Mathematics

Mathematics#

Cartesian to Spherical Coordinates : $$ \begin{gather} z = r \cos \theta\\ x = r \sin \theta \cos \phi \\ y = r \sin \theta \sin \phi \end{gather} $$

$dV$ in Polar Coordinates : $$ dV = r^2 \sin\theta ~ dr d\theta d\phi $$

$dV$ Rotating Circle : Imagine rotating a circle with some area mapping out the sphere $$ dV = 2 \pi r^2 \sin\theta ~ dr d\theta $$

$dV$ Radial Spherical Shell : Imagine a spherical shell with some surface area expanding out to a bigger sphere $$ dV = 4\pi r^2 dr $$

$dV$ Cylindrical Coordinates : $$ dV = r ~dzdrd\theta $$

Sine-Cosine Relative Phase : Sine and Cosine are off by $\pi/2$ phase,

$$
\cos(x) = \sin(\pi/2 - x)
$$

Derivative of Logarithm : $$ \frac{\partial }{\partial x}\Big[\ln f(x)\Big] = \frac{f'(x)}{f(x)} $$

Hyperbolic Sine to Exponential : $$ \sinh x = \frac{e^x - e^{-x}}{2} $$

Hyperbolic Cosine to Exponential : $$ \cosh x = \frac{e^x + e^{-x}}{2} $$

Stirling Formula : The factorial as $n \rightarrow \infty$ goes as,

$$
\log(n!) \sim n \log n - n
$$

Rotational Mechanics#

Solving Conservation of Momentum : 1. Try dimension analysis 2. Try direction conservation 3. Otherwise, algebra

Angular Momentum (Linear) : $$ \vec L = \vec r \times \vec p $$

Angular Momentum (Rotational) : $$ \vec L = I \vec \omega $$

Moment of Inertia : The second mass moment of radius (the first mass moment is the center of mass) $$ I = \int r^2 dm $$

Mass From Density : $$ M = \int \rho dV $$

Parallel Axis Theorem : The moment of inertia at the center of mass can describe any moment of inertia at another parallel axis $$ I_\parallel = I_\text{CM} + Mr_\perp^2 $$

Torque from Force : $$ \vec \tau = \vec r \times \vec F $$

Torque from Angular Momentum : $$ \vec \tau = \frac{d}{dt} \vec L $$

Center of Mass : Think the expectation value of the radius with respect to mass, $$ r_\text{CM} = \avg{r}_m = \frac{1}{M}\int{r ~ dm} $$

Lagrangian and Hamiltonian#

Lagrangian : 1. Solve for the kinetic term $$ K = \frac{1}{2}m(\dot x^2 + \dot y^2 + \dot z^2) $$ 2. Convert the kinetic term and potential term into natural and easier coordinate system (e.g., polar) 3. Write the Lagrangian $$ \mathcal{L}(q, \dot q, t) = K - U $$

Euler-Lagrange Equation : $$ \frac{d}{dt}\frac{\partial \mathcal L}{\partial \dot q} = \frac{\partial \mathcal L}{\partial q} $$

Conjuagte Momentum : $$ p = \frac{\partial \mathcal L}{\partial \dot q} $$

When is the Conjugate Momentum Conserved? : When the Lagrangian or Hamiltonian is independent of $q$ $$ \frac{\partial \mathcal L}{\partial q} = \frac{\partial \mathcal H}{\partial q} = 0 $$

Hamiltonian : $$ \sum_{i}p_i \dot q_i - \mathcal L $$

When is Hamiltonian Time-Independent? : When potential energy is indepndent of time or velocity. $$ \frac{d}{dt} U = 0 \quad \text{or} \quad \frac{d}{d\dot q} U = 0 $$

Time-Independent Hamiltonian : $$ H = K + U $$

Orbits#

Lagrangian (Polar) : Used heavily in orbital mechanics $$ \mathcal L = \frac{1}{2}m\dot r^2 + \frac{1}{2} mr^2 \dot\theta^2 + \frac{1}{2}mr^2 \sin \theta \dot\phi^2 - U(r, \theta, \phi) $$

