Adnan: Are you confused because of the maths involved in these equations or is it the physics what confuses you?

I would suggest you to read a nice little book: Werner Heisenberg, "The physical principles of quantum theory". Also perhaps helps: THE ORIGINS OF THE QUANTUM THEORY

by CATHRYN CARSON (in the web).

Adnan: I would also suggest you:

ERWIN SCHRÖDINGER

The fundamental idea of wave mechanics

Nobel Lecture, December 12, 1933

If you got an idea of the role of the Hamilton operator H in quantum mechanic (which parallels the role of the Hamilton function in classical mechanic) the rest is easy:

The time dependent Schrödinger equation determines the time evolution group U(t) = exp(-itH) generated by H, and the time independent Schrödinger equation determines the eigen vectors of H.

Of course, each of the many decent text books on quantum mechanics has much more detail on the matter.

Physically, the difference is much like statics and dynamics in classical mechanics. In statics, you determine in which positions you must put the objects of  a system so that they do not move. In other words, you determine which states of the system do not depend on time. The equations involve setting sums of forces equal to zero, and do not involve time. In dynamics, you use Newton's equations, which do involve time and tell you about motion.

In quantum mechanics it is similar: the time independent Schroedinger equation tells you which states are time independent. The equation, as in statics, does not involve time. On the other hand, the time-dependent Schroedinger equation tells you how a general state moves. The consistency of both view points is  maybe not quite obvious: if you have a solution psi of the time-independent Schroedinger equation, H psi = E psi, then the function psi exp(i E t) is a solution of the time-dependent equation. This does not look like an equilibrium. It is, however, because, in quantum mechanics, states are only defined up to a multiplicative phase factor.

Adnan, I am afraid the answers you've got so far may further confuse you. Let me try it in a simple way. Time-dependent Schrödinger equation (TD-SE) is THE master equation, the FULL one; it may describe any non-relativistic system, without being concerned about specific states of the system, whatever they might be. Now, in good part of quantum mechanics (QM), people are interested in  specific states -- most often the states in a CONFINED potential, the so called eigen-states. Those are very much like an electromagnetic (EM) eigen-modes of a EM-resonator, or eigen-modes of a violin string (theoretically speaking, there are infinite number of them, as long as we neglect damping). Here comes an important point related to you question.

Since we do know (or just assume) that those modes/states exist, we can assign a FREQUENCY (omega) of oscillations to each one of them (often we even don't know that frequency apriory, and aim to find it). In terms of QM then we can claim that the wave-function of such a state is psi(r vec,t) = exp(- i omega t) * PSI(r vec), where "r vec" is a radius vector, "t" is time. We thus simply separated a solution of TD-SE into a TIME-dependent component, " exp(- i omega t) ", and strictly SPACE-dependent component, "PSI(r vec)". This is a procedure well familiar to any engineer -- the separation of variables, and it is great for the system with discreet spectrum.

Now, following the ages-known procedures, we plug this solution into TD-SE, and since the frequency "omega" is fixed, all the " exp(- i omega t) " get canceled out, and we have a new equation, this time for only SPACE-dependent component, "PSI(r vec)".  Here you are -- this a so called time-INDEPENDENT  Schrödinger equation for "PSI(r vec)". It is much narrower equation, and strictly speaking, we should've been assigning a subscript "N" to those function, something like "PSI_N(r vec)", where "N" signified a specific state; but following a "lazy" tradition in QM, we usually don't, which is not nice and confuse people. That is why T-independent SE look like something on the same footing as a full-blown TD-SE. It is not; it just a tool for us to work with specific states. And there is nothing "quantum-mechanical"  to it; the same way you'd look for eigen-modes of a string, or a rod, or any acoustic resonator, or waves in a bath-tube.  

When you know all the solutions for eigen-functuons, and IF your system allows for a FULL solution consisting ONLY eigen-solutions, you can say: all right , now I can build the FULL solution as a linear sum of ALL the eigen-solutions (mutiplied by their respective time-dependent harmonic components, exp(- i omega_N t), each one coming in with its own amplitude, "a_N". Those amplitudes are in general complex, and their absolutes, "|a_N|^2" are essentially the probabilities of a particle to be found in that specific N-th state, it is a so called "population" of the state. Since the particle should be found somewhere with the probability "1", the total sum of those state probabilities is to be "1".

In general, in QM one can even talk about "eigen-states" of a system with non-discreet spectrum, but it is a different story, and requires going into math-subtleties. 

I hope this helps.

--Alex

Alexander: I like very much your didactic explanation.

Nevertheless I don't agree with the restriction in your statement: "THE master equation, the FULL one; it may describe any non-relativistic system". Don't forget that Dirac relativistic equation can also be expressed as a time dependent Schrödinger equation. 

Now a question to Adnan: Are you more confused with our answers? Or is it clear?

Ivan: "Dirac relativistic equation can also be expressed as a time dependent Schrödinger equation" 

No, it doesn't. Dirac eqn has a second derivative of wavefunction in time, whereas the Schrödinger eqn has a first derivative. Of course one can play with Dirac eqn (and that is how he's got his eqns for spin), yet it will be not a SE. One can also play with words and terminology, but historically, pedagogically, and in the main-stream literature, SE is SE, and Dirac eqn is Dirac eqn.  Anyway, I think this particular subject doesn't belong here; no need to wave terminology flags while trying to help the author of the question with simple things. 

