## # Problem #

[ E=mc^{2} ]

[ frac{partial u}{partial t} = h^2 left( frac{partial^2 u}{partial x^2} + frac{partial^2 u}{partial y^2} + frac{partial^2 u}{partial z^2}right) ] # Proof # [ e^{isigma_{z}phi}sigma_ye^{-isigma_zphi}=e^{2isigma_zphi}sigma_y ]

# Solution

[ begin{align*} e^{isigma_{z}phi}sigma_ye^{-isigma_zphi} &=e^{isigma_{z}phi}(e^{-isigma_zphi}sigma_y+[sigma_y,e^{-isigma_zphi}])\ &=sigma_y+e^{isigma_{z}phi}[sigma_y,cos(phi)-i sin(sigma_zphi)]\ &=sigma_y+e^{isigma_{z}phi}[sigma_y,-isigma_zsin(phi)]\ &=sigma_y-i sin(phi) e^{isigma_{z}phi}[sigma_y,sigma_z]\ &=sigma_y+2isin(phi) e^{isigma_zphi}sigma_zsigma_y\ &=e^{isigma_zphi}(e^{-isigma_zphi}sigma_y+2isigma_zsin(phi)sigma_zsigma_y)\ &=e^{isigma_zphi}(cos(sigma_zphi)+isigma_zsin(phi))sigma_y\ &=e^{2isigma_zphi}sigma_y end{align*} ]

[ begin{align*} x &= sqrt{4^2-1^2} \ &= sqrt{15}. end{align*}] \$\$

[ begin{align} dot{x} & = & sigma(y-x) \\ dot{y} & = &rho x - y - xz \\ dot{z} & = &-beta z + xy end{align} ] [ begin{align} nablacdotvec{E} &=& frac{rho}{epsilon_0} \\ nablacdotvec{B} &=& 0 \\ nablatimesvec{E} &=& -frac{partial B}{partial t} \\ nablatimesvec{B} &=& mu_0left(vec{J}+epsilon_0frac{partial E}{partial t} right) end{align} ]

Useful formulas [ begin{align} hat{A}hat{B}hat{C}&=hat{A}(hat{C}hat{B}+[hat{B},hat{C}])\\ hat{I}&=e^{isigma_zphi}e^{-isigma_zphi}\\ e^{isigma_zphi}&=cos(sigma_zphi)+isin(sigma_zphi)\\ &=cos(phi)+isigma_zsin(phi)\\ [sigma_i,sigma_j]&=ivarepsilon_{ijk}sigma_k\\ sigma_isigma_j&=ivarepsilon_{ijk}sigma_k end{align} ]