Effective Lagrangian of Orbits : By conservation of angular momentum, $\dot L=0$ allows us to choose a plane where the motion is only within the plane $\theta = \frac{\pi}{2}$. $$ \mathcal L = \frac{1}{2}m \dot r^2 + \frac{1}{2} m \dot r^2 \dot \phi^2 - U(r, \phi) $$

Orbital Angular Momentum : The conjugate angular momentum $$ \begin{align*} \ell &= \frac{\partial L}{\partial \dot \phi} \\ &= mr^2 \dot \phi \\ &= mvr \end{align*} $$

Effective Force of Orbits : From the Euler-Lagrange equation, $$ F = m \ddot r = \frac{\ell^2}{mr^2} - U'(r) $$

Effective Potential of Orbits : $$ U_\text{eff}(r) = \frac{\ell^2}{2mr^2} + U(r) $$

Reduced Mass : The mass of the barycenter in the reduced mass frame. $$ \mu = \frac{m_1m_2}{m_1 + m_2} $$

Springs#

Hooke’s Law : $$ F = -kx $$

EOM of Hooke’s Law : $$ x(t) = A \cos(\omega t + \phi); \qquad \omega = \sqrt{\frac{k}{m}} $$

Potential Energy of Hooke’s Law : $$ U = \frac{1}{2}kx^2 $$

Springs in Series and Parallel : Same rule as capcictors in electricity

Solving System of Springs : For a system of springs with mass $m_i$ and $k_i$, the EOM has the LHS depending on the mass diagonal tensor and RHS on the spring stiffness tensor

$$
M\ddot x = -Ax\\
$$

$$
\begin{gather*}
x(t) = ae^{i \omega t} \tag{anzatz}\\
\Downarrow\\
M \omega^2 a = Aa\\
\end{gather*}
$$

Solving for $\omega^2$ by taking the determinant of,

$$
\det\left(A - M\omega^2 \right) = 0
$$

Synchronous Oscillation Frequency : For a system of springs, the lowest frequency mode is the synchronous oscillation.

Force of Dampening : $$ F_\text{damp} = -b\dot x $$

Dampened Spring Solutions : For $\beta = b/2m$ and $\omega_0 = \sqrt{k/m}$, 1. Underdamped ($\beta^2 < \omega_0^2$) $$ \begin{gather*} x(t) = Ae^{-\beta t}\cos(\omega t - \delta)\\ \omega = \omega_0^2 - \beta^2 \end{gather*} $$ 2. Critically Damped ($\beta^2 = \omega_0^2$) 3. Overdamped ($\beta^2 > \omega_0^2$)

Harmonic Driven Spring : We can only write the differential equation,

$$
\ddot x + 2\beta \dot x + \omega_0^2 x = A\cos\omega t
$$

Harmonic Resonating Frequency : $$ \omega_R^2 = \omega_0^2 - 2\beta^2 $$

Harominc Driven Amplitude : We can only write the proportionality $$ D \propto \frac{1}{\abs{\omega_0^2 - \omega^2}} $$

Fluid Mechanics#

Bernoulli’s Principle : $$ \frac{v^2}{2} + gz + \frac{P}{\rho} = \text{constant} $$

Fluid Conservation : $$ \rho v A \Delta t = \text{constant} $$

Pressure Force : $$ F = PA $$

Buoyant Force : $$ F_B = \rho V_d g $$

* $V_d$ : Volume dispersed

Electrostatics#

Maxwell Equations for Electrostatics : $$ \begin{gather*} \nabla \cdot \vec E = \frac{\rho}{\epsilon_0}\\ \nabla \times \vec E = 0 \end{gather*} $$

Electric Field : $$ \vec E(\vec r) = \frac{1}{4\pi\epsilon_0}\frac{Q}{r^2} $$

Coulomb Force : $$ \vec F = q \vec E $$

Electric Potential Field : Because the elecric field is conservative, its electric potential which is its gradient is a scalar field. $$ \vec E = - \nabla V $$ Alternatively you may use, $$ V(\vec r) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\vec r')}{\abs{\vec r - \vec r'}} d^3\vec r $$

Voltage : The electric potential to create a electrical configuration moving a charged particle from often $a=\infty$ to $b$. $$ V_{ab} = \int_a^b \vec E \cdot d \vec l $$

