PX262 - B1 - schrödinger equation

the time-dependent equation

a fundamental equation in QM to describe the system and the state in which it is, the schrödinger equation:
$i ℏ \frac{\partial ψ (x, t)}{\partial t} = - \frac{ℏ^{2}}{2 m} \frac{\partial^{2} ψ (x, t)}{\partial x^{2}} + V (x, t) ψ (x, t)$
where,
$m =$ mass of the particle
$V (x, t) =$ the potential in which it is moving
$ψ (x, t) =$ the wavefunction which encodes all information about the state of the particle
there is a correspondence between classical energy an schrödinger equation
- in classical mechanics: $E = \frac{p^{2}}{2 m} + V$
- swapping $p, V, E$ with operators:

\begin{array}{r} E \to i ℏ \frac{\partial}{\partial t} \\ \vec{p} \to - i ℏ \vec{\nabla} \end{array}

- putting these into the energy relation, and adding :

classically, it is real, and it gives the displacement
in QM, it can be complect, and does not directly give the displacement
the QM wavefunction is connected to the probability of finding a particle at some position
in some interval, between $x$ and $x + d x$ , at a time, $t$ , the probability of finding the particle is given by the bohr postulates: $P (x, t) d x = | ψ (x, t) |^{2}$
the wavefunction has to be normalized, ie. the probability of finding the particle in the system is fixed:

\begin{matrix} \int_{- \infty}^{+ \infty} P (x, t) d x = 1 \\ ⟹ \int_{- \infty}^{+ \infty} | ψ (x, t) |^{2} d x = \int_{- \infty}^{+ \infty} ψ^{*} ψ d x = 1 \end{matrix}