PX262 - M2 - recap

i ℏ \frac{\partial ψ}{\partial t} = \hat{H} ψ

where, $\hat{H}$ is the hamiltonian

\hat{H} ϕ_{n} (x) = E_{n} ϕ_{n} (x)

where, $ϕ_{n}$ and $E_{n}$ are the eigenfunctions and eigenvalues of the hamiltonian

ψ (x, t) = \sum_{n} c_{n} ϕ_{n} (x) \exp (\frac{- i E_{n} t}{ℏ})

for an observable, $A$ , described by an operator, $\hat{A}$ , there will be eigenfunctions, $u_{a} (x)$ , corresponding to real eigenvalues, $λ_{a} :$

\hat{A} u_{a} (x) = λ_{a} u_{a} (x)

\int u_{a_{1}}^{*} (x^{'}) u_{a_{2}} (x^{'}) d x^{'} = δ_{a_{1}, a_{2}} = {\begin{cases} 1 & a_{1} = a_{2}, \\ 0 & a_{1} \neq a_{2} . \end{cases}

c_{a} = \int u_{a}^{*} (x^{'}) g (x^{'}) d x^{'}

if a system is described by the wavefunction, $ψ (x)$ , measuring $A$ will give one of the eigenvalues, $λ_{a}$ , with probability:

P (λ_{a}) = c_{a}^{*} c_{a} = | c_{a} |^{2}

where, $ψ (x) = \sum_{n} c_{a} u_{a} (x)$

ψ (x) = \sum_{a} \int u_{a}^{*} (x^{'}) ψ (x^{'}) d x^{'} u_{a} (x)

to show: $\sum_{a} u_{n} (x) u_{a}^{*} (x^{'}) = δ (x - x^{'})$ , ie: $ψ (x) = \int δ (x - x^{'}) ψ (x^{'}) d x^{'}$
from $\int ψ^{*} ψ d x = 1 :$

\begin{aligned} = \int \sum_{a} c_{a}^{*} u_{a}^{*} (x) ψ (x) d x \\ = \sum_{a} c_{a}^{*} c_{a} \\ = \sum_{a} P (λ_{a}) \\ = 1 \end{aligned}

vector space analogy

thinking of vector in 3D space: $$\vec v = v_{x} \hat i + v_{y}\hat j + v_{z} \hat k$$
the unit vectors satisfy: $\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$ and $\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$
the scalar products: $v_{x} = \vec{v} \cdot \hat{i}$ , $v_{y} = \vec{v} \cdot \hat{j}$ and $v_{z} = \vec{v} \cdot \hat{k}$
$v_{x}$ , $v_{y}$ and $v_{z}$ are analogs of $c_{a}$ 's
the projection of $ψ (x)$ along a basis 'vector', $u_{a} (x)$ , is $c_{a} :$

c_{a} = \int u_{a}^{*} (x^{'}) ψ (x^{'}) d x^{'}

\begin{matrix} \vec{v} = \sum_{i} v_{i} {\hat{e}}_{i} \\ ψ (x) = \sum_{a} c_{a} u_{a} (x) \end{matrix}