Gauss’ Law : $$ \oint \vec E(\vec r) \cdot d \vec S = \frac{Q_\text{enc}}{\epsilon_0} $$

E-field of an Infinite Plane : $$ \vec E = \frac{\sigma}{2 \epsilon_0} \hat n $$

E-field of a Line and Cylinder : $$ \vec E = \frac{\lambda}{2\pi \epsilon_0 r} \hat r $$

Facts about Conductors : * Electric field inside is zero * Net charge density inside is zero * Any net charge is at the surface * Electric field is always perpendicular to the surface * Electric potential is continuous at all boundaries

Method of Images : Follow two rules:

* Don't count the energy created in the image
* Directly calculate the electric field inside the image

Electric Work : For a single charge $Q$,

$$
W = Q \int_a^b E \cdot dr = QV_{ab}
$$

For multiple charges, a double counting factor is corrected. It is much easier and intuitive to write this for the discrete case

$$
W = \frac{1}{2} \sum q_i V(r_i)
$$

Electric Field Energy : $$ U_E = \frac{\epsilon_0}{2} \int \abs{E}^2 ~ d^3r $$

Electric Power : $$ P = IV $$

Ohm’s Law : $$ V = IR $$

Capacitance : $$ C = \frac{Q}{V} $$

Parallel Plate Capacitance : $$ C = \frac{A\epsilon_0}{d} $$

Capcitance Energy : $$ U_C = \frac{1}{2}CV^2 $$

Magnetostatics#

Maxwell Equations for Magnetostatics : $$ \begin{gather*} \nabla \cdot \vec B = 0\\ \nabla \times \vec B = \mu_0 \vec J \end{gather*} $$

Ampere’s Law : $$ \oint_C \vec B \times d\vec l = \mu_0 I_\text{enc} $$

Lorentz Force : $$ \begin{align*} \vec F &= q (\vec v \times \vec B)\\ &= I (d\vec l \times \vec B) \end{align*} $$

Biot-Savart Law : $$ \vec B(\vec r) = \frac{\mu_0 I}{4 \pi}\int\frac{d\vec l \times \hat{\vec r'}}{r'^2} $$

3 Standard Problems of Magnetostatics : 1. Find B-field given current configuration Use Ampere’s law if symmetric else Biot-Savart 2. Find forces on a wire given charge in a B-field Use Loretnz force 3. Find energy of the B-field Integrate

B-field of Infinite Wire : Using Ampere’s law on a cylinder $$ \vec B = \frac{\mu I}{2 \pi r} $$

B-field of Solenoid : Using Ampere’s law on a square loop of size $l$, where $n$ is the number of windings per length. $$ B = \mu_0 n I $$

B-field of Toroid : A curved and closed loop solenoid with $N$ windings, $$ B(r) = \frac{\mu_0 N I}{2\pi r}; \qquad R_\text{in} < r < R_\text{out} $$

B-field Work : The magnetic field does no work as the Lorentz force is perpendicular to the magnetic field.

B-field Energy : $$ \mu_B = \frac{1}{\mu_0}\int{B^2 ~d^3\vec r} $$

Boundary Condition : Opposite of the electric field, so only the parallel component exists. For a surface current $\vec K$ adjacent to the surface $\hat n$,

$$
\Delta B_\parallel = \mu_0 \vec K \times \hat n
$$

Cyclotron Force : A charged partricle moving non-parallel to a uniform magnetic field experience a force on the axis perpendicular to both the velocity and magnetic field,

$$
\begin{gather*}
\text{dir}(B) = \hat z, \quad \text{dir}(v) = \hat y\\
\vec F = qvB(\hat y \times \hat z) = qvB\hat x
\end{gather*}
$$

The last equation is very useful relating to circular force

$$
\begin{gather*}
qvB = m\frac{v^2}{r}\\
B = \left(\frac{m}{q}\right)\frac{v}{r}
\end{gather*}
$$

With the RHS, you can solve for the cyclotron radius $r$ and the angular frequency $v/r$.

