Quantum Physics

Black Body Radiation

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

Compton Effect: $\Delta\lambda = \frac{h}{m_0c}(1-\cos\theta)$

 

Bohr의 가정(각운동량 양자화): $L=mvr=n\hbar$

수소 원자

• $r_n = \frac{\epsilon_0n^2h^2}{\pi me^2}$
• $v_n = \frac{e^2}{2\epsilon_0 nh}$
• Bohr Radius $a_0=\frac{\epsilon_0h^2}{\pi me^2}$
• $r_n = a_0n^2$
• $K = \frac{me^4}{8\epsilon_0^2n^2h^2}$
• $U = -\frac{me^4}{4\epsilon_0^2n^2h^2}$
• $E=U+K = -\frac{me^4}{8\epsilon_0^2n^2h^2}$
• $R_H=\frac{me^4}{8c\epsilon_0^2h^3}$
• $\frac{1}{\lambda}=R_H\left(\frac{1}{{n_1}^2}-\frac{1}{{n_2}^2}\right)$

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

 

불확정성 원리: $\Delta x\Delta p\ge \hbar$, $\Delta E\Delta t\ge \hbar$

 

Schrodinger Equation(Time Independent)

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

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

Inifinite Potential Well (1D)

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

Infinite Potential Square Well (3D)

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

Harmonic Oscillator

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

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