• $\underline{\underline{\nu}} \quad \underline{\underline{\lambda}} \quad \underline{\underline{v}}$

• $v=\nu \lambda \text {. }$

• $E=\underline{\underline{h \nu}} ; \quad \lambda=\frac{h}{p}$

• $U=\pi c$

• $\pi = Momentum$

• U = Energy density

• $U = n h \nu$

• n = number density

• $\Pi= n \times \underline{p} = \frac{n h \nu}{c}=n \frac{h}{\lambda}$

• $c=\nu \lambda$

• $n \lambda=2 d \sin \theta$

• Provided we assign

• $\lambda = \frac{h}, {p}$ p = known $\lambda = associated$

• $p ; \quad \varepsilon \longrightarrow particle$

• $\nu; \quad\lambda \longrightarrow wave$

• $\vec{p}=m \vec{v}$

• $\varepsilon=\frac{1}{2} m v^2=\frac{{p}^2}{2 m}$

• $v=\nu \lambda$

• ${v_{\text {par }}} \quad v_{\text {wave }}$

• $E=h \nu=\frac{p^2}{2 m}$

• $\nu \rightarrow \underline{\underline{\omega}}$ : angular frequency

• $\lambda \rightarrow k \rightarrow$ Wave number

• $E=h \nu=h$

• $\frac{\omega}{2 \pi} = h \omega$

• $\omega=2 \pi \nu$

• $p=\frac{h}{\lambda}=\frac{2 \pi \hbar}{\lambda} = h(\frac{2 \pi}{\lambda}) \equiv h k$

• $\text( \omega, k) \quad v=\nu \lambda$

• $\omega=c k \rightarrow \text{dispersion relation}$

• ${\omega}=\underline{\underline{c(k^{\prime}})} k^{\prime}$

More subtle matters

• What is the frequency?
• What is the relationship between frequency, wavelength and the speed?

• $\lambda_m=\frac{h}{p} ; \nu_m=\frac{E}{h}$

• Non -relativistic Case

• $E=\frac{p^2}{2 m}$

• $\lambda_m \nu_m=V_{\text {deBroglie}} = (\frac{E}{p})$

• $v_{\text {matterwave }} v_{m \omega}=\lambda_{m \omega} v_{m \omega} = \frac{E}{p}=\frac{1}{2} \frac{m v^2}{m v}$

• $v \rightarrow particle$

• $=\frac{v}{2}$

• $E =h \nu ; \quad p=\frac{h}{\lambda}$

• $\nu =\frac{E}{h} ; \quad \lambda=\frac{h}{p}$

• $p=\frac{m_0 v}{\sqrt{1-v^2 c^2}}$

• $v = \underline{\underline{\lambda \nu}} = \frac{E}{p}=\frac{m_0 c^2}{\sqrt{1-v^2 / c^2}}$

• $v=\frac{E}{p} ;$

• $E=\frac{m_0 c^2}{\sqrt{1-v^2 / c^2}}$

• $p=\frac{m_0 v}{\sqrt{1-v / c^2}}$

• $v_{wave}=\frac{c^2}{v_{\text {particle }}} \quad v<c$

• $v_{wave} >C$

• Early Speculations

• India: Kanada (Vaisheshika School)

• Greece: Democritus

• England: Isaac Newton, Dalton, ...

• Other Schools : The great elements (Greece, India)

• Chemistry, Thermodynamics

• Priestley

• Lavoisier

• Dalton

• Mendeleev

• Mary and Pierre Curie, Becquerel

• Atoms are fundamental units of chemical reaction Consequence: May or may not be the ultimate constituents of matter.

• Must explain chemical properties

• Two major Contenders

• Plum pudding model (Thompson)

• Planetary Model (Rutherford)

• Figure: Illustrations of Atomic models

• Details

• Source: ${ }_{83}^{214} \mathrm{Bi}$;

• Energy of $\alpha$ particles $5.5 \mathrm{MeV}$

• Target: A very thin gold foil $\left(2.1 \times 10^{-7} \mathrm{~m}\right)$

• Detector: Zinc Sulphide (scintillation) + Microscope