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Lecture14(1)

# Lecture14(1) - Particle in three-dimensional space The...

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1 Physical Chemistry Lecture 14 Solving multidimensional problems Functions of multiple independent variables Functions can be Very complex Very simple Independent variables Each is required, independent of the others, to describe any function Functions of multiple variables may be Inseparable Separable dy cx dy cx e e e y x F Example y g x f y x F Separable y g x f y x F e Inseparabl r z y x iables var t independen common of Sets x z xy x z y x F 2 2 ) ( 2 ) , ( : ) ( ) ( ) , ( : ) ( ) ( ) , ( : ) , , ( ) , , ( ) , , ( Equality of functions of different variables Two functions of two different independent variables Equality means that they must be equal to a constant , some number that is not a function of either variable C y g and C x f y g x f ) ( ) ( ) ( ) ( Particle in three-dimensional space The world has three spatial dimensions Generalize simple problem to reflect the three- dimensional aspects of problem Example: a free particle in 3-D space Appropriate model for a gas molecule in an unconfined space The particle’s motions in the x, y, and z dimensions are considered to be independent of each other Particle in three-dimensional space In free space, there is no potential energy All energy is kinetic Three contributions to the kinetic energy Hamiltonian function is parsed into a sum of three terms Each term depends on an independent variable (x, y, or z) Substitution by correspondence gives a Hamiltonian operator that has three terms, each depending on a different co-ordinate only z y x z y x H H H H m p m p m p m T T H 2 2 2 2 2 2 2 p p z y x H H H H z y x m H ˆ ˆ ˆ ˆ 2 ˆ 2 2 2 2 2 2 2

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Lecture14(1) - Particle in three-dimensional space The...

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