Lecture13 - Free particle in one dimension For a value of E...

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1 Physical Chemistry Lecture 13 Solving Schroedinger’s Equation for Simple Systems Simple constant-energy model systems Unrestricted translation Free particle in one dimension Restricted translation Particle in a one- dimensional box Vibrational motion Particle in a harmonic potential The free particle in one dimension The free particle only has kinetic energy No potential energy Schroedinger’s equation is a second- order differential equation For a specific value of E there are two solutions 2 2 2 2 2 2 2 2 2 2 2 ˆ 2 ˆ ˆ mE dx d E dx d m E H dx d m T H Free particle in one dimension For a value of there are two solutions Particular solutions Can be described in two different, but related, ways Use exponential functions Represent planes waves in space, one moving in the positive and one in the negative direction General solution for the energy problem is a linear combination of particular solutions No limits on values of k All values possible No quantization of energy m k E x x x e A x e A x k mE dx d k ikx ikx 2 ) ( ) ( ) ( ) ( ) ( 2 2 2 2 2 2 2 Eigenfunctions of other operators Consider the particular states of the free particle The particular states are eigenfunctions of the linear momentum operator The momentum is precisely known for this state, Corresponds to a state with the particle traveling in the positive direction with a precise speed, v = /m  ikx x ikx ikx x ikx ikx x ikx x x x e p e k e p ike i e p e dx d i p p equation Eigenvalue dx d i p ˆ ˆ Orthogonality and completeness Eigenfunctions of a hermitian operator corresponding to different eigenvalues are orthogonal .
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This note was uploaded on 02/02/2012 for the course CHEM 419 taught by Professor Staff during the Fall '10 term at University of Delaware.

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Lecture13 - Free particle in one dimension For a value of E...

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