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024_p4c_lec_particles_as_waves_2 - Wavefunction What really...

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4/28/2010 1 What really is a quantum (matter) wave, anyway? Ψ (x, t) contains all possible knowable information about the particle. Every particle has a wavefunction , Ψ (x, t) . Wavefunction Ψ 2 (x, t) represents the probability that the particle will be found in a particular state. Ψ (x, t) can be determined by solving Schrödinger’s Equation Ψ (x, t) represents the superposition of many different waves of possibilities The Schrödinger’s equation is used to determine the wave function ψ (x, t) associated with a non-relativistic particle : 2 2 ( , ) ( , ) d x t x t ψ ψ = Schrödinger’s Equation 2 ( ) ( , ) 2 U x x t i m dx t ψ + = = This can be rearranged to look like a wave equation: 2 2 2 0 d k dx ψ ψ + = The most general solution to Schrödinger’s equation is: ( ) ikx ikx x Ae Be ψ = + Schrödinger’s Equation spatial solution ( ) i t t e ω ψ = time evolution Schrödinger’s equation plays the role of Newton's laws and conservation of energy in classical mechanics - i.e., it predicts analytically and precisely the future behavior of a dynamic system in terms of probability of the outcomes of its events.
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4/28/2010 2 A particle of mass m and total energy E is confined to an infinite potential energy well such that 0 0 0 and x L U x x L < < = a) Draw an energy well diagram. b) Show that (2/L) ½ sin(kx) is a solution to the Schrödinger’s equation, where k = 8 π 2 m/h 2 . c) Determine the wavelengths, and the d) allowed energies for this particle, and its e) probability density. The wave functions ψ (x) for a particle in a rigid box are analogous to standing waves on a string that is tied at both ends. 2 sin n n x L L π ψ = 2 n L n λ = n is referred to as the quantum number of the particle. ψ 1 (x) ψ 2 (x) ψ 3 (x) x x x L L L 0 0 0 Energy is quantized . A confined particle is allowed only certain discrete levels of energy! When confined to an infinite potential well, the energy of the particle is given by: Energy 2 1 n E n E = where n = 1, 2, 3, . . . , is the quantum number of the particle 2 1 2 8 h E mL = is the energy of the ground state.
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