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Key Concepts for Exam #1 Correspondence Principle Quantum mechanics approaches classical mechanics in the limit of large quantum numbers. Photoelectric Effect If the frequency of incident light is above the threshold frequency, then as the intensity of light increases, the kinetic energy of ejected electrons remains constant and the number of electrons increases. In addition, as the frequency of light increases, the kinetic energy of ejected electrons increases and the number of electrons remains constant. If the frequency of the light is below the threshold frequency, then each individual photon lacks enough energy to remove an electron from the metal and no electrons are released from the metal. Blackbody Radiation c T max 2 where the constant c 2 1.44 10 2 m 5 Bohr Atom: Spectral Lines 1 1 1 R 2 2 n1 n2 where Rydberg's constant, R, is 1.097 107 m1 DeBroglie Waves h p where h is Planck's constant, and p is the momentum of the particle Well-behaved Wave Functions 1. The function must be single-valued; i.e. at any point in space, the function 2 must have only one numerical value. 2. The function must be finite and continuous at all points in space. The first and second derivatives of the function must be finite and continuous. 3. The function 2 must have a finite integral over all space. all space * d a finite number Harmonic Oscillators: Hermite Polynomials The Schrodinger equation for the harmonic oscillator is 2 d 2 1 2 ˆ H kx E 2 dx 2 2 The energy eigenvalues for the harmonic oscillator are given by 1 h k 1 E v v h 2 2 m 2 where Planck's constant, h,is 6.626 1034 J s; k is the force constant; is the frequency of the oscillator, and v = 0, 1, 2, 3, … 1 k 2 where k is the force constant, is the reduced mass, and is the frequency of the oscillator. Kronnecker delta nm nm *n m d The eigenfunctions n and m are orthogonal when nm 0 . The other possible value of the Kronnecker delta is 1, and it occurs when n = m. The Kronnecker delta can only equal zero or one; no other values are possible. Expectation Values * ˆ d ˆ * d or ˆ * ˆ d , where is normalized Orthogonality Two functions f and g are orthogonal to each other if f * gdq g * fdq 0 where f * is the complex conjugate of f and g * is the complex conjugate of g. Particle In a Box The potential energy, V, is zero inside the box, and infinite outside the box. The wave function must be zero outside the box. Therefore, the wave function must equal zero at x = a and x = 0 (where a is the length of the box). h2n2 The energy eigenvalue for a one dimensional box of length a is E where n 1,2,3,... 8ma 2 The energy eigenvalue for a three dimensional box of sides a, b, and c is 2 2 h 2 n x n y n z2 E where n x , n y , n z 1,2,3,... 8m a 2 b 2 c 2 Linear Momentum d d pˆ x i dx i dx Angular Momentum The angular momentum operators have the following eigenvalue equations Lˆ2Yl ,m l l 1Yl ,m where l 0,1,2,... where m 0,1,...,l Lˆ Y m Y z l ,m l l ,m Spin Operators Sˆ z s s m s s 2 Sˆ 2 s ss 1 2 s l where m s where s 1 2 1 (for electrons) 2 Other Concepts Complex conjugates, probability density Operators, commutators, eigenvalues, eigenfunctions How to normalize wave functions Orbitals and quantum numbers: m, l , ml , ms Heisenberg's Uncertainty Principle Pauli Exclusion Principle Legendrian Operator, Laplacian Operator Diatomic Rigid Rotor Plots of radial distribution functions (e.g. Rn2,l r 2 vs. r) Bohr magneton Aufbau Principle for Atomic Electron Configurations Closed vs. open shell of electrons Relevant pages in Mortimer are Chapters 14, 15, 16, and Sections 19.1 and 19.2. Please note that the sections from Chapter 16 that address symmetrization, Russell-Saunders coupling, and Slater determinants will NOT be covered by Exam #1. Note: This list is only a guide to help you study. It is NOT comprehensive, and the exam may cover any topics discussed in class.