Formulae for Quantum Mechanics, Quantum Field Theory, and Perturbation Theory

[Ouabache]

To allow easier access to formulae we commonly refer to within various topics and as a time saver when wanting to generate them in LaTex for discussions, I propose this thread as a convenient place to store useful formulae (to copy and paste as appropriate within threads).

 

[Ouabache]

Maxwell's Equations - Integral Form

Maxwell's Equations - Integral Form

Gauss' Law for Electricity
$$\epsilon_o \oint E \cdot dA = \sum q$$
Ampère’s Law
$$\oint B\cdot ds =\mu_o\int J \cdot dA+ \mu_o \epsilon_o \frac{d}{dt} \int E \cdot dA$$
Faraday's Law of Induction
$$\oint E \cdot ds = -\frac{d}{dt}\int B\cdot dA$$
Gauss' Law for Magnetism
$$\oint B \cdot dA = 0$$

alternate forms see ,[/URL] ref2

"Maxwell's Equations are the set of four equations, attributed to James Clerk Maxwell (written by Oliver Heaviside), that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter." [/URL]

 

[Ouabache]

Maxwell's Equations - Differential Form

Maxwell's Equations - Differential Form

Gauss' Law for Electricity
$$\nabla \cdot E = \frac{\rho}{\epsilon_0}$$
Ampère’s Law
$$\nabla \times B = \mu_0 J + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}$$
Faraday's Law of Induction
$$\nabla \times E = -\frac{\partial B}{\partial t}$$
Gauss' Law for Magnetism

$$\nabla \cdot B = 0$$

The above differential and integral forms (previous post) may be used in the absence of magnetic and polarizable media.
Alternate forms see ,[/URL] ref2

 

[Ouabache]

Euler's Equation - for engineering

Two Forms of Euler's Equation - commonly used in electrical engineering

$$e^{+j \theta}= \cos \theta + j \sin \theta$$
$$e^{-j\theta}= \cos \theta - j \sin\theta$$Using the above expressions, $\cos\theta$ and $\sin\theta$ can be derived

$$\cos\theta = \frac{1}{2}(e^{j\theta} + e^{-j\theta})$$

$$\sin\theta = \frac{1}{2j}(e^{j\theta}-e^{-j\theta})$$

Alternate form of Euler's Formula, see ref

 

[Ouabache]

Mech Engr - Heat Transfer across Cylindrical Tube

Start with Fourier's Law of Heat Conduction ref1

$$\renewcommand{\vec}[1]{\mbox{\boldmath #1 }} \vec{Q} =-k \bar{\nabla} T$$

For this geometry (cylindrical tubing) by Fourier's Law, ref2

$$Q=k A \left (\frac {\Delta T}{\Delta r} \right )$$

Heat Transfer Across Length of Cylindrical Tubing

$$\mbox {\Huge Q= \frac {2 \pi k L (T_i-T_o)}{ln (\frac{r_o}{r_i}) } }$$

$k$ - thermal conductivity of material [BTU/(hr-ft-deg F)]
$L$ - length of tube (ft)
$T_i$ - temperature along inside surface of tube (deg F)
$T_o$ - temperature along outside surface of tube (deg F)
$r_o$ - outside tube radius (ft)
$r_i$ - inside tube radius (ft)
$Q$ - heat transfer (BTU/hr)

Heat Flux - Heat Transfer Rate per Unit Area ref3

$$Q^{''} = \frac {Q}{A} \ \ \ \ \ \ \ \left ( \frac {BTU}{hr \cdot ft^2} \right )$$

For this geometry

$$A = 2 \pi r_o L \ \ \ \ \ \ (ft^2)$$

 

[Ouabache]

Average convection heat transfer coefficient*

$$\bar{h} = \frac{\dot{m}c_p(T_{m,o}-T_{m,i})}{\pi D L \ \Delta T_{lm}} \ \ \ \ \ equ. (i)$$

$\dot{m}$ flow rate of fluid (kg/s)
$c_p$ specific heat at constant pressure [J/(kg-K)]
$T_{m,i}$ mean temperature outside cyl. tube [deg C]
$T_{m,o}$ mean temperature inside cyl. tube [deg C]
$D$ diameter of cyl. tube [m.]
$L$ length of cyl. tube [m.]
$\Delta T_{lm}$ change in the log mean temperature [deg C]
$\bar{h}$ ave. conv. heat transfer coef.[W/(m^2 - deg K)]

Change in Log Mean Temperature*

$$\Delta T_{lm}=\frac {(T_s - T_{m,o})-(T_s - T_{m,i})}{ln \frac {T_s - T_{m,o}}{T_s - T_{m,i}}} \ \ \ \ equ. (ii)$$

$T_s$ constant surface temperature [deg C]* from Fundamentals of Heat Transfer by Incropera and DeWitt

 

[Perturbation]

Hell, I'm bored, so why not.

