How Can You Find ψ(x,t) from ψ(x) for a Free Particle?

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Homework Help Overview

The discussion revolves around finding the time-dependent wavefunction ψ(x,t) for a free particle, given the spatial wavefunction ψ(x) = (pi/a)^(-1/4) * exp(-ax^2/2). Participants are exploring methods to compute the necessary integral for the coefficient expansion function θ(k).

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the normalization of the wavefunction and the computation of the integral θ(k) using various methods, including error functions and complex analysis. There are suggestions to utilize Gaussian integral techniques and to consider the properties of the error function at specific limits.

Discussion Status

The discussion is active, with participants offering different approaches to evaluate the integral. Some express uncertainty about the feasibility of solving the integral directly, while others suggest alternative methods that could simplify the process. There is no explicit consensus on the best approach yet.

Contextual Notes

Participants note the potential challenges posed by the integral and the importance of understanding the techniques involved in solving it, indicating a learning-focused environment. There is an emphasis on the value of manual computation skills in this context.

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Homework Statement


Consider the free-particle wavefunction,
ψ(x)=(pi/a)^(-1/4)*exp(-ax^2/2)

Find ψ(x,t)


The Attempt at a Solution



The wavefunction is already normalized, so the next thing to find is coefficient expansion function (θ(k)), where:

θ(k)=∫dx*ψ(x)*exp(-ikx) from -infinity to infinity

But this equation seems to be impossible to solve without error function (as maple 16 tells me).

Is there any trick to solve this?
 
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The error function has very nice properties at the infinity, so you should be able to compute the integral. Alternatively, you could use the apparatus of complex analysis to evaluate the integral.
 
Why not use some of the simple Gaussian integral tricks (like completing the square in the exponential)? Or am I missing something?
 
You can avoid the error function because of the limits on the integral, or equivalently, use the nice properties of the error function at those limits.

You should really learn to crank this integral out by hand, though. The techniques used are useful to know.
 

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