# Wave Aspect of Matter

## The De Broglie Wavelength

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, $$\lambda=h/(mv)$$.

## Statistical Interpretation, the Wave Function

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. SchrÃ¶dinger equation. $$\hat{H}\Psi=E\Psi$$. Where $$\hat{H}$$ is the Hamiltonian of the system. Square-integrable, continuous, finite, $$\int |\psi(\vec{r},t)|^2\text{d}^3r=1$$.

Created: 2023-06-25 Sun 02:35

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