Wave Aspect of Matter

A particle is described as a frequency \(\nu\) and a wave fector \(\vec{k}\). \(E=h\nu=\hbar\omega,\vec{p}=\hbar\vec{k}=\hbar\frac{2\pi}{\lambda}\hat{k}\).

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

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

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