@Julio. Time is periodic in nature T=1/f. This is the only fundamental accurate definition of time. Not recognizing this fact has caused all sorts of problems in quantum mechanics, general relativity and quantum gravity. The following link is to a quantum gravity article which associates Hubble parameter with the Planck frequency and eliminates time from the theory. This theory can generate entire table of standard model particles from a single formula.

Article Periodic quantum gravity and cosmology

I dont think there is any derivation because it is a fundamental principle of quantum mechanics. Just like there is no derivation to newton's laws. 

It is a form of conservation of energy equation written for a wave. 

But if you feel that substituting operator forms of position and momentum is a derivation of sorts, then yes. But again that is not a formal derivation. A derivation is when you start with the basic principles and move ahead using logical steps and schrodinger equation itself is the most basic principle.

My reaction to this question is:  if it can be derived, what is a legitimate starting-point from which to begin your derivation?  Presumably, one which gives better insight than does Schrödinger itself.  What would you propose? 

then the same can be said about dirac equation and klein gordon equation Is not it?

The KG eqn and the Dirac eqn can be derived from the principle of conservation of relativistic energy. You can look up Dirac's own approach at arriving at his eqn involving 4 parameters: one for mass (A) and three for momenta along 3 orthogonal axises(B1, B2, B3). Using these parameters in the principle of conservation of relativistic energy and solving for them he argued that these parameters cannot be just numbers, but they have to be matrices. These were the 4 gamma matrices. I don't remember the exact derivation, but you can look it up. 

So Dirac equation can be derived.

A nice derivation of the general form of the Schrödinger equation, based on the symmetries of space-time, is presented in Chapter 3 of Ballentine's book Quantum Mechanics: A Modern Development (World Scientific, Singapore, 1998). For the full picture, I suggest reading the first three chapters.

so both schrodinger equation and dirac equation can be derieved using the conservation of energy then why is it said that schrodinger equation is a postulate and dirac equation is derieved

There are several ways to derive Schrödinger's, at bottom it's simply a wave equation, which you can derive in different ways.

There is a relativistic version of Schrodinger wave equation which can be derived from fundamentals. This wave equation includes spin as a part of the Laplacian operator and it yields Dirac energy levels.

Article Quarkonium and hydrogen spectra with spin-dependent relativi...

The time dependent Schrodinger equation is one of  6 postulates of quantum mechanics. Schrodinger's equation cannot be derived. It was thought up using logical arguments and so far it has seemed to work experimentally.

The equations is essentially a re-write up for energy conservation:

E=T+V

Where T is the Kinetic Energy for a particle given by the KE  operator and V is the potential.

From Feynman's lectures:

"Where did we get that from? It's not possible to derive it from anything you know. It came out of the mind of Schrödinger."

Issam,

if you write the amplitude equation for a vibrating string or rope (a rope that you move up and down at one end, and irrespective of whether the other end is loose or affixed, you generate a system of standing waves) you get Schrödinger's equation

Chris,

Do you call this derivation!

Charles,

why have so many students derived it? Because teachers asked.

Issam,

postulate is the whole system of views. In the frames of this system this equation inevitable. Any derivation is formalism. There are always many possible formalisations.

Regards,

Eugene.

Expressing standing waves is just one of many ways, which basically shows that time-independent Schrödinger's describes a standing wave system, with peaks and troughs of amplitude (an amplitude which can then stand in for a number of properties).

Time evolution then requires factoring in the time dependency.

Chris,

the problem is: Standing wave of what? Moreover, time-indepependent Schreodinger is not enough. The main problem of QM is: What is the time in QM?

Regards,

Eugene.

Let us assume that 'in context of quantum mechanics' schrodinger equation can be derived. Then any starting point for the derivation should give a much broader and general perspective of QM. That principle would then become a fundamental postulate of QM. But still that postulate could be derived using some other mechanism, but out of context of QM.

What I want to say is there has to be a starting point, and in context of QM schrodinger equation was a starting point. It simply cannot be derived using Quantum mechanical postulates because it itself is one of them. 

Even if we do a derivation using standing waves as mentioned above, how do we know that the result we get is valid for quantum mechanics? We only know so because the equation we get is similar to schrodinger equation and not the other way around. At least that is what i feel.

Dear Jaskaran,

eigen values is the simplest way to introduce discrete observables into physics (that was the real problem). Tere are many other possibilities, for example, limiting cycles in non-linear mechanics.

Regards,

Eugene.

Charles could you post a link or name some text where I could study the mathematical derivation of Schrodinger's eqn? I would very much love to study it in mathematical rigor. Thanks for your answer. 

The "standard" view is that the time-dependent Schrodinger equation (TDSE) is the postulated fundamental equation of motion of non-relativistic QM. Then, the  time-independent SE can be easily derived from the TDSE. (Analogous remarks for the Dirac equations of relativistic QM). However, the role of time in QM is very problematic. This problem becomes even more acute in quantum gravity. A plausible solution is to recognize that there is no place for time in a purely quantum world, and to postulate the time-independent SE as the fundamental equation. Then, it can be shown that the TDSE emerges from the correlation of the quantum system with a classical system that acts as a clock. Then, the TDSE is an approximate quantum-classical equation! For details see arXiv:1202.4638v3 [quant-ph]  and references therein.  

@Julio. Time is periodic in nature T=1/f. This is the only fundamental accurate definition of time. Not recognizing this fact has caused all sorts of problems in quantum mechanics, general relativity and quantum gravity. The following link is to a quantum gravity article which associates Hubble parameter with the Planck frequency and eliminates time from the theory. This theory can generate entire table of standard model particles from a single formula.

Article Periodic quantum gravity and cosmology

Charles,

QM says nothing about space and time. It simply takes them from Newton. The attempts to take them from Einstein have difficulties.

Regards,

Eugene.

I'll think, Charles. Of course, the main problem is measurement. See Steve Faulkner.

Regards,

Eugene.