Schrödinger equation: P(r)>1 ?

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Discussion Overview

The discussion revolves around the interpretation of the probability density derived from the Schrödinger equation for the ground state of the hydrogen atom. Participants explore the implications of calculating probability values and the conditions under which these values are meaningful.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant presents the solution to the Schrödinger equation for the hydrogen atom and calculates a probability density, noting that it exceeds 1 at a specific radius.
  • Another participant clarifies that the value calculated is a probability density, suggesting that to find the actual probability, one must integrate over a region.
  • A different participant inquires about the functions used in graphical representations of the charge distribution and mentions an alternative formula for charge distribution based on the wave function.
  • Another participant expresses uncertainty regarding the specific graphs and suggests looking into other mathematical functions that might relate to the hydrogen atom.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretation of the probability density and its implications. There are competing views on how to properly calculate and understand the probability associated with the wave function.

Contextual Notes

There is an assumption that the probability density must be integrated over a region to yield meaningful probabilities, but the specifics of this integration and the implications of exceeding 1 are not fully resolved.

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Schrödinger equation: P(r)>1 ?

I have the solution of the Schrödinger equation for the ground state of the hydrogen electron. The solution ist:

u100(r)=sqrt(1/(pi*a^3))*exp(-r/a)

If I want to calculate some probability values I do this with:

P(r)=4*pi*r^2*|u100|^2

If I set r=10^-13 I get a value that is greater than 1, I get P(10^-13)=10^34.

This cannot be. Whats wrong with my probability formula?
 
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P is a probability... density. To get the probability that it lies in a region, you need to integrate over that region.
 
With what function are the graphs plotted? (I mean one like this:
http://panda.unm.edu/courses/finley/P262/Hydrogen/img106.gif )

But to calculate the charge distribution of the electron I can take:

Charge(r)=-e*abs(u100)^2

thanks for your help
 
Last edited by a moderator:
I'm not familiar with those graphs -- I might look at the Bernoulli or Airy functions, but if it's not them, I don't think I could guess. (Also, I think I've read that the Airy functions appear in dealing with the hydrogen atom)
 

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