Electrodynamics#

Maxwell Equations Corrections : $$ \begin{align*} \nabla \times \vec E &= -\frac{\partial \vec B}{\partial t}\\ \nabla \times \vec B &= \mu_0 \vec J + \mu\epsilon_0 \frac{\partial \vec E}{\partial t} \end{align*} $$

Faraday’s Law : A changing magnetic field produces an electric field,

$$
\oint \vec E \cdot d \vec l = -\frac{d \Phi_B}{d t}
$$

Electromotive Force : A misnomer for the electric potential produced by a changing magnetic field,

$$
\varepsilon = \frac{d\Phi_B}{dt}
$$

Inductance : $$ L = \frac{\Phi_B}{I} $$

Solenoid Inductance : $$ L = \frac{\mu_0 N^2 A}{l} $$

Solenoid Energy : The energy stored in the solenoid is generated from the magnetic field $$ U_L = \frac{1}{2}LI^2 $$

Ampere’s Law for Electrodynamics : The charge enclosed is now dependent on the electric flux,

$$
\oint_C \vec B \cdot d \vec l = \mu_0 \epsilon_0 \dot\Phi_E
$$

Electric Dipoles : $$ \vec p = \int \vec r ~ dQ $$

Electric Dipole Potential : $$ V(\vec r) = \frac{1}{4\pi\epsilon_0} \frac{\vec p \cdot \vec r}{r^2} $$

Torque of Electric Dipole in External E-field : $$ \tau = \vec p \times \vec E $$

Electric Dipole Energy : $$ U = -\vec p \cdot \vec E $$

E-field of an Electric Dipole : $$ E \propto \frac{\vec p}{r^3} $$

Magnetic Dipole : $$ \vec m = I\vec A $$

Torque of Magnetic Dipole in External B-Field : $$ \tau = \vec m \times \vec B $$

Magnetic Dipole Energy : $$ U = -\vec m \cdot \vec B $$

B-field of a Magnetic Dipole : $$ B \propto \frac{\vec m}{r^3} $$

Charge Density from Polarization : For the poalrization vector $\vec P$,

$$
\begin{gather*}
\frac{dq}{dA} = \vec P \cdot \hat n\\
\frac{dq}{dV} = -\nabla \cdot \vec P
\end{gather*}
$$

Dielectric Capacitance : $$ \begin{gather*} \epsilon = \kappa \epsilon_0\\ C = \frac{\kappa \epsilon_0 A}{d} \end{gather*} $$

Electromagnetic Waves#

Wave Equation : The laplacian if the E-field and B-field is related to its own acceleration

$$
\begin{gather*}
\nabla^2E = \mu_0 \epsilon_0 \ddot E\\
\nabla^2B = \mu_0 \epsilon_0 \ddot B
\end{gather*}
$$

Speed of Light : $$ c = 1/\sqrt{\mu_0\epsilon_0} $$

Wave Solution : $$ \begin{align*} \vec E &= E_0 e^{i(kr - \omega t)} \\ \vec B &= \vec E / c \end{align*} $$

Poynting Vector : The vector of propogration that points along the wave’s momentum

$$
\vec S = \frac{1}{\mu_0}(\vec E \times \vec B)
$$

Radiant Flux : Magnetude of the poynting vector

$$
F = \abs{\vec S}
$$

Intensity : Time-average flux

$$
\avg{F} = \frac{1}{2}c\epsilon_0E_0^2
$$

Radiation Power of Accelerating charge : $$ P \propto q^2 \ddot x^2 $$

Oscillating Electric Dipole Intensity : $$ \avg{S} \propto \frac{p_0^2\omega^4\sin^2\theta}{r^2} $$

Oscillating Electric Dipole Average Power : $$ \avg{P} \propto P_0^2\omega^4 $$

Oscillating Magnetic Dipole Average Power : $$ \avg{P} = m_0^2\omega^4 $$

Circuits#

Waves#

Wave Equation : $$ \ddot f = \dot x^2 f'' $$

Wave Solutions : The solution of the wave equation is either traverse waves or standing waves:

