Normalization of a gaussian wavefunction

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Discussion Overview

The discussion revolves around the normalization of a Gaussian wavefunction in the context of quantum mechanics. Participants are exploring the mathematical process required to find the normalization constant for the wavefunction given by $$\psi(x)=Ne^{-\frac{|x-x_o|}{2a}}$$ and the associated integral for normalization.

Discussion Character

  • Mathematical reasoning
  • Homework-related
  • Technical explanation

Main Points Raised

  • One participant presents the wavefunction and the normalization condition, expressing uncertainty about how to integrate the resulting expression due to the absolute value in the exponent.
  • Another participant suggests separating the integral into two parts to eliminate the absolute value, which could simplify the integration process.
  • A participant confirms that the variable \(x\) is likely real, indicating that complex variables are not expected knowledge for the class.
  • Further clarification is provided on how to handle the absolute value by defining cases based on the value of \(x\) relative to \(x_0\).
  • One participant acknowledges the helpfulness of the suggested approach and expresses appreciation for the clarification.

Areas of Agreement / Disagreement

Participants generally agree on the approach of separating the integral to facilitate the normalization process. However, there is no consensus on the specific integration technique or the handling of the absolute value, as some uncertainty remains regarding the integration of the modified expression.

Contextual Notes

Participants note that the integration involves handling absolute values, which complicates the integral. There is also mention of potential complexities if \(x\) were to be treated as a complex variable, although this is not expected in the current context.

Who May Find This Useful

This discussion may be useful for students studying quantum mechanics, particularly those working on wavefunction normalization and integration techniques involving absolute values.

skate_nerd
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I'm given a wavefunction (I think it's implied this is some sort of solution to the Schrödinger equation) in my quantum mechanics class, and I need to normalize it to find its constant coefficient.
So I have
$$\psi(x)=Ne^{-\frac{|x-x_o|}{2a}}$$
And the formula for normalizing this to find \(N\) would be
$$\int_{-\infty}^{\infty}\bar{\psi(x)}\psi(x){dx}=1$$
Plugging in \(\psi(x)\) gives
$$1=\int_{-\infty}^{\infty}N^{2}e^{-\frac{|x-x_o|}{a}}dx$$

At first I was thinking I could just take the derivative of the exponent and divide by that to solve the integral but I realized that wouldn't work out right, and this integral behaves somewhat like a gaussian integral like when you need to integrate \(e^{-x^2}\).

I know the process of how to integrate \(e^{-x^2}\) from negative infinity to infinity (defining the integral as I and then squaring it, changing to polar coordinates, u-subbing and then taking the root of that solution to get \(\sqrt{\pi}\)) but when I tried to do that with \(\frac{|x-x_o|}{a}\) instead of \(x^2\) I end up with a weird expression in the exponent that I don't know what to do with. I was hoping changing to polar coordinates would work but I don't see how to do that with this.
Any guidance would be really appreciated! Thanks
 
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Separate the integral to integrate $(-\infty, x_0]$ and $[x_0, \infty)$ separately. Then the absolute value disappears, you can factor out the constant $x_0$ and are left with a standard exponential. Or is $x$ a complex number or something like that? It's been a while since my introductory quantum mechanics class.
 
I don't think we are expected to know how to work with complex variables in this class so \(x\) is probably real.
But yeah I see what you're saying, and breaking up the bounds to make two integrals will be helpful if I can figure out how to integrate this crazy integrand...
 
skatenerd said:
I don't think we are expected to know how to work with complex variables in this class so \(x\) is probably real.
But yeah I see what you're saying, and breaking up the bounds to make two integrals will be helpful if I can figure out how to integrate this crazy integrand...

With complex numbers it could probably be worked about the same since the integral would be spherically symmetric, but I wouldn't know. Anyway once you've broken it up it becomes simple because the annoying absolute values disappear, as:

$$|x - x_0| = \begin{cases}x - x_0 ~ ~ ~ \mathrm{if} ~ x > x_0 \\ x_0 - x ~ ~ ~ \mathrm{if} ~ x < x_0\end{cases}$$

Furthermore since $|x - x_0|$ is symmetric you only need to compute one side of the integral, the whole integral is just twice that.
 
Ahhh I see what you're saying now! Thanks that helps a lot, neat little trick...
 

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