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Chapter1_12 - emitted and absorbed 2 Assumes that all...

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PHY4604 R. D. Field Department of Physics Chapter1_12.doc University of Florida Schrödinger’s Equation The Classical Hamiltonian: Classically the energy is the sum of the kinetic energy plus the potential energy as follows ( in one dimension ): ) ( 2 2 x V m p E x + = , and hence corresponding Quantum Mechanical Hamiltonian operator is ) ( 2 2 2 2 x V x m H op + = h and t i H op = h . Time-Dependent Schrödinger Equation: We now operate on the wave function Ψ (x,t) with both forms of H op yielding t t x i t x x V x t x m Ψ = Ψ + Ψ ) , ( ) , ( ) ( ) , ( 2 2 2 2 h h . Time-Independent Schrödinger Equation: Look for solutions of the form h / ) ( ) , ( iEt e x t x = Ψ ψ . Solutions of this form correspond to states with definite energy since H op | Ψ > = E| Ψ > . Substituting Ψ (x,t) into the time dependent equation yields ) ( ) ( ) ( ) ( 2 2 2 2 x E x x V dx x d m ψ ψ ψ = + h . Schrödinger Equation: 1. Ignores the creation and annihilation of particles, but photons may be
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Unformatted text preview: emitted and absorbed. 2. Assumes that all relevant velocities are much less than the speed of light ( i.e. non-relativistic). Free Particle: A free particle ( i.e. V(x) = 0) with energy E must satisfy ) ( ) ( 2 2 2 2 x E dx x d m = − h and hence ikx Ae x = ) ( where A is a constant and E m k = ) 2 /( 2 2 h . The state ψ (x) has k p x h >= < and 2 2 2 k p x h >= < which means Δ p x = 0 ( i.e. no uncertainty in p x ). Thus, ) ( ) , ( t kx i Ae t x ω − = Ψ corresponds to a free particle with definite momentum k p x h = and definite energy h = E , but the position of the particle is completely uncertain ( i.e. the particle is equally likely to be anywhere)....
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