Can someone explain the Schrödinger equation?

[bleucanary256]

Sorry, I'm new, can someone provide a simplified (i.e. dumbed-down) explanation of the Schrödinger equation? i understand some of the basics of quantum mechanics, and i find this thread fascinating. at the base level, this seemed like a fairly straightforward question, but it seems there are many subtleties I'm not quite grasping. however, it's my understanding that in general, electrons don't just fall into the nucleus because of their rotational velocities, which pull them outwards in a straight line even as their attraction to the nucleus pulls them inward, keeping them in orbit. at least, for larger, less physically confusing objects, that would be the general principle. correct?

 

[smallphi]

The problem with the classical picture of an electron 'ball' orbiting the nucleus is that even if its speed has the necessary value to keep it in a steady orbit,a rotating charge is an accelerated charge (centripetal acceleration provided by Coulomb force). According to electrodynamics, accelerated charges radiate their energy in terms of electromagnetic waves so eventually the electron will fall on the nucleus. There are problems where they calculate the time for the electron to fall on nucleus in that totally classical picture.

The 'Old quantum mechanics' (1912-1924) created by Niels Bohr postulated that the electron is still orbiting the nuclues like the above classical picture but for some reason it doesn't radiate energy in that state like a normal charge would do. Yes they 'solved' the problem by postulating it didn't exist LOL The old quantum mechanics predicted correctly the energy spectrum of the Hydrogen atom but had severe problems with Helium and other multi-electron atoms.

The biggest experimental evidence something strange is happening inside the atoms was the discrete emission spectra of the chemical elements. It suggested that atoms have correspondingly discrete internal energy. The old quantum mechanics of Niels Bohr simply postulated that only certain orbits of the electrons with certain values of the angular momentum were 'allowed' thus effectively postulating the discrete energy levels of atoms.

Around 1924 the De Broygle thesis appeared in which he made the bold hypothesis that particles can behave like waves. That was the logical reverse of the Plank and Einstein's idea that photons (thought to be 'waves' at that time) can have particle-like behavior.

Schroedinger was teaching the De Broygle ideas to some audience when somebody (whos name was forgotten by history) asked 'if the electron is a wave, where is its wave equation?'. That inspired Schroedinger that 'something is waving inside the atom' and to heuristically guess the differential equation for the wave function that bears his name. Schroedinger was pretty good in solid state calculations and he knew very well that discrete numbers (the energy levels of atoms) mathematically can arise from a differential equation through an eigenvalue problem. He wrote the equation, can't remember if he calculated something in the first paper but pretty soon people realized that they can actually calculate the energy levels of atomic system from that equation and that was a sign something 'was working'. At first they didn't know how to interpret the meaning of the wave function but that was found later.

Almost at the same time as Schroedinger created his 'wave mechanics', Heisenberg wrote another equation about some 'matrices' describing 'probabilities of transition' inside the atom. That was the second version of quantum mechanics called 'matrix mechanics'. Very soon Schroedinger proved the two versions equivalent.

That's how quantum mechanics was born after 10 something years of unsuccessfull trials to describe the electrons in atoms following trajectories like classical particles. At the end people just gave up on the classical picture and embraced the weird formalism of wave functions and matrix operators whose predictions are in agreement with experiment.

 

[bleucanary256]

First, thanks Smallphi for that history of the Schroedinger/Heisenberg equations, it really cleared somethings up for me. so basically, if I'm to understand this, electron uncertainty due to the Heisenberg (and Schroedinger) equation means that electrons don't always behave like point particles, they sometimes or in some aspects behave like waves, and their exact positions at any point in time can only be expressed as a probability...so basically, the electromagnetic wave energy emitted by electrons as they orbit the nucleus is the main contradiction of why they can't stay in indefinite orbit around the nucleus, because it means that they are losing energy as they orbit, meaning they will eventually not be able to maintain speed and therefore, centripetal velocity, and will fall inwards to the atom.

My question then is, if electrons can behave as waves, how does that help to explain why they usually stay in orbit? i know I'm asking pretty basic stuff, but i just want to understand? perhaps I'm mistaken here, but did you say they behaved like electromagnetic waves? then their 'current' creates a magnetic field that repels the nucleus, meaning it is positively charged?

did that make sense? please tell me why not I'm pretty sure I'm wrong...just want to know why.

 

[Hootenanny]

In quantum mechanics, we consider the electron as a particle, not an EM wave. You may be confused by what we mean by a 'wave function'. A wave function isn't a physical observable, it is mearly an equation we can use to describe a particle, it doesn't mean that the particle behaves as a wave (although in some cases it may).

You should understand that in quantum mechanics, electrons do not classically orbit around the nucleus, rather there is some finite probability that they exist at some distance from the nucleus. For an electron of a certain energy, there is corresponding a probability density function (the square of the magnitude of the wave function), i.e. the probability that the electron can be found in a certain volume. The most probable distance for an electron to exist from the nucleus [of a hydrogen atom] coincides with Bohr's predictions. So, if you like it is most probable that the electron will be found on some circular trajectory around the nucleus. However, there is also some non-zero probability that it is also found at some distance very close to the nucleus or at some distance further away. Therefore, the electron's trajectory is really unknown, it can exist anywhere in this 'cloud' of uncertainty defined by the probability density. We can only determine the electron's position if we observe it. Here is a an image which shows the rough 'shape' of the probability densities, or 'electron cloud' of an H atom; http://phycomp.technion.ac.il/~phr76ja/viz1.gif

I hope this was helpful and made some kind of sense.

 

[Rade]

...Here is a an image which shows the rough 'shape' of the probability densities, or 'electron cloud' of an H atom; http://phycomp.technion.ac.il/~phr76ja/viz1.gif...

Hootenanny--I have an observation and question. In your helpful link http://phycomp.technion.ac.il/~phr76ja/viz1.gif were you aware that the geometric configuration difference between quantum energy states [1s,m=0] and [2p,m=0] of the electron in the H atom are exactly as predicted (and shown on the cover of a recent issue of Science journal) by the Perelman equation that solves the Poincaré Conjecture ?

So my question--do you see any possibility that this Perelman equation supports a hypothesis that the motion of electron back-forth between energy states [1s,m=0] <----> [2p,m=0] in H-atom follows a discrete and continuous set of geometric probability density configurations (as shown on the cover of Science) from sphere [1s,m=0] to dumbbell [2p,m=0] ? Is it possible that Perelman equation informs that what we observe as discrete quanta are nothing more than stops along the way of a more fundamental non-discrete reality of motion of entities with dual wave-particle essence ?

Thank you for your time.