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Equations – Atomic Physics Energies E hf E 2 p2 c2 m2 c 4 Potential energy between two charges: qq V r 1 2 4 o r Potential due to Angular Momentum: 1 2 V r 2mr r 2 Rydberg Energy: 2 e mr 8 o ao ' 4 o 2 2 Photoelectric effect ( W Work function): Ek,max E W ER e 2 2 Energy of electron level: En n2h2 8mL2 Operators Operator x Eigenfunction = Eigenvalue x Eigenfunction Ô O Expectation value: O * Ôdx Commutator: Ô1, Ô2 Ô1Ô2 Ô2Ô1 X direction: x̂ x Momentum (x-direction): P̂x i x 2 P̂x 2 2 2 x Momentum (y-direction): P̂y i y 2 P̂y 2 2 2 y Momentum (z-direction): Pz i z 2 Pz 2 2 2 z Angular momentum (x direction): Lx i sin cot cos Spin Ŝ s s 1 Angular momentum (y-direction): Ly i cos cot sin Angular momentum (z-direction): Lz i Energy: Ê 2 2m 2 v r Quantum Mechanics Schrödinger's Equation (TDSE 1D): 2 2 V i 2 2m x t i kz t Ae Schrödinger's Equation (TISE, 1D): 2 d 2 V x E 2m dx 2 dT i ET dt T Solution for TISE 1D: 1 n x n n!2 a 1 2 x2 x 2 H n e 2a a Gaussian Hermite Polynomial Normalization Exponential a m Schrödinger's Equation (TISE, 2D): 2 2 2 V x, y E 2m x 2 y 2 E En, x En, y nx ny 1 Solution for TISE 2D: x 2 y2 x y 2 nx ,ny x, y H nx H ny e 2a a a Schrödinger's Equation (TISE, 3D): 2m 2 r, , 2 r E V x r, , 0 Normalization of Waveform. Overall probability = 1 all n, , n, ,m r,, 2 * 1 2 all n, , Orthonormal waveforms: 1 m n * dx m n 0 m n 1 Equations – Atomic Physics Linear superposition of two waveforms C11 C2 2 x1, x2 Normalized if C12 C2 2 1 Represents 2 energy states. Probability of E1 is C12 , E2 is C2 2 1 Quantized energy: E n 2 Quantized energy difference: 1 1 E ER Z 2 2 2 n n 1 Fermions: x1 , x2 x2 , x1 1 x1, x2 a x1 b x2 a x2 b x1 2 Quantized angular momentum: L2 1 2 n1 Degeneracy of an atom: 2 2 1 n2 Pauli exclusion principle No two electrons in the same atom can have all quantum numbers the same 0 Magnetic Moment of particle in atom e L L g M 2m g and M depend on the particle being looked at. Rotational Energy: I 2 L2 n n 1 Erot 2 2m 2 2 n 1 E I 1 Spin of a fermion: s 2 gs Heisenberg Uncertainty Principle: Et Uncertainty: y y Identical particles: 2 2 y Sizes Bohr Radius 4 o 2 ao me2 ao ' when reduced mass is used 2 Quantized spin: S 2 s s 1 1 a x1 b x2 a x2 b x1 2 Quantum Numbers Principle Quantum Number: n 1,2,3,... Orbital angular momentum quantum number: 0,1,...,n 1 Azimuthal / Magnetic Quantum Number: m ,...,0,..., 1 Spin Quantum Number: ms 2 j s 2 2 x1, x2 ,t A x1 B x2 e E EA EB Indistinguishable particles: iEt x1, x2 x2 , x1 De Broglie Wavelength h h h p mv 2mEk Compton wavelength if v c 1 Vibrational energy: Evib n 2 Bosons: x1 , x2 x2 , x1 2 2 2