Solving Schrödinger's equation for a hydrogen atom with Euler's method

In summary, @Tuca said that you need to know the energy eigenvalues beforehand or find the correct energy by trial and error (which is called "shooting method" and can utilize Euler's method or some other finite difference scheme). The correct eigenvalues make the wavefunction approach zero when ##r\rightarrow\infty##, as should be for physically possible states. @Hello said that you can use Euler's method only for spherically symmetric eigenstates of the H atom, and you also need to either know the energy eigenvalues beforehand or find the correct energy by trial and error (which is called "shooting method" and can utilize Euler's method or some other finite difference scheme
  • #1
Tuca
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Hi, first-time poster here

I'm a student at HS-level in DK, who has decided to write my annual large scale assignment on Schrödinger's equation. My teacher has only given us a brief introduction to the equation and has tasked us to solve it numerically with Euler's method for the hydrogen atom. He rewrote the equation on to a form I simply cannot seem to find online or anywhere else. My knowledge of using Euler's method for second-order ODEs is very sparse too. I have attached the equation. If someone can please help me understand, how I must use Euler's method on this specific form, I would be very thankful.

Thank you
1614103604353.png
 
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  • #2
You can use Euler's method only for spherically symmetric eigenstates of the H atom, and you also need to either know the energy eigenvalues beforehand or find the correct energy by trial and error (which is called "shooting method" and can utilize Euler's method or some other finite difference scheme). The correct eigenvalues make the wavefunction approach zero when ##r\rightarrow\infty##, as should be for physically possible states.
 
  • #3
Hello @Tuca ,
:welcome: !​

I didn't reply to your post, since I don't have a ready answer. What I see in the picture seems to be the equation for the radial part of the wave function, suitably manipulated and simplified. I suppose there is no harm in peeking forward to the solutions, e.g. here . Picture e.g. https://users.aber.ac.uk/ruw/teach/327/hydrogen.php

1614337909744.png

You have a second order differential equation, so you need initial conditions.

Suppose you have ##R(0)## and ##S(0)## (and, for starters, ##n=1##) given, then

1614338332912.png

gives you ##S'(0)## which, in combination with ##R'= S## allows you to take an Euler step.

I suppose you can start with ##R(0) = A = 1## and figure out a better value afterwards from the normalization requirement ##\int \text {probability} = 1##.

Now the weakness in my reply and the reason (*) it comes so late: I still don't know how to deal with ##S(0)## or some equivalent.

Maybe you or someone else has an idea ?

(*) :smile: Actually: one of the reasons. Another being that I tried to reproduce your radial equation from the separated Schrödinger equation -- and ended up with a mess.
Tuca said:
My knowledge of using Euler's method for second-order ODEs is very sparse too
Not much knowledge needed to upgrade from order 1 to two: Given ##f(0)## and ##f'(0)## you treat the latter as a 'new' function ##g##: ##g(x) = f'(x)## and then step both ##f## and ##g## :
##f_{n+1} = f_{n} + g_n \Delta x##​
##g_{n+1} = g_{n} + g'_n \Delta x##​
where g'_n follows from the second order equation.​

Hope this helps at least a little bit; keep us posted !

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FAQ: Solving Schrödinger's equation for a hydrogen atom with Euler's method

1. What is Schrödinger's equation?

Schrödinger's equation is a mathematical equation that describes the behavior of quantum particles, such as electrons, in a given system. It is a fundamental equation in quantum mechanics and is used to calculate the probability of finding a particle in a certain location at a specific time.

2. What is a hydrogen atom?

A hydrogen atom is the simplest and most abundant atom in the universe. It consists of one proton in the nucleus and one electron orbiting around it. It is often used as a model system in physics and chemistry studies.

3. What is Euler's method?

Euler's method is a numerical method used to approximate solutions to differential equations, such as Schrödinger's equation. It involves breaking down the equation into smaller steps and using iterative calculations to find an approximate solution.

4. How does Euler's method help solve Schrödinger's equation for a hydrogen atom?

By using Euler's method, we can approximate the solution to Schrödinger's equation for a hydrogen atom. This allows us to calculate the probability of finding the electron in different locations around the nucleus, which is important in understanding the behavior of atoms and molecules.

5. What are the limitations of using Euler's method to solve Schrödinger's equation?

Euler's method is a simple and efficient way to approximate solutions to differential equations, but it has its limitations. It can only provide an approximate solution, and the accuracy of the results depends on the step size used in the calculations. Additionally, it may not be suitable for more complex systems with multiple particles or interactions.

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