ParticleinBoxLect3ME501F2011

# - Purdue University ME 501 Statistical Thermodynamics Lecture 3 Particle in a Box Solution of the SWE Prof Robert P Lucht Room 2204 Mechanical

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Patterned Border Template 1 Purdue University School of Mechanical Engineering ME 501: Statistical Thermodynamics Lecture 3: Particle in a Box Solution of the SWE Prof. Robert P. Lucht Room 2204, Mechanical Engineering Building School of Mechanical Engineering Purdue University West Lafayette, Indiana [email protected] , 765-494-5623 (Phone) August 26, 2011

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Patterned Border Template 2 Purdue University School of Mechanical Engineering Lecture Topics Solution of the SWE by separation of variables. Particle in a box solution of the SWE. Translational energy states.
Patterned Border Template 3 Purdue University School of Mechanical Engineering SWE Solution by Separation of Variables • The Schrödinger Wave Equation is    2 2 , ,, , 2 rt i V rt tm     • Assume that the wavefunction ( r ,t) can be written as the product of two functions:       , r Tt  where ( r ) is purely a function of r and T(t) is purely a function of t . Substituting into the SWE gives    2 2 () () (,) () () 1 2 rTt i Vrt E  • For consideration of atomic and molecular structure, we will consider potential functions that are functions of r only,     , Vr o n l y

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Patterned Border Template 4 Purdue University School of Mechanical Engineering SWE Solution by Separation of Variables • Substituting into (E1) and rearranging gives • The RHS of (E2) is a function of r only, and the LHS is a function of t only. Therefore, the equation can be satisfied only if the LHS and RHS are equal to the same constant. Call this separation variable and rearrange to give Equation (E3) is often called the time-independent SWE. Equation (E4) describes the time-dependence of the wavefunction for the special case of a time-independent potential function.         2 2 2 2 dT t r i Vr E Tt d t m r         2 2 3 2 rV rr r E m      4 dT t iT t E dt
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## This note was uploaded on 12/26/2011 for the course ME 501 taught by Professor Na during the Fall '10 term at Purdue University-West Lafayette.

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- Purdue University ME 501 Statistical Thermodynamics Lecture 3 Particle in a Box Solution of the SWE Prof Robert P Lucht Room 2204 Mechanical

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