How do I prove that the wave function R_{10}(r) is normalized?

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Homework Help Overview

The discussion revolves around proving the normalization of the wave function R_{10}(r) = \frac{2}{a_{0}^{3/2}}\cdot e^{-\frac{r}{a_0}}. Participants are examining the mathematical requirements for normalization in the context of quantum mechanics.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • One participant attempts to compute the normalization integral directly but finds the result does not equal 1. Another participant suggests the inclusion of the volume element r²dr for spherical coordinates, which seems to resolve the issue. A request for clarification on the origin of this requirement is also made.

Discussion Status

The discussion is active, with participants exploring the necessary components for the normalization integral. A potential resolution is suggested regarding the volume element, but further clarification is sought on the underlying principles.

Contextual Notes

Participants are working within the framework of quantum mechanics and addressing the specifics of spherical coordinates in the context of wave function normalization.

karkas
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Homework Statement


Its simple, I've encountered this problem in Beiser's book and it doesn't seem right:

Prove that the function R_{10}(r) = \frac{2}{a_{0}^{3/2}}\cdot e^{-\frac{r}{a_0}} is normalized.

Homework Equations


\int_{-\infty}^{\infty} |\psi|^2 dV = 1

The Attempt at a Solution


I figured it's simply just taking the integral
R_{10} (r)=\int^{\infty}_{0}( \frac{2}{a_{0}^{3/2}}\cdot e^{-\frac{r}{a_0}})^2dr = 1
but the result is not 1, it's 2/a_0^2
 
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Don't you need r²dr instead of just dr?
(volume integral, spherical-polar coords.)
 
Yes it appears that solves the problem, thanks!
 
I would enjoy a pointer as to where this comes from exactly, maybe a link?

edit: just standard triple integration in 3 coordinates?
 
Last edited:

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