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Unformatted text preview: 1 PHYS809 Class 10 Notes Separation of variables in Cartesian geometry Separation of variables in Cartesian geometry, with ( ) ( ) ( ) ( ) , , , x y z X x Y y Z z = leads to the set of equations 2 2 2 2 2 2 , , , d X X dx d Y Y dy d Z Z dz = = = (1.1) where , and are constants that satisfy 0. + + = (1.2) We see that the constants cannot all have the same sign. The sign of the constants is to some extent dictated by the particular problem under consideration. The problem of a rectangular box with potential specified on the sides of the box can be solved by using superposition of solutions for the box in which the potential is nonzero only on one side of the box. For a box with sides on which x = 0 , x = a , y = 0 , y = b , z = 0 and z = c , suppose the potential is nonzero only on the z = c side. The boundary conditions on the adjacent sides require that and are both negative which makes positive. Furthermore, these boundary conditions require discrete values for these constants such that...
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This note was uploaded on 12/02/2011 for the course PHYS 809 taught by Professor Macdonald during the Fall '11 term at University of Delaware.
 Fall '11
 MacDonald

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