Schrodinger equation is working horse of quantum mechanics, rather wave-mechanics, in spite of the  initial controversies between  Schrodinger's formulation of wave mechanics and Heisenberg's formulation of quantum mechanics. For removing your confusion about the mathematical formulation an the physical interpretation of time-independent (stationary states) and time dependent (dynamical states) Schrodinger equation I suggest you to consult first chapter of  my book entitled,' Advanced Quantum Mechanics' published by M/s Pragati Prakashan Meerut (India) Eleventh Edition, 2013 

Sorry Alexander, but you are confusing Dirac equation with Klein-Gordon eqn. Precisely Dirac eqn is a first order equation! 

It is not too difficult to see that the Schrodinger's equation appears when a system is submitted to laws that do not change : the system represented by maps X(t) of X'(t+a) with a constant a are the same : X(t)=x'(t+a). This is the case when time is only an argument used to locate the state of the system in time. This is different when the time is used to represent the duration since an event (the elapsed time). When a system is submitted to external influences, these should be known (or measurable) and incorporated in the system.  So the Schrodinger equation still apply, but in the variables representing the system the external constraints should appear with the same footing as the other, and follow the same equation, and clearly we have a compatibility condition.

There is the Pauli-Schrödinger equation as well (a two-component form of the Schroedinger equation with spin, which actually allows for an easy derivation of the Dirac equation....

It is understood that the Schrödinger eqution is:  ihΨ = HΨ 

The configuration of the Hamilton opertor H and the nature of Ψ (scalar, spinor, bispinor) makes the difference (Schrödinger, Pauli, Dirac). The operator  is the first order derivative with respect to time.

@Niels R. Walet. There is another form of Schrödinger equation which includes spin as a part of the Laplacian operator. This wave equation also yields Dirac energy levels.

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

Schrodinger equation follows from Brownian motion on imposing space-momentum uncertainty relation. while Dirac equation follows from Brownian motion on imposing time-energy uncertainty relation. Spin is automatically incorporated in Dirac equation but not in Schrodinger equation. It is not possible to make Schrodinger equation consistent with relativity while Dirac equation is a relativistic quantum equation.

when the Hamiltonian operator acts on a certain wave function, and the result is proportional to the same function, then that constant is the stationary state

I believe that several important answers (and questions) are vitally linked to a Comment Ivan made about 5 years ago, regarding Minkowski’s “ict” rotation, versus the simpler forms (which cause us to forget the roots of the material). Especially when working with frameworks so incredibly interlinked, yet dissonant with one another until a proper subsumation or effective, consistent unified map occurs, we are extremely unwise to forget the history of any aspect of our field(s), if at all possible. These elementary components seem to work together in very important ways, achieving many subtle hints (and some less subtle that I’m seeing), which — if perceived for what they are — produce a strange congruity that would appear to match most experimental results and bring such questions back into focus. I do think we lose something by eliminating the ”ict” formulation. The set of isomorphisms that I may see on the horizon in many cases imply and rely upon this result, although I could express something comparable with more tortured functions, they would merely be disguising a very natural rotation operation into a space including an imaginary dimensional component.

If I recall the last discussion, someone pointed out that the same could be achieved with hyperbolic tangents, etc, all of which — as Ivan pointed out — nevertheless say the same thing as the trigonometric formulation, which is the formulation Minkowski presented and Ivan re-emphasized, simply using “i” in the appropriate location, producing what amounts to related rotations. Perhaps some undergraduates simply perform better on tests of symbol manipulation than on tests of symbol interpretation when addressing Minkowski formalism, but I don’t think it is wise to leave our “theoretical roots‘ so early.

Freeman Dyson was kind enough to grant a remote Q&A session for my Particle Physics students and other students not very long ago. He is, as expected, a wonderful, humble, wise, and gracious person. He made several salient points: (1) He went to High School in WWII and was lucky to get 7 hours of class in a week, (2) If he hadn’t had the opportunity to make mistakes, he would have never made his discoveries, almost certainly, (3) The PhD/Tenure process today is plagued (by natureof the system) by gross overspecialization, whereas his insights and those of his greatest colleagues came from somewhat frenetically being genuinely excited about many disciplines all at once and finding suprising connections, and (4) We rush to force students to learn very Advanced Mathematics, but that same Mathematics and though process has shown itself incapable of resolving the problems remaining with QFT, which limit its interpretive power over large distances. He would rather people experiment, make (safe) mistakes, then find the Mathematics that seems like it could work. In other words, although he said it much better, “If we don’t let out students get out of the box, and if we don’t get out of the box, we’ll never again make any breakthroughs.”

I am increasingly convinced that Ivan’s point has far greater merit than may be immediately apparent to some, especially given the equations and transformations I’m seeing apparently work well together. I’m looking into Minkowski’s original motivation, and I would be very grateful for any suggestions, thoughts, or papers that any of you have. Thank you, Ivan, for persisting in raising a question I suspect we all should have kept asking, rather than smugly celebrating our ignorance with poorly-understood symbols that, perhaps very helpful for people in specialized subdisciplines, possess the mere illusion of understanding for thise looking with a macroscopic perspective!