Schrodinger equation significance

[geft]

Hi, could someone please explain the equation to me (its significance, application and the how the equation itself works) in simple terms? I find most other resources too complex for me to digest. I hope this is not too much to ask.

 

[Ken G]

The key idea behind the Schroedinger equation is that a single particle can perform in ways we normally associate with waves, and that things we associate with wave behavior can be broken down into the action of many particles. This is called "wave-particle duality", but to have that, we need some dynamical equation that tells us how the wave nature of particles behaves with time. That's what the Schroedinger equation does.

Now, you might ask, why do we need a wave equation for particles, if we already have a classical wave equation? The problem is that the classical wave equation doesn't have the right "dispersion relation" for nonrelativistic particles, which means it doesn't correctly associate the wavelength of the wave with its frequency. The classical wave equation connects the square of the inverse wavelength to the square of the frequency, but for nonrelativistic particles, the deBroglie relations tell us that what we want is to associate the square of the inverse wavelength with just the frequency to the power 1. That means we want a single time derivative in the equation, to draw out a single power of the frequency, rather than the second time derivative that is in the classical wave equation. But a single time derivative won't get the phase right, so we also need to use complex amplitudes for the wave, so we can tack an "i" onto the first time derivative, and get the right phase. This necessity to invoke complex amplitudes has the curious effect of forcing the wave amplitude itself to never be an observable, but the Born rule tells us how to get from the wave amplitude to the values of the things we can really observe.

 

[jewbinson]

http://en.wikipedia.org/wiki/Schrödinger_equation#Short_heuristic_derivation

 

[sweet springs]

Hi, geft.

In the beginning of the last century, wave like behavior ( diffraction, superposition, etc) of light particles ( electron, neutron, etc.) were found. Wave function, usually noted Psi(x,t) was proposed to describe such wave behaviors of particle. Evolution of wave function is determined by Shroedinger equation, i.e. time derivative of wave function equals to -i hbar H Psi(x,t) here i is imaginary number unit, hbar is constant 1.05E-34 [Js] called reduced Planck constant and H is function expressing energy of the system where momentum of the particle p should be replaced by operator -i hbar d/dx. For example, H of free particle is H = p^2/2m = - hbar^2 (d/dx)^2 where m is mass of particle.

Regards 　

 

[jtbell]

I hope this is not too much to ask.

Frankly, I think it is too much to ask, without telling us anything about what, specifically, you don't understand about what you've read already. Anybody who answers has to "shoot blindly," so to speak. You're more likely to get useful answers if you ask better-focused questions.

 

[geft]

Well, I suppose my question pertains mostly to its significance and application. Such as why it's often hailed as one of the most important discoveries in the 20th century.

 

[Ken G]

I would answer that by saying that equation underpins the unification of particle and wave dynamics, which were the two fundamentally disjoint ways to discuss dynamical principles prior to deBroglie. There were plenty of purely classical hints of this underlying unity, such as the fact that Newton's laws could be recast in terms of a principle of "least action", which had a direct analog to how a "least action" principle also works for waves. The related importance of resonant phenomena were also present in both wave and particle dynamics, another hint of a deeper underlying unity. But it was the Schroedinger equation that quantified this unification into a single dynamical model. Ironically, its importance is often stressed in terms of now nonclassical it is, but I would argue it is more insightful to see its importance as super-classical or meta-classical, i.e., classically unifying.

 

[geft]

Pretty much what I was looking for. Thanks!

 

[Neandethal00]

Hi, could someone please explain the equation to me (its significance, application and the how the equation itself works) in simple terms? I find most other resources too complex for me to digest. I hope this is not too much to ask.

If you want it explained in layman's terms, hear it from a layman.

Physicists need to know energy, momentum, position, etc. of a particle(s). All these together is known as the 'state' of a particle. They found all these properties of a particle can be described by a thing called 'wavefunction'. This wavefunction contains all above information about a particle when it is in a certain state, ready to extract the needed 'info'. I do not think a wavefunction resemble any physical wave we are used to.

So how do they extract info from a wavefunction? They use what they call 'an operator'. Yes, the operator operates (like a surgeon operates and pull out the tumor from a lung) on the wavefunction and what comes out after the operation are 'info' physicists wanted. Now, how do they make an 'operator'? That's when Schrödinger comes into the scene. I think it was Schrödinger who made the first operator using Total Energy.

Total Energy = Kinetic + Potential energy.

Mathematical looks of this operator are shown in a few posts above.

To all experts, I'm sure I have made many mistakes in my description. Correct me please.