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lecture_16

# lecture_16 - 1 A par ticle confined to a one-dimensional...

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d 2 Ψ dx 2 + 2 m h 2 ( E - V ( x )) Ψ= 0 d 2 Ψ dx 2 + 2 m h 2 ( E - V 0 ) Ψ = 0 d 2 Ψ dx 2 - α 2 Ψ = 0 2 = 2 m h 2 ( V 0 - E ) d 2 Ψ dx 2 2 Ψ= 0 d 2 Ψ dx 2 + 2 m h 2 E Ψ = 0 d 2 Ψ dx 2 + k 2 Ψ = 0 k 2 = 2 m h 2 E Ψ I = Ae x x < 0 Ψ III = Be - x x L Ψ II = C sin kx + D cos kx 0 < x < L In the infinite square well D=0. Here Ψ II =A at x=0 and II =B at x=L , so both sin and cos are possible. Match wave functions and its derivatives at x=0 and x=L ; 4 eqs., 4 unkowns. The matching occurs for discrete values of particle energy, E .

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• The wave function penetrates the walls, which lowers the energy levels relative to the infinite square well. • The wavelengths are longer, the corresponding particle momenta are lower.
6.5: Three-Dimensional Infinite-Potential Well Degeneracy Analysis of the Schrödinger wave equation in three dimensions introduces three quantum numbers that quantize the energy. A quantum state is degenerate when there is more than one wave function for a given energy. Degeneracy results from particular properties of the potential energy

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lecture_16 - 1 A par ticle confined to a one-dimensional...

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