TIme-independent schrodinger equation

[AbsoluteZer0]

I've been looking into the time independent Schrödinger equation (E\Psi = Ĥ\Psi.)

I know that \Psi is the wave function and Ĥ is the hamiltonian operator. I know that Ĥ is the total of all the energies in a system. What exactly is the wave function? Is it a quantum state? And what does the E represent?

Thanks,

 

[nonequilibrium]

The following is for the one-particle case (i.e. a system with one particle).

Mathematically psi is a function on space, expressed as \psi(x,y,z) and it is complex-valued. Mathematically \psi : \mathbb R^3 \to \mathbb C. Its minimal physical meaning is that its modulus squared, mathematically | \psi |^2, gives the probability density of finding a particle at a certain point in space.

I know that Ĥ is the total of all the energies in a system.

This is true, in a sense, but I think it's prone to misconception. It is true that knowing \hat H is equivalent to knowing all the energy levels of the system (this is known in mathematics as the spectral theorem for operators) but that is not what \hat H is itself. It is an operator meaning that is a function which takes a function like psi as its argument/its input, and gives another such function as its image/output. A very simple Hamiltonian (that is its name) is \hat H = -\frac{\partial^2}{\partial x^2} (ignoring constants and expressing it in the one-dimensional case). As input it takes a function \psi and as output it gives minus its second derivative with respect to x, this is of course a new function from \mathbb R^3 to \mathbb C.

E is the specific energy level your system has (in other words you have to choose one; the idea is that you choose the energy level(*) of your system and then calculate what \psi satisfies that equation, and then its modulus squared gives the probability density as discussed above).

(*) Note that you can't choose just any value for E; the possible list is determined by \hat H, as you said.

EDIT: yes, another name for \psi is the so-called "quantum state". Also called "the quantummechanical wavefunction".