I Particle in a potential well

[Malvina]

TL;DR Summary

What are energies responsible for?

Hello. Help me understand the energies. The energy of a particle in a potential well, formula 1. An electron in the field of a nucleus, as an example of a particle in a potential well, and it turns out according to this formula that the further from the nucleus, the more energy is needed. And more for what?
And according to formula 2, it turns out that the further the electron is from the nucleus, the less energy it needs to break away from the nucleus.
What is the energy from the first formula "responsible" for (directly proportional to n) and what is the energy from the second formula "responsible" for (inversely proportional to n)

 

[BvU]

formula 1. An electron in the field of a nucleus, as an example of a particle in a potential well, and it turns out according to this formula that the further from the nucleus, the more energy is needed. And more for what?

I don't think so. Where did you find this $$E_n = {n^2\pi^2\hbar^2\over 2 ml^2}\quad ?$$ and what relates to the distance from the nucleus ?

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[Malvina]

Я так не думаю. Где вы нашли это $$E_n = {n^2\pi^2\hbar^2\over 2 ml^2}\quad ?$$ и как это связано с расстоянием от ядра?

Пожалуйста, оставляйте ссылку, когда вы цитируете/утверждаете что-либо, чтобы мы могли помочь вам более эффективно, и используйте $\LaTeX$ для ввода ваших уравнений (см. руководство — кнопка в левом нижнем углу).

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https://en.m.wikipedia.org/wiki/Schrödinger_equation
chapter: Examples: Particle in a box

I don't think so. Where did you find this $$E_n = {n^2\pi^2\hbar^2\over 2 ml^2}\quad ?$$ and what relates to the distance from the nucleus ?

Please leave a link when you quote/claim something so we can help more efficiently, and please use $\LaTeX$ to type your equations (see guide -- button at lower left.

$\$

I don't think so. Where did you find this $$E_n = {n^2\pi^2\hbar^2\over 2 ml^2}\quad ?$$ and what relates to the distance from the nucleus ?

Please leave a link when you quote/claim something so we can help more efficiently, and please use $\LaTeX$ to type your equations (see guide -- button at lower left.

$\$

 

[Malvina]

https://en.m.wikipedia.org/wiki/Schrödinger_equation

chapter: Examples: Particle in a box

 

[Malvina]

What does this formula say?

 

[BvU]

chapter: Examples: Particle in a box
direct link (from contents) here

(i.e to.: https://en.m.wikipedia.org/wiki/Schrödinger_equation#Particle_in_a_box)

It says that the particle can only have distinct kinetic energies. (Since the potential energy is absent, it only has kinetic energy).

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[PeterDonis]

the further from the nucleus, the more energy is needed.

"More" in the sense that the electron's total energy is less negative--it is less tightly bound.

the further the electron is from the nucleus, the less energy it needs to break away from the nucleus.

Yes, because it's less tightly bound; less energy has to be added to it to make its total energy positive, meaning that it's no longer bound to the nucleus.

 

[snorkack]

Note the differences.
In case of a box, the energy is constant inside the box, and infinitely big everywhere outside. Therefore the selected zero point of the energy is the flat bottom of the box.
In a hydrogen-like atom, the energy is not constant anywhere and diverges to infinitely small at the nucleus. However, it asymptotically approaches constant at infinite distance from the nucleus. That infinite distance is therefore selected as the zero point of energy.

 

[PeterDonis]

In case of a box, the energy is constant inside the box, and infinitely big everywhere outside. Therefore the selected zero point of the energy is the flat bottom of the box.
In a hydrogen-like atom, the energy is not constant anywhere and diverges to infinitely small at the nucleus. However, it asymptotically approaches constant at infinite distance from the nucleus. That infinite distance is therefore selected as the zero point of energy.

To be clear, here you are describing the potential energy. But the total energy of the particle includes both potential and kinetic energy.

 

[Malvina]

Спасибо!

 

[Malvina]

"More" in the sense that the electron's total energy is less negative--it is less tightly bound.


Yes, because it's less tightly bound; less energy has to be added to it to make its total energy positive, meaning that it's no longer bound to the nucleus.

 

[Malvina]

Thank you!