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Probabilistic Modeling in Quantum Theory Jennifer Hartmann Classical Mechanics vs. Quantum Mechanics • Basic Assumptions of Classical Mechanics: 1) No limit to accuracy with which one or more dynamical variables can be simultaneously measured. 2) No limit to the number of dynamical variables that can be accurately measured simultaneously. 3) Expressions for velocity are continuously varying functions of time, the kinetic energy can vary continuously. • • However, with the involvement of very small particles, these assumptions are not valid. Classical mechanics fail to explain the experimental observations of: – – Atomic Spectra Photoelectric Effect • Atomic Spectra – Atoms only emitted light at specific wavelengths with a specific energy. Photographic Plate Atomic Lamp Light through slit Prism λ or frequency – Balmer (1885) fit the data for a hydrogen atomic lamp to: 1/λ=R((1/nf2)-(1/ni2)) where R is the Rydberg constant (109677.8 cm-1) and “n” are integers called “quantum numbers”. – Energy is “quantized” meaning it can only exist at certain energy levels and not in between • Like rungs on a ladder • Correspondence Principle • Photoelectric Effect – Emission of an electron from a metal surface by light eMetal • Intensity of light has no effect on K.E., just number of electrons ejected. K. E. of the electrons excited off the metal surface •K.E. of the ejected electrons is independent of light intensity, only a function of frequency. 0 •Classical physics has no explanation. Frequency of Light Wave/Particle Duality • An explanation for the previous experimental observations is that although light is primarily thought of as a wave, can sometimes act like a particle as well. – i.e. photons • De Broglie Wavelength – Louis de Broglie suggested that although electrons have been previously regarded as particles also show wave-like characteristics with wavelengths given by the equation: λ = h/p = h/mv – h (Planck’s constant) = 6.626 x 10-34 J·s Quantum Mechanics • Taking into account the idea that energy is quantized and the wave/particle duality properties of very small particles, Quantum Theory provides the following postulates: 1) The state of a system is fully described by a “wavefunction” Ψ(x, t) such that the quantity Ψ* Ψ dx is proportional to the probability of finding the system or the particle in dx. 2) For every observable property of the system, there exists a corresponding operator. 3) The value of an observable is determined by applying the corresponding operator to the wavefunction of a system. Finding the Wavefunction for a System • Hamiltonian Operator (Energy, H): (-h2/2m) 2 + V(x) where 2=(δ2/δx2 + δ2/δy2 + δ2/δz2) and h=h/2π • Find the wavefunction that is an eigenfunction of the energy operator. –HΨ=εΨ – Easy to measure energy – Easiest to solve Atomic Structure • Consider simplest atom with one electron: z Electron θ Proton r y φ x With proton of mass (M) and charge (+Ze), and electron with mass (m) and charge (-e). Schrödinger Equation (-h2/2μ) 2Ψ-(Ze2/4πε◦r)Ψ=εΨ where μ is the reduced mass and the term (4πε◦) accounts for vacuum permittivity Rewritten in polar coordinates as: (1/r2)(δ/δr(r2(δΨ/δr)))+(1/r2sinθ)(δ/δθ sinθ(δΨ/δθ))+(1/r2sin2θ)(δ2Ψ/δφ2)+(2μ/h2 )(ε+(Ze2/4πε◦r))Ψ=0 Solve the Schrödinger Equation • Separation of Variables Ψ=Ψ(r)Ψ(θ)Ψ(φ)=(R)(Θ)(Φ) • Term I (1/φ)(δ2Φ/δφ2)=-m2 • Term II (1/sinθ)(d/dθ(sinθ(dΘ/dθ)))-(m2/sin2θ)Θ+λΘ=0 • Term III (1/r2)(d/dr(r2dR/dr)+[(2μ/h2)(ε+(Ze2/4πε◦)-(λ/r2)]R=0 • The solutions of Terms I and II are the spherical harmonics Y(Θ,Φ) Φ=(2π)eimφ m=0,±1,±2,±3… Θ are the Associated Legendre Polynomials: • Function of two quantum numbers: l & m • l=0,1,2,3… (angular momentum quantum number) • m=0,±1,±2,±3…|m|≤l (z-component of angular momentum quantum number) • The solutions of Term III are the Associated Laguerre Polynomials: • Function of quantum number n=1,2,3… (principle quantum number) Atomic Orbitals • Once the wavefunction(Ψ) is determined, according to postulate 1, Ψ2 gives the probabilistic distribution of the electron with respect to nuclear distance. • This distribution is then corrected for a three dimensional space forming 3-D probabilistic distributions called orbitals. References • Hanna, Melvin W.. Quantum Mechanics in Chemistry. 3. Menlo Park, Ca: Benjamin/Cummings, 1981. • Hanna, Melvin W.. Quantum Mechanics in Chemistry. 3. Menlo Park, Ca: Benjamin/Cummings, 1981.