(There are implicit summations over repeated indices throughout; units with $\hbar =c=1$)

Quantum Mechanics:

$$H|\psi (t)\rangle =i\frac{\partial}{\partial t}|\psi (t)\rangle$$

Non-relativistic in coordinate representation: $$H=-\frac{\nabla^2}{2M}+V(\vec{x})$$
Relativistic in coordinate representation:$$H=\gamma^0\left(-i\mathbf{\gamma}\cdot\mathbf{\nabla} +\gamma^{\mu}V_{\mu}(x)+m)$$ (Dirac)

Clifford algebra defined by Dirac matrices: $$\{\gamma^{\mu}, \gamma^{\nu}\}=2g^{\mu\nu}$$

Dirac equation: $$\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi=0$$
Klein-Gordon equation: $$\left(\partial^2 +m^2\right)\phi =0$$

Born approximation: $$i\mathcal{M}=-i\tilde{V}(\vec{x})$$

Quantum Field theory:

Dirac Lagrangian: $$\mathcal{L}=\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi$$ ($\bar{\psi}=\psi^{\dagger}\gamma^0$)

Complex Klein-Gordon Lagrangian: $$\mathcal{L}=\tfrac{1}{2}\left(|\partial_{\mu}\phi |^2-m^2|\phi |^2\right)$$

Complex Phi-four Lagrangian: $$\mathcal{L}=\tfrac{1}{2}\left(|\partial_{\mu} \phi |^2-m^2|\phi |^2\right)+\frac{\lambda}{4!}|\phi |^4$$

Yukawa Lagrangian: $$\mathcal{L}=\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi +\tfrac{1}{2}\left(|\partial_{\mu}\phi |^2-m_{\phi}^2|\phi |^2\right)+g\phi\bar{\psi}\psi$$

QED Lagrangian: $$\mathcal{L}=\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi-\frac{1}{4}F_{\mu\nu}^2+e\bar{\psi}\gamma^{\mu}A_{\mu}\psi$$

$$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$$

Yang Mills Lagrangian:

$$\mathcal{L}=\bar{\psi}\left(i\gamma^{\mu}D_{\mu}-m\right)\psi+F^a_{\mu\nu}^2$$

$$D_{\mu}=\partial_{\mu}-igA^a_{\mu}t^a$$
$$F^a_{\mu\nu}=\partial_{\mu}A^a{\nu}-\partial_{\nu}A^a{\mu}+gf^{abc}A^b_{\mu}A^c_{\nu}$$

Where $t^a$ are the n dimensional matrices representing the Lie algebra

$$[T^a, T^b]=if^{abc}T^c$$

Feynman functional intergal form of propogation amplitude:
$$\langle \psi_b|e^{-iHT}|\psi_a\rangle =\int\mathcal{D}\psi\mathcal{D}\pi\exp\left(i{{\textstyle \int^T_0 d^4x\mathcal{L}\left[\psi\right]}\right)$$

$$\left(\pi =\frac{\delta S}{\delta\dot{\psi}}\right)$$

Perturbation Theory:

$$\langle\Omega |T\left\{\psi (x_n)\cdots\psi (x_1)\right\}|\Omega\rangle =\lim_{T\rightarrow\infty (1-i\epsilon )}\langle 0 |T\left\{\psi (x_n)_I\cdots\psi (x_1)_I\right\exp\left[{\textstyle -i\int^T_{-T} d^4x H_I}\right]\}|0\rangle}\left(\langle 0 |\exp\left[{\textstyle -i\int^T_{-T} d^4x H_I}\right]|0\rangle\right)^{-1}$$

$$=\lim_{T\rightarrow\infty (1-i\epsilon )}\frac{\int\mathcal{D}\psi\exp\left(i{\textstyle \int^T_{-T} d^4x\mathcal{L}\left[\psi\right]}\right)\psi_H(x_n)\cdots\psi_H(x_1)}{\int\mathcal{D}\psi\exp\left(i{\textstyle \int^T_{-T} d^4x\mathcal{L}\left[\psi\right]}\right)}$$

Schwinger-Dyson equations of motion:
$$\left\langle\left(\frac{\delta}{\delta\psi (x)}\int d^4x'\mathcal{L}\right) T\left\{\psi (x_n)\cdots\psi (x_1)\right\}\right\rangle =\sum^n_{i=1}\left\langle\psi (x_n)\cdots\left(-i\delta^{(4)}(x-x_i)\right)\cdots\psi (x_1)\right\rangle$$

I'd put some Feynman rules and the Ward-Takahashi identity and stuff up but there's no Latex for Feynman diagrams.