How Was the Schrödinger Wave Equation Derived?

[wave41]

Hello, everyone I am new to this forum, I hope I am posting this at the right place. I am in my first year of college at Concordia University, and taking chemistry right now. But my main interest is physics. So when we were learning about the equation, I wanted to know more detail information on it . I was wondering if anyone has a site they can tell me about that equation? On how the formula was derived?
Thanks a lot![/color]

 

[Doc Al]

First off, welcome to Physics Forums!

The web has gazillions of sites that treat the Schrödinger Equation. Here's one to start you off: http://

Enjoy and be sure to come back with your questions. There are plenty of very knowledgeable folks here.

 

[wave41]

Thank you for the welcome! I was looking for a place where I can ask the many questions that I come up with!
Thanks for the link!

 

[SchrodingersCat]

http://scienceworld.wolfram.com/physics/SchroedingerEquation.html

Hope that helps. It just gives a brief overview of the equation. The textbook Introduction to Quantum Mechanics by Griffiths is rather helpful. Is that the textbook your class is using?

 

[Janitor]

It is always good to know the limitations of things. In the case of the equation you are asking about, it is appropriate in the low energy limit, i.e. it is non-relativistic. (I once read that Schrödinger actually was well on his way to a relativistic form, only to get discouraged by some technical difficulties, and so he deliberately restricted his derivation to the non-relativistic case.) It does not handle changing particle number, i.e. creation or annihilation of particles. Also, the Schrödinger equation is, strictly speaking, appropriate for a spinless particle.

Despite the limitations, it still does a pretty good job of handling spin-1/2 electrons in a variety of bound states, including of course atomic electrons!

Exact solutions are possible only for very simple cases. When you are dealing with two or more interacting particles in some sort of potential field, you are likely to have to resort to numerical approximation methods to solve the probability distribution of the particles. Probability density for finding a particle (or system of particles) somewhere in configuration space is proportional to the squared magnitude of the amplitude function that you get as your solution to the Schrödinger equation.

 

[Javier]

History

To extend a bit on what was said:
Schrödinger, according to Dirac, wrote down the relativistic equation for a spinless particle first. He arrived at this by noticing the experimental relationship between the energy of a particle (like an electron or a photon) and the frequency of the wave-behavior of that particle, and similarly between the momentum and the wavelength. For example, Einstein found that individual photons carried an energy E=hv, where v is the frequency of light used. Imposing the relativistic relationship
E^2=p^2+m^2 (with c=1), he arrived at the relativistic equation that describes a function that obeys a wave equation, so that this function would describe the state of a particle with mass m with "wavelike" properties as experimentally observed.
However, Schrödinger noticed that the while the equation agreed with the "rough" spectrum of Hydrogen, it did not agree with the "fine structure" corrections. He decided that the non-relativistic form of his equation would at least give an agreement with the rough spectrum of Hydrogen, so it was capturing something right...the relativistic form would go too far and make false predictions.
The resolution came with the realization that there must be an intrinsic angular momentum associated with the electron in the bound state with the proton making up hydrogen. The corrections to the energy spectrum of hydrogen (using the original *relativistic* form of Schrodingers equation) due to spin effects gave the right results for the fine structure.
Dirac had a problem, though, with the relativistic equation, and the same problem existed with his own equation (which took into acount the spin of the electron from the start). The problem is that the Hamiltonian would be unbounded below...this meant that electrons would seek lower energy configurations by tunneling to negative energy states and keep going down in energy, never reaching a stable energy configuration.
A resolution to this is that there are antiparticles, particles with opposite "quantum numbers", and that (e.g.) an electron can't spontaneously become an anti-electron. Therefore, Dirac's equation described two types of particles with energies bounded below by mc^2.
However, as experiments became more sensitive, it was found that there were further splittings that the new relativistic quantum mechanics didn't explain. The resolution was QED (and more generally, quantum fields): there were "radiative corrections" to the spectrum of Hydrogen due to the interaction of the electron with the quantum "Maxwell" field. That is, an electron near a proton interacts with discrete bits of a field, which are photons, on top of the classical electromagnetic background field (which can, in turn, be seen as a bunch of photons in a "coherent quantum state"). The interaction with these photons is in a quantum fashion, meaning the effect is, in a specific sense, due to all numbers of photons interacting in all possible ways with the electron, the overall effect being that the electron feels some effective fuzzy influence of it. Another example of this fuzzy influence that is more relevant to what you see in chemistry, e.g., is the state of a single electron in a simple covalent molecule: the electron is delocalized around the two nuclei (being in all places allowed at once) forming a glue for the molecule (if the electron didn't delocalize in this fahion, the molecule would not be stable).
By the way, from a modern perspective, quantum field theory is the necessary way to combine quantum mechanics and special relativity...the need to use fields arises naturally when you try to do this. Also, antiparticles are an immediate consequence in qft when you simply demand conservation of a set of quantum numbers (like electric charge).

 

[wave41]

wow thank you for the fine discription and history, we were briefly learning this in chemistry, I wanted to know more about the equation because our teacher did not tell us anything further about it because it was too complicated and she didn't know how to do it. So I though of looking up this formula because I was very interested in it. But thank you again for a great piece of history about it, to all of you
I am now looking at some books to and the sites you remecomended me.
Thanks