1. Traverse Waves
	$$
	f(x,t) = A\cos(kx - \omega t + \delta)
	$$

2. Standing Waves
	$$
	g(x,t) = \frac{1}{2}\Big[f(x+vt) + f(x-vt)\Big]
	$$

Wavenumber and Wavelength Relation : The wavenumber is the number of waves in the length of $2\pi$ $$ k = \frac{2\pi}{\lambda} $$

Frequency and Angular Frequency Relation : The angular frequency is the number rotations in $2\pi$ per second. $$ \omega = 2\pi f $$

Dispersion Relation : The dispersion relation is the relation between $k$ and $\omega$

$$
\omega(k) = vk
$$

Phase Velocity : The dispersion relation for a single wavenumber

$$
v_\text{phase} = \frac{\omega}{k}
$$

Group Velocity : The dispersion relation for each wavenumber

$$
v_\text{group} = \frac{d\omega}{dk}
$$

Index of Refraction : By the dispersion relation,

$$
\frac{\omega}{k} = \frac{c}{n}
$$

The index of refraction is the amount at which light is slowed down to a velocity at some medium $v$ with index of refraction $n$.

$$
n = \frac{c}{v}
$$

Malus’ Law : Light is comes out a polarizer in one direction giving the intensity a cosine relation due to the dot product

$$
I = I_0 \cos^2\theta
$$

Brewster’s Angle : The angle at which incident light is split into a polarized reflected light and the refracted light is perpendicular to the reflected light

$$
\theta_B = \arctan\left(\frac{n_2}{n_1}\right)
$$

Phase Difference from Path Length Differnce : Two idendical waves in frequency emitted at the same time but end up travelling a different path length of difference $\Delta x$ has a phase difference of

$$
\Delta \delta = k \Delta x
$$

Interference Phase and Path Length : Two identical waves on collision will constructively or destructively intefere with each other at the even or odd $\pi$ phase difference respectively.

$$
\begin{gather*}
\Delta \delta = 2m \pi \tag{constructive}\\
\Delta \delta = (2m + 1)\pi \tag{destructive}\\
\Big\Downarrow \\
\Delta x = \lambda m \tag{constructive}\\
\Delta x = \lambda \left(m + \tfrac{1}{2}\right) \tag{destructive}\\
m \ge 0;
\end{gather*}
$$

Double Slit Interference : The path length for Young’s double slit experiment is

$$
\Delta x = d\sin\theta
$$

It's interference maxima and minima are whole and halve factors of the wavelength respectively

Single Slit Interference Minima : It’s interference maxima and minima are opposite of the classical rule. This time the minima is at whole wavelengths

$$
\begin{gather*}
a \sin \theta = m\lambda \tag{minima}\\
m \ge 1;
\end{gather*}
$$

Optical Path Length : Wave going through a mediums of index of refraction $n$ has a phase shift depending on $n$ and how far its traveled in the medium $d$,

$$
\Delta \delta = knd
$$

This implies an effective or optical path length of,

$$
\Delta x = nd
$$

Thin-Film Phase Shift : In addition to optical path length, a wave traveling from a mediums $n_1$ to a thin film of $n=n_2$ can experience a $\pi$ phase shift if,

$$
\begin{gather*}
n_2 > n_1 \tag{phase shift}\\
n_2 < n_1 \tag{$\pi$ phase shift}
\end{gather*}
$$

Thus the total phase shift for a wave through $n_2 > n_1$ of thickness $d$ and exitting is,

$$
\Delta \delta = 2kn_2d + \pi
$$

Rayleigh Criterion (Circular Aperture) : An emitter of two light sources of wavelength $\lambda$ separated at some distance $D$ is separable only if the angle follows,

$$
\sin\theta = 1.22 \frac{\lambda}{D}
$$

More often seen the time reversal of the setup, how close $d$ does two light sources separated by $\Delta x$ has to be for an aperature of diameter $D$.

$$
\tan \theta = \frac{\Delta x}{d} = 1.22 \frac{\lambda}{D}
$$

Braggs Diffraction : A crystal lattice with each node separated by distance $d$ has a index of refraction $n$ and experiences interference maxima at,

$$
d\sin\theta = \frac{n\lambda}{2}
$$

Angle of Reflection : The angle of incident is also the angle of reflection

$$
\theta_i = \theta_r
$$

Angle of Refraction (Snell’s Law) : $$ \frac{n_1}{n_2} = \frac{v_1}{v_2} = \frac{\sin \theta_2}{\sin \theta_1} $$

