Chapter6_8 - PHY3063 R. D. Field The Infinite Square Well...

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PHY3063 R. D. Field Department of Physics Chapter6_8.doc University of Florida The Infinite Square Well (2) Normalized Wavefunctions: Now we require that 1 ) ( ) ( = dx x x n n ψ , and hence 1 2 sin 4 ) / ( sin 4 2 0 2 2 0 2 2 = = = LA d n LA dx L x n A n L π θ . Thus , L A 2 / 1 = and h / ) ( ) , ( t iE n n n e x t x = Ψ with ) / sin( 2 ) ( L x n L x n = . Probability Density: The probability density for the n th state is ) / ( sin 2 | ) , ( | ) ( 2 2 L x n L t x x n n ρ = Ψ = . Note that ρ n and not a function of time! These states are “stationary states” ( i.e. independent of time ). Average Value of x: The average position for the n th state is 2 4 2 ) ( sin 2 ) / ( sin 2 ) ( ) ( 2 2 2 2 0 2 2 2 0 2 * L n n L d n L dx L x n x L dx x x x x n L n = = = = Ψ Ψ = > < Average Value of x 2 : The average value of x 2 in the n th state is =
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