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

In summary, the conversation discusses the idea of creating a thread to store useful formulae for easier access and time-saving in discussions. It also mentions various forms of Maxwell's Equations, including the integral and differential forms, as well as other equations such as Gauss' Law for Electricity and Ampère’s Law. The conversation also includes discussions on heat transfer and quantum mechanics, with equations such as Euler's Equation, the Dirac equation, and perturbation theory being mentioned. Additional concepts such as Feynman functional integral and the Schwinger-Dyson equations of motion are also brought up.
  • #1
Ouabache
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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).
 
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  • #2
Maxwell's Equations - Integral Form

Maxwell's Equations - Integral Form

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

alternate forms see [URL [Broken],[/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 [Broken][/URL]
 
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  • #3
Maxwell's Equations - Differential Form

Maxwell's Equations - Differential Form

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

[tex] \nabla \cdot B = 0 [/tex]

The above differential and integral forms (previous post) may be used in the absence of magnetic and polarizable media.
Alternate forms see [URL [Broken],[/URL] ref2
 
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  • #4
Euler's Equation - for engineering

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

[tex]e^{+j \theta}= \cos \theta + j \sin \theta [/tex]
[tex]e^{-j\theta}= \cos \theta - j \sin\theta [/tex]Using the above expressions, [itex]\cos\theta[/itex] and [itex]\sin\theta[/itex] can be derived

[tex]\cos\theta = \frac{1}{2}(e^{j\theta} + e^{-j\theta}) [/tex]

[tex]\sin\theta = \frac{1}{2j}(e^{j\theta}-e^{-j\theta}) [/tex]

Alternate form of Euler's Formula, see ref
 
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  • #5
Mech Engr - Heat Transfer across Cylindrical Tube

Start with Fourier's Law of Heat Conduction ref1

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

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

[tex]Q=k A \left (\frac {\Delta T}{\Delta r} \right )[/tex]

Heat Transfer Across Length of Cylindrical Tubing

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

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

Heat Flux - Heat Transfer Rate per Unit Area ref3

[tex] Q^{''} = \frac {Q}{A} \ \ \ \ \ \ \ \left ( \frac {BTU}{hr \cdot ft^2} \right )[/tex]

For this geometry

[tex]A = 2 \pi r_o L \ \ \ \ \ \ (ft^2) [/tex]
 
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  • #6
Average convection heat transfer coefficient*

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

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

Change in Log Mean Temperature*

[tex]\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) [/tex]

[itex]T_s[/itex] constant surface temperature [deg C]* from Fundamentals of Heat Transfer by Incropera and DeWitt
 
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  • #7
Hell, I'm bored, so why not.

(There are implicit summations over repeated indices throughout; units with [itex]\hbar =c=1[/itex])

Quantum Mechanics:

[tex]H|\psi (t)\rangle =i\frac{\partial}{\partial t}|\psi (t)\rangle[/tex]

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

Clifford algebra defined by Dirac matrices: [tex]\{\gamma^{\mu}, \gamma^{\nu}\}=2g^{\mu\nu}[/tex]

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

Born approximation: [tex]i\mathcal{M}=-i\tilde{V}(\vec{x})[/tex]

Quantum Field theory:

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

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

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

Yukawa Lagrangian: [tex]\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[/tex]

QED Lagrangian: [tex]\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[/tex]

[tex]F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}[/tex]

Yang Mills Lagrangian:

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

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

Where [itex]t^a[/itex] are the n dimensional matrices representing the Lie algebra

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

Feynman functional intergal form of propogation amplitude:
[tex]\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) [/tex]

[tex]\left(\pi =\frac{\delta S}{\delta\dot{\psi}}\right)[/tex]

Perturbation Theory:

[tex]\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}[/tex]

[tex]=\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)}[/tex]

Schwinger-Dyson equations of motion:
[tex]\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[/tex]

I'd put some Feynman rules and the Ward-Takahashi identity and stuff up but there's no Latex for Feynman diagrams.
 
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1. What are the most commonly used formulae in physics?

Some of the most commonly used formulae in physics include Newton's second law of motion, Ohm's law, the formula for gravitational force, and the formula for kinetic energy.

2. How can I remember all the formulae for chemistry?

One way to remember formulae in chemistry is to create a cheat sheet or flashcards to review regularly. You can also break down complex formulae into smaller parts and practice applying them to different problems.

3. Are there any shortcuts or tricks for remembering mathematical formulae?

Yes, there are some tricks for remembering mathematical formulae, such as creating mnemonic devices or using visual aids to associate the formula with a specific image or concept. Practice and repetition can also help with memorization.

4. What is the process for deriving a formula?

The process for deriving a formula involves using mathematical principles and equations to solve a problem and find a relationship between different variables. The formula is then tested and refined through experimentation and observation to ensure its accuracy.

5. How can I apply formulae from different subjects to real-world situations?

To apply formulae to real-world situations, it is important to have a good understanding of the underlying concepts and principles. Look for patterns and relationships between different variables and use the appropriate formula to solve the problem. It can also be helpful to practice applying formulae to different scenarios to improve your skills.

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