What is an operator

Question

since we are studying schrodinger's equation , we met with the Hamiltonian operator , so I was wondering what's an operator , and particulary the Hamiltonian operator ?

Answer 1

Being a bit simplistic, the idea is like this:

A function is something which eats some number-like object (a real number, a vector, a matrix, etc) and spits out another number-like object (not necessarily the same type).

Analogously, an operator eats a function and spits out another function. An elementary example of this is the differentiation operator $\frac{d}{dx}$. Apply this operator to a function $f$ and you get its derivative $\frac{df}{dx}$:

$$\frac{d}{dx}f(x) = \frac{df(x)}{dx}$$

The Hamilton operator, $H$, is a linear operator associated with a given quantum system. It acts on (eats) a wavefunction (let's specifically consider an eigenfunction $\psi$) and results in (spits out) the wavefunction times a scalar (the energy $E$).

$$H\psi = E\psi$$

The above equation is called the time-independent Schrodinger equation. It really is just an eigenvalue equation like the type you saw in linear algebra. Only now your vectors are complex-valued functions and your linear transformation is an operator.