Hydrogen atom. Simple question I think.

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Discussion Overview

The discussion revolves around the energy levels of the hydrogen atom as derived from the Schrödinger equation, specifically addressing why electrons remain bound to the atom at certain energy values and the distinction between bound and unbound states.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes that the energy values obtained from the Schrödinger equation for the hydrogen atom are approximately ##E_n \approx \frac{1}{n^2}## and questions why electrons do not leave the atom at certain energy levels.
  • Another participant explains that the minimum energy required to remove the electron, known as the ionization energy, is derived from the limit of the energy values as ##n## approaches infinity, indicating that there are infinitely many bound states with total energy lower than the energy needed for removal.
  • A different perspective suggests that a square wave is not a solution to the Schrödinger equation, implying a potential limitation in the approach to understanding the energy states.
  • One participant clarifies that the bound states have negative energies, specifically ##E_n \approx -\frac{1}{n^2}##, while scattering states correspond to unbound electrons with positive energies.

Areas of Agreement / Disagreement

Participants express differing views on the nature of energy states and the implications for electron binding, indicating that multiple competing perspectives remain without a consensus.

Contextual Notes

There are unresolved aspects regarding the definitions of bound and unbound states, as well as the implications of different energy values on electron behavior.

LagrangeEuler
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When you solving Sroedinger eq for hydrogen atom you get energy values ##E_n\approx \frac{1}{n^2}##. Why for some value of this energy electron don't leave atom?
 
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Because the minimum energy needed to remove the electron, the ionization energy, is obtained from
$$
IE = \lim_{n \rightarrow \infty} E_n
$$

Another way to put it is that you have an infinite number of bound states, where the total energy (kinetic + potential) is lower than the energy necessary to remove the electron.
 
Maybe because a square wave is not a solution to the equation.
 
LagrangeEuler said:
When you solving Sroedinger eq for hydrogen atom you get energy values ##E_n\approx \frac{1}{n^2}##. Why for some value of this energy electron don't leave atom?
The bound states have negative (!) energies ##E_n\approx -\frac{1}{n^2}##

The (continuum) scattering states have positive energies; they correspond to umbound electrons.
 

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