p39_073 - 73. (a) The wave function is now given by (x, t)...

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73. (a) The wave function is now given by Ψ( x, t )= ψ 0 h e i ( kx ωt ) + e i ( kx + ) i = ψ 0 e iωt ( e ikx + e ikx ) . Thus | Ψ( x, t ) | 2 = ¯ ¯ ψ 0 e iωt ( e ikx + e ikx ¯ 2 = ¯ ¯ ψ 0 e iωt ¯ ¯ 2 ¯ ¯ e ikx + e ikx ¯ ¯ 2 = ψ 2 0 ¯ ¯ e ikx + e ikx ¯ ¯ 2 = ψ 2 0 | (cos kx + i sin )+(cos i sin ) | 2 =4 ψ 2 0 (cos ) 2 =2 ψ 2 0 (1 + cos 2 ) . (b) Consider two plane matter waves, each with the same amplitude ψ 0 / 2 and traveling in opposite directions along the x axis. The combined wave Ψ is a standing wave: Ψ( x, t ψ 0 e i ( kx ) + ψ 0 e i ( kx + ) = ψ 0 ( e ikx + e ikx ) e iωt =( 2 ψ 0 cos ) e iωt . Thus, the squared amplitude of the matter wave is | Ψ( x, t ) | 2 =(2 ψ 0 cos ) 2 ¯ ¯ e iωt ¯ ¯ 2 ψ 2 0 (1 + cos 2 ) , which is shown below. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2468 kx (c) We set
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This note was uploaded on 11/12/2011 for the course PHYS 2001 taught by Professor Sprunger during the Fall '08 term at LSU.

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