Schrödinger's time-dependent equation (general)

[Paradox101]

The Hamiltonian operator in the equation i×h/2π×∂/∂t×ψ=H×ψ(where 'i' is the imaginary no.,'h/2π' is just expanded form of the reduced Planck constant,'∂/∂t' is the partial derivative with respect to time 't' and ψ is the wave function) is,as I recall,H=I+V(i don't know how to get those carets on top of them) implying the "set of possible outcomes when measuring the total energy of a system".What does this mean?Does it relate to state of the system,i.e,the wave function at time t?or can it also be used for a non-relativistic particle(particle not at the speed of light) with a given position or perhaps a harmonic oscillator?

 

[Matterwave]

The Hamiltonian $H=-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x)$ is the "energy operator" just like $p=i\hbar\frac{\partial}{\partial x}$ is the momentum operator. So, eigenstates of the Hamiltonian are eigenstates of energy, and the average energy of a state is simply $\langle E\rangle = \langle\psi|H|\psi\rangle$.

 

[bhobba]

What does this mean?Does it relate to state of the system,i.e,the wave function at time t?or can it also be used for a non-relativistic particle(particle not at the speed of light) with a given position or perhaps a harmonic oscillator?

Its simply an equation describing how the state (when expanded in terms of position eigenstates) varies in time

That may be a bit of gibberish right now. To fully understand it you need to see a complete development of QM from first principles.

It likely is more mathematically advanced than you are at present, but the book to have a look at is Ballentine - Quantum Mechabics - A Modern Development:
https://www.amazon.com/dp/9814578584/?tag=pfamazon01-20

In particular have a look at the first 3 chapters. I think its likely you will get the gist even if the mathematical detail is a bit obscure.

Thanks
Bill

 

[vanhees71]

More precisely: It's the time evolution of the position-representation components of the state vector, representing a pure state. The time evolution of the state vector itself depends on the picture of time evolution chosen. E.g., in the Heisenberg picture the state vector doesn't change at all (by definition), while in the Schrödinger picture it's evolving with the full Hamiltonian.