Deriving Wave Function: Confused about (ix/a) & (-x^2/2a)?

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SUMMARY

The discussion centers on deriving the time-independent wave function in quantum mechanics, specifically addressing the confusion surrounding the terms (ix/a) and (-x^2/2a). Participants emphasize the importance of changing variables to k' = k - k0 and completing the square to arrive at the correct solution. The conversation highlights a common misunderstanding regarding variable substitution in the context of Fourier transforms, particularly with Gaussian functions.

PREREQUISITES
  • Understanding of time-independent wave functions in quantum mechanics
  • Familiarity with Fourier transforms and their applications
  • Knowledge of variable substitution techniques in calculus
  • Basic concepts of Gaussian functions and their properties
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  • Study the process of completing the square in mathematical expressions
  • Learn about variable substitution in integrals and its implications
  • Explore the properties and applications of Fourier transforms in quantum mechanics
  • Review Gaussian functions and their significance in wave function analysis
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Students and professionals in quantum mechanics, physicists working on wave function derivations, and anyone seeking to deepen their understanding of Fourier transforms and variable substitution in mathematical physics.

sus1234
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Homework Statement
Trying to understand how the time independent wavefunction is derived from this
Relevant Equations
?
1644783404516.png


It is asking to derive the time-independent wave function and has managed to get the answer of
1644783451034.png

and i am very confused as where (ix/a) and (-x^2/2a) came from ?

Thanks.
 
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The answer's wrong.

First, change variables to ##k' = k-k_0##, and then complete the square.
 
vela said:
The answer's wrong.

First, change variables to ##k' = k-k_0##, and then complete the square.
Hi,

Thank you for your reply, I'm not sure where I'm meant to sub in k' = k-k0, could you provide more guidance ?

thanks
 
sus1234 said:
Thank you for your reply, I'm not sure where I'm meant to sub in k' = k-k0, could you provide more guidance ?
You've never changed the variable of integration?
 

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