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Physical Chemistry Lecture 9 Introduction to Quantum Mechanics The energy “problem” Early experiments Observed only certain emission frequencies Spectra of atoms H and other atoms Colors of mineral solutions Scientists involved have very familiar names Angstrom Balmer Bunsen Fraunhofer Kirchoff Rydberg Other “problems” E = hν Planck’s theory of blackbody radiation Inferred quantization of modes of a cavity Bohr’s theory of the H atom Rejected classical electrodynamics to interpret Rydberg’s observation Quantization of energy DeBroglie’s hypothesis Quantized electron momentum to explain a variety of effects ν 1 1 = ℜ 2 − 2 m n p = N Photoelectric effect Light ejects electrons from a metal plate No emission below a minimum frequency Number of electrons emitted is proportional to the intensity Electron kinetic energy Does not depend on light intensity Depends linearly on the frequency above the threshold frequency Slope of line gives Planck’s constant Diffraction by particles Diffraction is a property of waves Davisson and Germer demonstrated that electrons (particles) interacting with matter act as if they diffract like waves Further example of wave-particle duality Diffraction pattern from electrons interacting with crystalline cobalt Classical mechanics and states In mechanics, one describes precisely Where particles are Where particles are going subject to forces The system’s state is completely defined by the positions and momenta of all particles -- trajectories Everything is absolutely determined by forces and initial conditions t v (t ) = v (0) + 1 F (t ' )dt ' ∫ m0 t r (t ) = r (0) + ∫ v(t ' )dt ' 0 Heisenberg’s uncertainty principle There is no way to measure simultaneously with infinite precision the position and the conjugate momentum of a particle Cannot know trajectories exactly Cannot describe a system as precisely as with classical mechanics Inherent property of all systems Theory must reflect this property Quantum mechanical state Heisenberg’s principle eliminates the correspondence of position and momentum to “state” inherent to classical physics Obvious that one needs a definition of state of a quantum system The “state” of a system is given by a function of the positions of particles, but it is not the trajectory under definite forces Function gives all possible information on a system Properties found by operation Definition builds in the restriction of limited information, consistent with Heisenberg’s observation Born’s idea: State is related to the probability of a particular condition, since we cannot know positions and momenta precisely. Schroedinger’s equation Chemists primarily interested in systems with constant energy Need a means (like Newton’s equations for classical mechanics) to determine the state of constant energy Functions corresponding to constant qualities found from eigenvalue equations HΨ = EΨ Summary Early experiments showed that many systems, particularly small systems, could only have certain energies Agreement with observation achieved by “violating” classical mechanics Uncertainty principle recognizes the wave nature of particles and the inability to use deterministic mechanics Totally new theory required Definition of state Operators for properties Eigenvalue equations for property values