Radial Schrödinger Equation

In summary, the key assumption in the problem statement is that l has a maximum value. When you substitute this into the radial schrödinger equation, you get a lot of terms that cancel each other out. However, if l has a maximum value, you can substitute n-1 into the equation and get a simpler equation.
  • #1
tuomas22
20
0
Sorry in advance my english, I tried to translate it to english as good as I can

Homework Statement



When [tex]l[/tex] has its maximum value, the hydrogen atom radial equation has a simple form of

R(r) = Arn-1e-r/na0, where a0 is Bohr's radius.

Write the respective radial schrödinger equation.

Homework Equations



Radial Schrödinger Equation:
[tex]-\frac{\hbar^2}{2*\mu*r^2}*\frac{d}{dr}(r^2*\frac{dR(r)}{dr})+\left[-\frac{kZe^2}{r}+\frac{\hbar^2*l(l+1)}{2*\mu*r^2}\right]*R(r) = ER(r)[/tex]

The Attempt at a Solution



I've attempted substituting the given R(r) to the radial equation, but I can't get anything out of it that makes sense. Too long to write it here with this hard latex thing :S

Should I get some nice reduced form or can I expect some equation monster?
 
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  • #2
tuomas22 said:
Sorry in advance my english, I tried to translate it to english as good as I can

Your translation is very good!:smile:

When [tex]l[/tex] has its maximum value...

This is the key assumption in the problem statement...What is the maximum value of [itex]l[/itex] for any given value of [itex]n[/itex] (remember, only certain values are allowed)?...Substitute that into your radial Schroedinger equation (along with the given function [itex]R(r)[/itex]).
Radial Schrödinger Equation:
[tex]-\frac{\hbar^2}{2*\mu*r^2}*\frac{d}{dr}(r^2*\frac{dR(r)}{dr})+\left[-\frac{kZe^2}{r}+\frac{\hbar^2*l(l+1)}{2*\mu*r^2}\right]*R(r) = ER(r)[/tex]
 
  • #3
Thanks for answer!
The maximum value of [itex]l[/itex] is [itex]n-1[/itex] right?
I tried to substitute it but it doesn't get much prettier :)

I'm doing the math with Maple here, and it freaks me out to even think that I should be able to do it without computer, which is the case actually...That's why I think I'm doing something wrong, maybe I understand the equation wrong or something.

Here's a screenshot of my maple session (left side of the equation)
http://img526.imageshack.us/img526/5863/tryw.jpg

If I count the right side in too, I can reduce the Arn-1e-r/na0 term from both sides but still I'm not so sure about it
 
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  • #4
You also have an equation for Bohr's radius [itex]a_0[/itex] right?...And [itex]Z=[/itex]____ for the Hydrogen atom?

When you substitute these things in, you should get a lot of terms canceling each-other out.
 
  • #5
aaah the bohr radius equation, didnt even think about it :)
(and Z=1 ofcourse)

the next part of the question was to prove that the given R(r) is the solution to the schrödinger equation with [itex]l=0[/itex], and I got it right now with that bohr radius tip you gave! :)
[tex]- \frac{1}{2} \frac{k^{2}e^{4}\mu}{\hbar ^{2}} = E[/tex]

But with [itex]l=n-1[/itex] I still get horrible monster equations...*cry* nothing seems to cancel out

there must be something I'm doing wrong since I can't be expected to do that kind of differentations without computer in a 2 hour exam ! No human could do it right! :D
(this is an old exam question)
 
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  • #6
Aaah I got it!
0 = 0 finally :)
There was some silly mistake in the equation.

The derivation is still inhumane to do with pen and paper though :)

Thanks much gabbagabbahey!
 

Related to Radial Schrödinger Equation

What is the Radial Schrödinger Equation?

The Radial Schrödinger Equation is a mathematical equation that describes the behavior of a quantum particle in a spherically symmetric potential. It is used to calculate the probability of finding a particle at a certain distance from the center of the potential.

What does the Radial Schrödinger Equation tell us about quantum particles?

The Radial Schrödinger Equation helps us understand the energy levels and wavefunctions of quantum particles. It also allows us to calculate the probability of finding a particle at a certain distance from the center of the potential.

How is the Radial Schrödinger Equation different from the standard Schrödinger Equation?

The Radial Schrödinger Equation is a simplified version of the standard Schrödinger Equation that is specifically designed for spherically symmetric potentials. It only considers the radial component of the wavefunction, while the standard equation takes into account all three spatial dimensions.

What are the variables in the Radial Schrödinger Equation?

The variables in the Radial Schrödinger Equation are the radial distance (r), the angular momentum (l), the principal quantum number (n), and the energy (E).

Why is the Radial Schrödinger Equation important in quantum mechanics?

The Radial Schrödinger Equation is important because it allows us to solve for the energy levels and wavefunctions of quantum particles in spherically symmetric potentials, which is a common scenario in quantum mechanics. It also helps us understand the behavior of particles in these potentials, which has applications in various fields such as solid-state physics and atomic and molecular physics.

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