Quantum Physics

Black Body Radiation

• Stephen-Boltzman Law: $P_{blackbody}=\sigma A{T}^4$
• Wien's Law: ${\lambda }_{peak}T=\text{Const.}$
• Rayleigh-Jeans Law: $R(\lambda ,T)=\frac{2\pi ckT}{{\lambda }^4}$
• Plank's Law: $R(\lambda ,T)=\frac{2\pi h{c}^2}{{\lambda }^5(e^{hc/\lambda kT}-1)}$

Compton Effect: $\mathrm{\Delta }\lambda =\frac{h}{m_{0}c}(1-\cos \theta )$

 

Bohr의 가정(각운동량 양자화): $L=mvr=n\hslash$

수소 원자

• $r_n=\frac{{ϵ}_0{n}^2{h}^2}{\pi m{e}^2}$
• $v_n=\frac{e^2}{2{ϵ}_{0}nh}$
• Bohr Radius $a_0=\frac{{ϵ}_0{h}^2}{\pi m{e}^2}$
• $r_n=a_0{n}^2$
• $K=\frac{m{e}^4}{8{ϵ}_0^2{n}^2{h}^2}$
• $U=-\frac{m{e}^4}{4{ϵ}_0^2{n}^2{h}^2}$
• $E=U+K=-\frac{m{e}^4}{8{ϵ}_0^2{n}^2{h}^2}$
• $R_H=\frac{m{e}^4}{8c{ϵ}_0^2{h}^3}$
• $\frac{1}{\lambda }=R_H(\frac{1}{{n_1}^2}-\frac{1}{{n_2}^2})$

De Broglie wave length: $\lambda =\frac{h}{p}$

 

불확정성 원리: $\mathrm{\Delta }x\mathrm{\Delta }p\ge \hslash$, $\mathrm{\Delta }E\mathrm{\Delta }t\ge \hslash$

 

Schrodinger Equation(Time Independent)

• Hamiltonian: $H=\frac{P^2}{2m}+U$
• $P=\frac{\hslash }{i}\mathrm{\nabla }=-{\hslash }^2{\mathrm{\nabla }}^2=-{\hslash }^2(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2})$
• Schrodinger Equation: $H\psi =E\psi$
• Probability Density function: $P(x)={\psi }^2(x)$
• normalization condition: ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\psi }^{2}dx=1$

Bohr's correspondence principle: 양자적 현상의 scale을 키우면 고전역학적 분석에 수렴한다.

Inifinite Potential Well (1D)

• $-\frac{{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}=E\psi$
• Boundary condition: $\psi =0$ at $x=0,L$
• $E=\frac{\hslash }{2m}\frac{n^2\pi }{L^2}=\frac{n^2{h}^2}{8m{L}^2}$
• ${\psi }_n=\sqrt{\frac{2}{L}}\sin \left(\frac{n\pi x}{L}\right)$

Infinite Potential Square Well (3D)

• $\psi (x)=A\sin \frac{n_x\pi x}{L}\sin n_y\pi yL\sin n_z\pi zL$
• $E=\frac{h^2}{8m{L}^2}({n_x}^2+{n_y}^2+{n_z}^2)$

Harmonic Oscillator

• $-\frac{{\hslash }^2}{2m}\frac{{\partial }^2\psi }{\partial x^2}+\frac{1}{2}m{w}^2{x}^2\psi =E\psi$
• $-\frac{{\hslash }^2}{2m}\frac{{\partial }^2\psi }{\partial x^2}+\frac{1}{2}m{w}^2{x}^2\psi =E\psi$
• $E_n=(n+\frac{1}{2})\hslash w$

Tunneling Effect: finite potential well의 경우 벽 너머서도 $\psi$가 nonzero -> 벽을 뚫고 외부로 나갈 확률 존재