Wave Aspect of Matter

A particle is described as a frequency $\nu$ and a wave fector $\overrightarrow{k}$. $E=h\nu =\hslash \omega ,\overrightarrow{p}=\hslash \overrightarrow{k}=\hslash \frac{2\pi }{\lambda }\hat{k}$.

$\psi (\overrightarrow{r},t)=A\exp i(\omega t-\overrightarrow{k}\cdot \overrightarrow{r})$.

$\lambda =h/p$. For non-relativistic cases, $\lambda =h/(mv)$.

1. Classical Trajectory $\Rightarrow$ Time-varying state
2. Quantum States $\Rightarrow$ Wave function $\psi (\overrightarrow{r},t,\cdots )$
3. $\psi (\overrightarrow{r},t)$ is the Probability Amplitude
4. Probability to find a particle at $t$ in volume $d^{3}r=dxdydz$ is $d\mathcal{P}(\overrightarrow{r},t)=C|\psi (\overrightarrow{r},t){|}^2{d}^{3}r$. $|\psi {|}^2$ is a Probability Density
5. Principle of spectral decomposition (Superposition of states) Measure $A$ $\Rightarrow$ eigenstates of $A$, $\{(a_i,{\psi }_i(\overrightarrow{r}))\}$. $\psi (\overrightarrow{r},t_0)={\psi }_{a_1}(\overrightarrow{r})\Rightarrow a_1$ and the wavefunction will remain in the state. Mixed states give a spectral decomposition of eigenstates, $\psi (\overrightarrow{r},t_0)=\sum _a{c}_a{\psi }_a(\overrightarrow{r})$, ${\mathcal{P}}_a=\frac{|c_a{|}^2}{\sum _a|c_a{|}^2},\sum _a{\mathcal{P}}_a=1$. Measure A, get $a_1$ implies the state is now ${\psi }_{a_1}(\overrightarrow{r})$.
6. Schrödinger equation. $\hat{H}\mathrm{\Psi }=E\mathrm{\Psi }$. Where $\hat{H}$ is the Hamiltonian of the system. Square-integrable, continuous, finite, $\int |\psi (\overrightarrow{r},t){|}^2{\text{d}}^{3}r=1$.