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Old May 28th 2008, 10:27 AM   #1
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The Schrodinger Equation.

What kind of maths do I need to understand to be able to solve the Schrodinger Equartion for a Hydrogen atom? What kind of information can one derive from the solution of this equation?

According to an Advanced Physics textbook the Schrodinger Equation gives the Principal Quantum number, the Azimuthal Quantum number and the Magnetic Quantum number, can someone please explain how you get these numbers?

Also, can you derive the shape of an orbital using the solution of Schrodinger?

Thanks.
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Old May 28th 2008, 03:27 PM   #2
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Originally Posted by AAKhan07 View Post
What kind of maths do I need to understand to be able to solve the Schrodinger Equartion for a Hydrogen atom? What kind of information can one derive from the solution of this equation?

According to an Advanced Physics textbook the Schrodinger Equation gives the Principal Quantum number, the Azimuthal Quantum number and the Magnetic Quantum number, can someone please explain how you get these numbers?

Also, can you derive the shape of an orbital using the solution of Schrodinger?

Thanks.
The level of Mathematics you need is differential equations. (Some few techniques involving partial differential equations are required, but are easy enough to understand and are typically demonstrated in a Physics text.) So that means College Algebra level plus all Calculus up through multidimensional equations. So you wouldn't be able to take QM until, say, your 5th semester at college if you started off with Calc I.

The Schrodinger wave equation gives the solution for the behavior of a non-relativistic particle under the influence of a given potential, not including any spin characteristics. An electron in a Hydrogen atom is mainly non-relativistic so we can use the S-equation to predict its properties. When solving this we do indeed get the principle, angular, and azimuthal quantum numbers. The magnetic number is related to the spin and has to be inserted into the solution "by hand." The S-equation does not predict this one.

I don't have the time right now to show the derivation of the Hydrogen atom solution. Just to give you a taste of the complexity of the solution go to this page and look under the "wavefunction" heading.

You can derive the shape of the orbitals, but it is a tricky process. The S-equation can be used to derive the probability of a particle being in a specific spot. By evaluating these probabilities for as many points in space as you can you can come up with a "map" of where the particle is most likely to be. Of course, you can use the symmetries involved in the problem to guess what some likely features of what the orbital will look like, like we know all of them have symmetry about the z axis for example.

-Dan
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