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Unformatted text preview: 1 Physics 316 Solution for homework 9 Spring 2005 Cornell University I. EXERCISE 1 In Electrodynamics the charge density ρ ( ~x, t ) and the current density ~ j ( ~x, t ) satisfy a continuity equation, ∂ ∂t ρ ( ~x, t ) + ~ ∇ · ~ j ( ~x, t ) = 0 . (1.1) Show that the probability density ρ =  Ψ( ~x, t )  2 satisfies the same continuity equation together with the probability current density ~ j = ¯ h m Im { Ψ( ~x, t ) * ~ ∇ Ψ( ~x, t ) } . The wavefunction Ψ( ~x, t ) is a solution of the Schrdinger equation: i ¯ h ∂ ∂t Ψ( ~x, t ) = ¯ h 2 2 m ∇ 2 Ψ( ~x, t ) + V ( ~x )Ψ( ~x, t ) . (1.2) The term on the lefthand side of (1.1) can be manipulated the following way: ∂ ∂t ρ = ∂ ∂t ΨΨ * = Ψ ∂ ∂t Ψ * + Ψ * ∂ ∂t Ψ = Ψ • ¯ h 2 im ∇ 2 Ψ 1 i ¯ h V Ψ ‚ * + Ψ * • ¯ h 2 im ∇ 2 Ψ 1 i ¯ h V Ψ ‚ = ¯ h 2 im £ Ψ * ∇ 2 Ψ Ψ ∇ 2 Ψ * / + 1 i ¯ h V ΨΨ * 1 i ¯ h V ΨΨ * = ¯ h m = ' Ψ ∇ 2 Ψ * “ , (1.3) where we have used the fact that ={ z } = z z * 2 i . The righthand side can be written as: ~ ∇ · ~ j = ¯ h m = n ~ ∇ Ψ * · ~ ∇ Ψ + Ψ * ∇ 2 Ψ o = ¯ h m = n ~ ∇ Ψ * · ~ ∇ Ψ o + ¯ h m = ' Ψ * ∇ 2 Ψ “ = ¯ h m = ' Ψ * ∇ 2 Ψ “ . (1.4) The last step uses the fact that ~ ∇ Ψ * · ~ ∇ Ψ is real ([ ~ ∇ Ψ * · ~ ∇ Ψ] * = ~ ∇ Ψ * · ~ ∇ Ψ). Combining (1.3) and (1.4), we find that (1.1) is verified. II. EXERCISE 2 Given a wave function Ψ( x, t ) at t = 0 as Ψ( x, 0) = A a e x 2 2 σ and Ψ( x, 0) = A b e β  x  , where A a and A b are chosen in each case to normalize the wave function to 1. For each case, find the momentum representation of the wave function, i.e. find F ( k ) so that Ψ( x, t ) can be written as Ψ( x, t ) = Z ∞∞ F ( k ) e i [ kx ω ( k ) t ] √ 2 π dk . (2.1) The Fourier transform of Ψ( x ) = A a e x 2 2 σ has already been computed in class. We found that the normalization factor is A a = ( πσ ) 1 4 , and that F ( k ) = ‡ σ π · 1 4 e k 2 2 σ 1 . (2.2) 2 The probability amplitude is the absolute square of this function. Hence, the probability of finding the particle’s...
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This note was uploaded on 09/28/2008 for the course PHYS 316 taught by Professor Hoffstaetter during the Spring '05 term at Cornell University (Engineering School).
 Spring '05
 HOFFSTAETTER
 Physics, Momentum, Work

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