Ars Numerandi

Black body. Wien's law, Stefan-Boltzmann's law

Wien's law
$$\frac{d}{d\lambda} I_{\lambda}(\lambda,T)=0$$
$$\frac{-5}{\lambda^6}\frac{1}{\exp\left(\frac{hc}{\lambda kT}\right)-1}+\frac{ hc }{kT\lambda^7}\frac{\exp\left(\frac{hc}{\lambda kT}\right)}{\left(\exp\left(\frac{hc}{\lambda kT}\right)-1\right)^2}=0$$
$$e^x(x-5)+5=0$$ …

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Black body. Planck's law

Electromagnetic waves in a box:
$$\nu_i=n_i \frac{ c}{2L}$$
$$dn^3/L^3=\frac{8\pi}{ c^3} \nu^2 d\nu$$ …

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Compton effect

Photon energy
$$E_{\gamma}=h\nu_{\gamma}=\frac{hc}{\lambda_{\gamma}}$$
and momentum
$$p_{\gamma}=\frac{h\nu_{\gamma}}{c}=\frac{h}{\lambda_{\gamma}}$$

The photon scatters on the …

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The 1D harmonic oscillator

$$\hat{H}=\frac{\hat{p}^2}{2m}+\frac{m\omega^2\hat{x}^2}{2}$$

$$E\psi(x)=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\psi(x)+\frac{m\omega^2x^2}{2}\psi(x)$$

Substitution \(x=\frac{y}{\sqrt{m\omega/\hbar}}\), \(\phi(y)=\psi\left(\frac{y}{\sqrt{m\omega/\hbar}}\right)\)
$$-\frac{d^2}{dy^2}\phi(y)+y^2\phi(y)=\frac{2E}{\hbar\omega}\phi(y)$$
$$\phi(y)=H(y)\exp\left(-\frac{y^2}{2}\right)$$
Hermite's differential …

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Heisenberg's uncertainty principle

For position \(\hat{x}\) and momentum \(\hat{p}_x\) operators the canonical commutation relation \({\left[\hat{x},\hat{p}_x\right]=i\hbar}\) implies that for any state \(\phi\) the standard deviation of position \(\Delta_{\phi}\hat{x}\) and the standard deviation of momentum \(\Delta_{\phi}\hat{p}_x\) are related by the inequality
$$\Delta_{\phi}\hat{x}\Delta_{\phi}\hat{p}_x\geq\frac{\hbar}{2}$$
Proof:
$$\left[\hat{A},\hat{B}\right]=i\hat{C}$$
An average value of \(\hat{D}^\dagger\hat{D}\), …

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Double pendulum

\(x_1=l_1\sin\left(\theta_1\right),\quad y_1=-l_1\cos\left(\theta_1\right)\)
\(x_2=l_2\sin\left(\theta_2\right)+l_1\sin\left(\theta_1\right),\quad y_2=-l_2\cos\left(\theta_2\right)-l_1\cos\left(\theta_1\right)\)

$$L\left(\theta_1,\dot{\theta}_1,\theta_2,\dot{\theta}_2\right)=\frac{m_1(\dot{x}_1^2+\dot{y}_1^2)}{2}+\frac{m_2(\dot{x}_2^2+\dot{y}_2^2)}{2}-m_1gy_1-m_2gy_2=$$
$$=\frac{m_1l_1^2\dot{\theta}_1^2}{2}+\frac{m_2\left(l_1^2\dot{\theta}_1^2+l_2^2\dot{\theta}_2^2+2l_1l_2\cos\left(\theta_2-\theta_1\right)\dot{\theta}_1\dot{\theta}_2\right)}{2}+$$
$$+\left(m_1+m_2\right)gl_1\cos\left(\theta_1\right)+m_2l_2\cos\left(\theta_2\right)$$

Lagrange's …

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Lagrange's equations

Action
$$S=\int_{t_1}^{t_2}dt\quad L\left(q_i,{\dot{q}}_i,t\right)$$
Lagrangian
$$L\left(q_i,{\dot{q}}_i,t\right)=E_{kin}-E_{pot}$$
Generalized coordinates and velocities \(q_i,{\dot{q}}_i\)
Lagrange's …

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