The Schroedinger Wave EquationDate: 09/14/98 at 04:36:22 From: Adam Chowanetz Subject: Schroedinger's wave equation I was wondering if you could help me understand the concepts behind Schrodinger's wave equation. I really want to understand the way that the equation models the atomic shells and sub shell orbitals. I do have a reasonably good background in calculus and the knowledge for solving differential equations and partial fractions, but the references I have read do not describe the variables very well. Is there any way you could explain to me these variables and how they are involved in the model of the atom? My chemistry classes have inspired an interest in the topic. Thank you so much for your time, and I do understand that it must take an enormous amount of time trying to answer mathematical questions from people all over the world. Thanks again, Adam Date: 09/14/98 at 06:20:20 From: Doctor Mitteldorf Subject: Re: Schroedinger's wave equation Dear Adam, Long ago, I learned this subject from a classic text by Pauling and Wilson, _Introduction to Quantum Mechanics_. Its approach is straightforward but rather old-fashioned. A somewhat more modern approach is in Shankar's _Principles of Quantum Mechanics_. Here are the basic steps: The first and perhaps most difficult thing is to grasp what the Schroedinger equation is about. It doesn't tell you where the electron is at a given time, or how it moves. This is the big shocker, because those are questions you'd want to ask, the more so in 1925. The solution of the equation is a "wave function" but it's not at all clear what it has to do with waves. In other circumstances, it looks like a sine wave, but here in the atom it has little in common with anything you'd call a wave. The wave function isn't measurable directly by any means. It is a complex number attached to every point in space and time, which must be multiplied by its complex conjugate to get its absolute square. The absolute square is proportional to the probability of finding the electron at a given spot. The equation itself is: -h^2/(2m) del^2 psi + V*psi = E*psi In this equation, h is Planck's constant over 2 pi, m is the mass of the electron, del^2 is the Laplacian operator, which I'll come back to, and V is the potential energy, a function of where you are in space. For the Hydrogen atom, V is (-e^2/r) where r is distance from the origin. E is just a constant number, equal to the energy of the electron. In principle, E can be anything, but you find when you solve the equation that there are only scattered values of E for which the equation can be solved at all. This corresponds to the fact that energy of the hydrogen atom is quantized, and can only take on certain values. (These energy values were known early on from spectroscopic experiments, and the fact that the Schroedinger equation reproduced them accurately was the first thing that gave people confidence in such a crazy system as quantum mechanics.) Back to the Laplacian. It is the second partial derivative with respect to x, plus the second partial derivative with respect to y plus the second partial derivative with respect to z: del^2 psi = d^2 psi /dx^2 + d^2 psi /dy^2 + d^2 psi /dz^2 This works for a psi that's a function of (x,y,z) coordinates in space. But the V part of the equation is a function of r = sqrt(x^2+y^2+z^2). This makes the equation very messy to solve. A standard trick for dealing with it is to transform variables, to write the Laplacian operator as an equivalent combination of partial derivatives in the spherical coordinate system (r,theta,phi). This looks like: del^2 psi = (1/r^2)d/dr (r^2*d psi/dr) + 1/(r^2sin^2(theta)) d^2 psi / d phi^2 + 1/(r^2*sin(theta)) d/dtheta (sin(theta)*d psi / d theta) This looks awfully messy, but it turns out to be easier to solve than if you wrote the whole thing out in terms of x, y and z, where V is proportional to 1/sqrt(x^2+y^2+z^2). The next step is to write the solution as a product of three functions, one a function of r alone, one a function of phi alone, and one a function of theta alone. There's no guarantee that this will work, but it simplifies the equation enormously, getting rid of all the problems associated with partial derivatives; so if you can find a solution this way it saves much work. Focus first on the r part of the equation, and notice that the partial derivatives with respect to r must be dealt with, but the partial derivatives with respect to theta and phi (2d and 3d terms) are zero by definition. I'll leave you to work through the solution to the three equations given in Pauling and Wilson. The solution to the (theta,phi) part of the equation is a standard set of polynomials in cos(theta) and cos(phi) called "spherical harmonics" that turn out to be useful in many contexts. The solution to the r part is a polynomial times a Gaussian e^(-(r/a)^2), where a is the "Bohr radius" of the atom = h^/(m*e^2), about half an angstrom. Even once you have the complete set of solutions, it's a long way from here to explaining chemistry. For one thing, the equation is for a single electron. If there are 2 electrons, the equations become so much more complicated that solutions weren't even attempted until recently, using numerical methods on a supercomputer. If there are 3 electrons, the equations are hopeless. So a standard thing to do is to make the very rough approximation that the electrons "don't see each other" and just form independent clouds of charge about the nucleus. It's a bad approximation, but it's very difficult to approach the equations without this assumption. The actual Schroedinger equation for two electrons is a partial differential equation in 6 dimensions, since the wave function depends on where both electrons are: (x1, y1, z1, x2, y2, z2). This is the one that's solved on a supercomputer. The wave function for 3 electrons depends on 9 variables and is intractable. A simple oxygen atom has 8 electrons and is hopeless. But it turns out that much about chemistry can be related in a general way to the solutions of the Schroedinger equation for a hydrogen atom. The electrons in more complex atoms behave as if they were roughly independent of each other just enough that the hydrogen solutions are recognizable in complex atoms. In hydrogen, the electron seeks the lowest energy solution to the Schroedinger equation, and once it gets there will remain stable for a long time. But if you have more than one electron, the "exclusion principle" sets in. It turns out that electrons are forbidden by quantum mechanical rules (that have nothing to do with the Schroedinger equation) from occupying the same energy state as one another. So in an oxygen atom with 8 electrons, the "real" wave function is a function of 24 different coordinates, describing where all 8 electrons are. But this is so difficult to deal with that we settle for the rough approximation of 8 different functions of 3 dimensional coordinates. And the 8 functions are different because the electrons refuse to occupy the same state. So only the first lucky 2 get to occupy the lowest energy state. (Two, not one, because they can have different spins. We haven't even talked about spin yet...) After that, the next 2 electrons occupy a higher energy level, and they keep piling up in higher and higher energy solutions to the Hydrogen Schroedinger equation. Some basic properties of the Periodic Table can be explained by the pattern of energies in the solutions to the r part of the Schroedinger equation. I'm going to stop here. I hope I've given you a taste of the journey that you're embarking on. - Doctor Mitteldorf, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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