TIme-independent schrodinger equation

[AbsoluteZer0]

I've been looking into the time independent schrodinger 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".