# Schrödinger's time-dependent equation (general)

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 dont 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
Gold Member
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##.

• vanhees71
bhobba
Mentor
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

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vanhees71