Explanation According to the Equation 2.176, the time dependent Schrodinger equation for a particle in a uniform gravitational field, ψ ( x , t ) = ψ 0 ( x + 1 2 g t 2 , t ) exp [ − i m g t ℏ ( x + 1 6 g t 2 ) ] (I) Here, ψ ( x , t ) is the free fall wave function, m is the mass of the particle, g is the acceleration due to gravity, and t is the time period. According to the free Gaussian wave packet, ψ 0 = ( 2 a π ) 1 / 4 1 γ e − a ( x + 1 2 g t 2 ) 2 / γ 2 ; γ = 1 + 2 i a ℏ t / m (II) Partially differentiate the relation with respect to time of equation (I). ∂ ψ ∂ t = [ ∂ ψ 0 ∂ t + ψ 0 ( − i m g ℏ ) ( x + 1 2 g t 2 ) ] exp [ − i m g t ℏ ( x + 1 6 g t 2 ) ] (III) Partially differentiate the relation with respect to time of equation (II). ∂ ψ 0 ∂ t = ( 2 a π ) 1 / 4 { − 1 γ 2 ( d γ d t ) + 1 γ [ − 2 a γ 2 ( x + 1 2 g t 2 ) g t − a ( x + 1 2 g t 2 ) 2 ( − 2 γ 3 d γ d t ) ] } e − a ( x + 1 2 g t 2 ) 2 / γ 2 = [ − 1 γ d γ d t − 2 a g t γ 2 ( x + 1 2 g t 2 ) + 2 a γ 3 ( x + 1 2 g t 2 ) 2 d γ d t ] ψ 0 (IV) But, d γ d t = 1 2 γ ( 2 i a ℏ m ) = i a ℏ γ m Substitute the above relation in equation (III). ∂ ψ 0 ∂ t = [ − i a ℏ γ 2 m − 2 a g t γ 2 ( x + 1 2 g t 2 ) + 2 i a 2 ℏ γ 4 m ( x + 1 2 g t 2 ) 2 ] ψ 0 ∂ ψ ∂ t = [ − i a ℏ γ 2 m − 2 a g t γ 2 ( x + 1 2 g t 2 ) + 2 i a 2 ℏ γ 4 m ( x + 1 2 g t 2 ) 2 − i m g ℏ ( x + 1 2 g t 2 ) ] ψ Conclusion: Mean while, partially differentiate the relation with respect to position of equation (I). ∂ ψ ∂ x = [ ∂ ψ 0 ∂ t − ψ 0 i m g t ℏ ] exp [ − i m g t ℏ ( x + 1 6 g t 2 ) ] (V) Partially differentiate the relation with respect to position of equation (II). ∂ ψ 0 ∂ x = − 2 a [ ( x + 1 2 g t 2 ) / γ 2 ] ψ 0 Substitute the above relation in equation (V)

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ISBN: 9781107189638

Author: Griffiths, David J., Schroeter, Darrell F.

Publisher: Cambridge University Press

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Chapter 2, Problem 2.51P

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Chapter 2 Solutions

Introduction To Quantum Mechanics

Ch. 2.1 - Prob. 2.1P Ch. 2.1 - Prob. 2.2P Ch. 2.2 - Prob. 2.3P Ch. 2.2 - Prob. 2.4P Ch. 2.2 - Prob. 2.5P Ch. 2.2 - Prob. 2.6P Ch. 2.2 - Prob. 2.7P Ch. 2.2 - Prob. 2.8P Ch. 2.2 - Prob. 2.9P Ch. 2.3 - Prob. 2.10P

Ch. 2.3 - Prob. 2.11P Ch. 2.3 - Prob. 2.12P Ch. 2.3 - Prob. 2.13P Ch. 2.3 - Prob. 2.14P Ch. 2.3 - Prob. 2.15P Ch. 2.3 - Prob. 2.16P Ch. 2.4 - Prob. 2.17P Ch. 2.4 - Prob. 2.18P Ch. 2.4 - Prob. 2.19P Ch. 2.4 - Prob. 2.20P Ch. 2.4 - Prob. 2.21P Ch. 2.5 - Prob. 2.22P Ch. 2.5 - Prob. 2.23P Ch. 2.5 - Prob. 2.24P Ch. 2.5 - Prob. 2.25P Ch. 2.5 - Prob. 2.26P Ch. 2.5 - Prob. 2.27P Ch. 2.5 - Prob. 2.28P Ch. 2.6 - Prob. 2.29P Ch. 2.6 - Prob. 2.30P Ch. 2.6 - Prob. 2.31P Ch. 2.6 - Prob. 2.32P Ch. 2.6 - Prob. 2.34P Ch. 2.6 - Prob. 2.35P Ch. 2 - Prob. 2.36P Ch. 2 - Prob. 2.37P Ch. 2 - Prob. 2.38P Ch. 2 - Prob. 2.39P Ch. 2 - Prob. 2.40P Ch. 2 - Prob. 2.41P Ch. 2 - Prob. 2.42P Ch. 2 - Prob. 2.44P Ch. 2 - Prob. 2.45P Ch. 2 - Prob. 2.46P Ch. 2 - Prob. 2.47P Ch. 2 - Prob. 2.49P Ch. 2 - Prob. 2.50P Ch. 2 - Prob. 2.51P Ch. 2 - Prob. 2.52P Ch. 2 - Prob. 2.53P Ch. 2 - Prob. 2.54P Ch. 2 - Prob. 2.58P Ch. 2 - Prob. 2.63P Ch. 2 - Prob. 2.64P

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