PX262 - C2b - momentum & position operator

momentum operator

the kinetic energy, classically given as $T = \frac{p^{2}}{2 M}$ , can we written as:

T = \frac{1}{2 M} {\hat{p}}^{2} = - \frac{ℏ^{2}}{2 M} \nabla^{2}

where, $\hat{p}$ is the momentum operator

\begin{matrix} \hat{\vec{p}} = - i ℏ \vec{\nabla} \\ {\hat{p}}_{x} = - i ℏ \frac{\partial}{\partial x} \end{matrix}

the eigenvalues and eigenfunctions of the momentum operator: $- i ℏ \frac{\partial}{\partial x} ϕ = p ϕ$
the solutions are plane waves: $ϕ = A \exp (i k x)$ , where $k = \frac{p}{ℏ}$
if $V (\vec{r}) \neq 0 :$ eigenfunctions of $\hat{H}$ and $\hat{p}$ are not the same

\hat{x} ϕ = x_{0} ϕ

the eigenfunctions in this case are non-zero at a given position, and zero everywhere else $⟹ δ (x - x_{0}) :$
$\hat{x} δ (x - x_{0}) = x_{0} δ (x - x_{0})$
above is satisfied by $\hat{x} = x$
back to the hamiltonian operator:
$\hat{H} = - \frac{ℏ^{2}}{2 m} {\vec{\nabla}}^{2} + V (\vec{r})$
it can be constructed using the momentum and the position operators
the relation between $\hat{H}$ , and $\hat{p}, \hat{x}$ is the same as in classical physics
only hermitian operators can be used
the operators can be complex, but the measurements have to be real
if two well behaved functions, $f, g$ , are taken, then the hermitian operator satisfies:
$\int f \hat{Q} g d τ = \int g {\hat{Q}}^{*} f d τ$
where, $\hat{Q}$ is the operator
eigenvalues:
$\hat{Q} ϕ_{n} = q_{n} ϕ_{n}$
${\hat{Q}}^{*} ϕ_{n}^{*} = q_{n}^{*} ϕ_{n}^{*}$
- multiplying by the complex conjugate:
  $ϕ_{n}^{*} \hat{Q} ϕ_{n} = ϕ_{n}^{*} q_{n} ϕ_{n}$
  $ϕ_{n} {\hat{Q}}^{*} ϕ_{n}^{*} = ϕ_{n} q_{n}^{*} ϕ_{n}$
- integrating:
  $\int ϕ_{n}^{*} \hat{Q} ϕ_{n} d τ = \int ϕ_{n}^{*} q_{n} ϕ_{n} d τ = q_{n}$
  $\int ϕ_{n} {\hat{Q}}^{*} ϕ_{n}^{*} d τ = \int ϕ_{n} q_{n}^{*} ϕ_{n} d τ = q_{n}^{*}$
from the definition of hermitian operators, the two integrals on the $L H S$ are the same, and therefore $q_{n} = q_{n}^{*} ⟹ q_{n} \in R$