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Class_10new

# Class_10new - PHYS809 Class 10 Notes Separation of...

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1 PHYS809Class10Notes Separation of variables in Cartesian geometry Separation of variables in Cartesian geometry, with ( ) ( ) ( ) ( ) , , , x y z X x Y y Z z φ = leads to the set of equations 2 2 2 2 2 2 , , , d X X dx d Y Y dy d Z Z dz α β γ = = = (1.1) where α , β and γ are constants that satisfy 0. α β γ + + = (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 non-zero 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 non-zero only on the z = c side. The boundary conditions on the adjacent sides require that α and β are both negative which makes γ positive. Furthermore, these boundary conditions require discrete values for these constants such that sin , sin , x X m a y Y n b π π = = (1.3) where m and n are integers. We then find that

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Class_10new - PHYS809 Class 10 Notes Separation of...

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