~ h/ao p cannot be much less than ∆p p ~ h/ao and K ~ p2/m ~ h2/mao2 Since ao ~ h2/me2, K ~ me4/h2 ~ -E1 (page 59, Eq. 2-13) Confining a particle more tightly increases its kinetic energy → The electron cannot collapse into the proton even though the Coulomb attraction drawing it in → Small particle in bound states show a lowest state where K is nonzero; the particle is said to possess zero point energy. Werner Heisenberg (1901-1976) Born’s Assistant 1927, Uncertainty Principle 1932 Nobel Prize in Physics “The battle with Born grew so intense in the early months of 1927 that Heisenberg reportedly burst into tears at one point,…” “The theory yields a lot, but it hardly brings us any closer to the secret of the Old One. In any case I am convinced that He does not throw dice” –Einstein- 44 3.6 Orbitals and Orbits: An Analogy Orbitals – analysis in terms of radial and angular motion meaning of n, ℓ, and m. Electronic motion → in-and-out, ① vibrational (r), and round-and-round, ② rotational (Θ,Φ)motions. i) Vibrational motion · distance r changes with the motion · determined by n - ℓ- 1, specifies the no. of nodes in the radial function · smaller ℓ for a given n → more vibration ii) Rotational motion · angles θ and Ф change with the motion · determined directly by ℓ Department of Chemistry, KAIST 45 iii) Orientations of Rotational Plane · z-axis projection of the angular momentum → determines the possible orientations of the plane of rotation. For fixed ℓ m = 0 → The rotational plane is contained in the z-axis. m = /ℓ/ → The plane is fixed in the x-y plane, perpendicular to the z-axis. m = 0-ℓ →The plane is “tilted”. m = -ℓ → The rotation is clockwise. m = +ℓ → The rotation is counter clockwise. 46 Classical analogous orbits of the electron in 1s, 2s and 2p orbitals Department of Chemistry, KAIST 47 Bohr’s correspondence principle As the quantum numbers get larger and larger, the classical analogy is found to get better and better. In the limit of large quantum numbers, the behavior of any mechanical system becomes classical. Quantum mechanics contains Newtonian mechanics as a special case. Department of Chemistry, KAIST 48 3.7 A Qualitative Description of Electronic Transition Imagine a hydrogen atom in a stream of photons (in a ray of light) along the x-axis: The oscillating electric field Ex will exert an electrical force on the electron by –eEx and the potential energy of eExx. e·x : electric dipole Electric dipole transition Selection rule for H: ∆ ℓ ≡ ± 1 Department of Chemistry, KAIST 49 3.7 Qualitative Description of Electronic Transition Electric dipole transition Selection rules En state Electric dipole transition hν Emission Δl ≡ l2 - l1 = ±1 1s to 2p NOT 1s to 2s Eℓ state Energy uncertainty xpx xpx energy uncertainty t x mv t mvv t t t time uncertainty (Δt τ- radiative lifetime) 50