PX262 - C2a - the dynamical variables

dynamical variables: position, momentum, energy, etc
in classical physics, they are represented by algebraic variables: $x$ , $p_{x}$ , $E$ , which can have all possible (continuous) values
in quantum mechanics, only some values are allowed
this also needs to work together with the wavefunction to obtain information

operators

\begin{matrix} [- \frac{ℏ^{2}}{2 M} {\vec{\nabla}}^{2} + V (\vec{r})] μ = E μ \\ \hat{H} μ = E μ \end{matrix}

where, $\hat{H}$ is the hamiltonian (energy) operator

a mathematical operator is an object which acts on a function and produces a new function, which need not be the same function
in the above case, the same function is obtained
the special cases are called eigenfunctions, and the values of $E$ are called eigenvalues:
$\hat{H} =$ operator
$μ =$ eigenfunction
$E =$ eigenvalue
eigenfunctions correspond to the single eigenvalue of the system
if there is a system described by the eigenfunction of the given operator, and measurements of the corresponding variable is taken, the given eigenvalue is obtained
it is reasonable to expect that if a second measurement is taken, the same answer is obtained
if the measurement of a given value is obtained, the wavefunction immediately after the measurement should be the corresponding eigenfunction ( $ϕ_{n}$ )
the dynamical variables are represented by operators, and the eigenvalues of these operators correspond to the possible measurement outcomes

if $\hat{Q}$ is a hermitian operator: $\int f \hat{Q} g d τ = \int g {\hat{Q}}^{*} f d τ$ where $f$ and $g$ are well-defined functions that vanish at infinity
the momentum operator: $$\begin{align*}
\int f \hat P_{x} g ,d\tau &= -i\hbar \int_{-\infty}^{\infty} f \frac{\partial g}{\partial x},dx \
&= -i\hbar \left( [fg]{-\infty}^{\infty}- \int{-\infty}^{\infty} g \frac{\partial f}{\partial x},dx \right) \
&= + i\hbar \int_{-\infty}^{\infty} g \frac{\partial f}{\partial x},dx \
\therefore \int f \hat P_{x} g ,d\tau &= \int_{-\infty}^{\infty} g \hat P_{x}^{} f,dx
\end{align}$$