Physics/Schrödinger equation/wave function representing orbitals
QUESTION: Hi Steve,
I am in a first year undergraduate chem course learning about atomic structure and I don't quite get Schrödinger equation.
Can you explain the Schrödinger equation to me in simple/layman terms.
In addition, I also don quite get how the Schrödinger equation relates to wave function.
I searched on the internet and I found out that the wave function describes the position/energy of an electron which confuses me.
Can you explain in simple layman terms what the wave function is and how does it relate to atom orbitals. As I know each orbital have their own wave function.
Many Many Thanks,
ANSWER: Let me start you with a little history. Light is a wave. It obeys a wave equation, actually a set of them known as Maxwell's equations (not all his, he just put them together in a recognized succinct form for the first time). Waves obey wave equations. When in a ridiculously tiny doctoral dissertation de Broglie postulated that some electron scattering data (British scientists, I believe) which was eerily similar to x-ray scattering data meant that electrons also behaved as waves then Schrodinger said that they must also behave as waves. You see, "particle" just means that something must be absorbed or emitted as a whole, not that it has some kind of solid mass like our intuition mistakenly leads us to. Wave means that it extends through space and behaves like a wave...and therefore probably obeys a wave equation. Someone at the meeting where Schrodinger proposed this heretical idea (so I heard in grad school), demanded that he go and solve that equation.
The story goes that he disappeared up into the mountains on a skiing trip with one of his many many mistresses...and she must not have been that great because he came down from that ski trip with the wave equation figured out. He figured it had to obey the typical rules of differential equations and came up with the simplest equation he could that satisfied the known physics...like the de Broglie wavelength. The details are unimportant to the level you're looking for.
Long story short, if you look at any wave...sound, light, electron...throughout space you must be able to create a function to describe its condition at that point in space. What the strength of the electric field is, how dense the air is, etc...for particles it's the wavefunction. That gives you a strange sort of density-related thing which you can relate to the probability of finding the particle at any point during an observation.
That's basically it as far as layman's terms you're asking for, actually. It's related to the probability, satisfies a bunch of boring math rules, but what it's used for in practice is to find the probability that a particle will be observed in a location at a time...when used properly. That gives you everything, like the probability that it will be found at a point near what it is bound to (like an electron around an atom, that gives the atom its quantum shape, if you will). Or the probability that a particle will be observed outside of a barrier (quantum tunneling, a more advanced topic).
But you asked in the context of chemistry. The wave equation and the wave function (the wave function is something you solve the wave equation...essentially just a set of math rules that the wave function is known to obey...in order to find) are both dependent on energy. Energy of the Coulomb force (nucleus to electrons) depends on position. The solution to the wave equation, when you take into account the space relationship of both, gives you both the energy and the spatial distribution of the electrons in an atom.
I won't lie, the details are ridiculously complex and involve a lot of quantum mechanics and angular momentum and such...but in layman's terms the above is what you need to know. You're trying to wrap your brain around something that is simultaneously too big and too simple to comprehend without a great deal of study and years of hard work in a short time... The equation determines the function. The situation determines how the equation will be solved and therefore the combination of the equation (the physical reality of how space works) determine the function. The function determines the probability of an electron being anywhere (think of it as its "reality density" if you can). Stable solutions are states which the electrons around atoms can be found in, that kind of thing.
Good luck. :)
[an error occurred while processing this directive]---------- FOLLOW-UP ----------
QUESTION: Hi Steve,
Thank you very much for the through answer.
I just have two additional questions.
I would like to know how does the wave function relate to Heisenberg uncertainty principle and are they quite similar ?
What is the definition of angular momentum in layman terms.
The uncertainty principle is something one derives from experiments and a consequence of the wave function. Griffiths' quantum mechanics book does a good job of explaining it. Waves have well-defined energies when they have a sinusoidal type of wave function (which is distributed in space). Waves with well-defined positions in space have ill-defined energies (Fourier transform, if you wish to read further) because they don't have a precisely defined frequency.
Angular momentum is the amount of "spin" that a thing or a wave has. That's as well as I can define it, the "amount of rotation" that something has. Whether that rotation is in the object itself (spin) or in how it orbits (orbital angular momentum...like a tiny hurricane).