Often time used is use angles from the horizon $\alpha = \frac{\pi}{2} - \theta$,

$$
\frac{n_1}{n_2} = \frac{v_1}{v_2} = \frac{\cos \alpha_2}{\cos \alpha_1}
$$

Focal Length : $$ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} $$

Lensmaker Equation : $$ (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) $$

Magnitifcation : $$ m = -\frac{d_i}{d_o}; \qquad \begin{cases}m > 0 & \text{upright}\\ m < 0 & \text{inverted}\end{cases} $$

Rayleigh Scattering : $$ I \propto I_0 \lambda^{-4} $$

Doppler Effect : $$ \frac{f}{f_0} = \frac{\Delta v_r}{\Delta v_s} = \frac{c+v_r}{c-v_s} $$

* $\Delta v_r$: Relative velocity of the wave to the receiver
* $\Delta v_s$: Relative velocity of the wave to the source

Thermodynamics#

Canonical Ensemble : The set of all possible states (or outcome space) of an esemble of particles or microsystems given the following is fixed:

* Number of particles
* Volume
* Temperature

Grand Canonical Ensemble : Same as canonical ensemble but the number of particles can change.

Maxwell-Boltzmann Statistics : The probability that a particle is in energy level $E_i$ is given by the Maxwell-Boltzmann distribution

$$
P(E_i) = \frac{e^{-\beta E_i}}{\sum_{i=1}^N e^{-\beta E_i}}
$$

Although inaccurate and prone to statistical errors, the LHS is often expressed as the expected proportion of particles in energy level $E_i$ or $P(E_i) = \avg{N_i}/{N}$

Boltzmann Factor : The numerator in the Maxwell-Boltzmann distribution. It’s the unormalized likelihood that a particle is in some energy level $E_i$

$$
P(E_i) \propto e^{-\beta E_i}
$$

Partition Function : Initially it’s a tool to normalize the Maxwell-Boltzmann distribution, but it’s found to be extremely useful in determining other quantities. The partition function is the sum of all Boltzmann factors

$$
Z = \sum e^{-\beta E_i}
$$

Continuous Partition Function : $$ Z = \frac{1}{N!h^{3N}}\int{\exp\Big[-\beta H(\Sigma p_i, \Sigma x_i)\Big]~\sum d^3p_id^3x_i} $$

Relativistic Non-Interacting Partition Function : Without potential energy, the Hamiltonian is just $E=\abs{\vec p}c$ which gives the partition function for one particle is

$$
\begin{align*}
Z 	&=	\frac{V}{h^3N}\int{e^{-\beta \abs{\vec p} c} ~ d^3 p}\\
	&\propto VT^3
\end{align*}
$$

In general the power rule is $T^{\dim \vec p}$ thus average energy is

$$
\avg{E} = (\dim \vec x) k_B T
$$

Expectation of Energy from Partition Function : The expectation of energy can be determined from the partition function.

$$
\avg{E} = \frac{\partial}{\partial \beta} \ln Z
$$

Entropy : Entropy is simply defined as the \ln of the number of possible states $\Omega$ with the factor of Boltzmann constant:

$$
S = k_B \ln \Omega
$$

Entropy from Partition Function : The entropy can be determined form the partition function.

$$
S = \frac{\partial}{\partial T}\left(k_B T \ln Z\right)
$$

Equipartition Theorem : The internal energy of a system is contributed by a factor of $\frac{1}{2}k_B T$ for each degrees of freedom

$$
E_i = \frac{D}{2}k_B T
$$

A degree of freedom can be determined by the number of quadratic terms in the Hamiltonian

First Law of Thermodynamic : Internal energy of a system is increased by heat and decreased from the system doing work. Energy is not created or destroyed instead transfered from an external large resevoir

$$
\Delta U = Q - W
$$

Second Law of Thermodynamic : Entropy cannot be decreased and obeys

$$
\Delta S \ge \int \frac{\delta Q}{T}
$$

Third Law of Thermodynamic : At absolute zero $T=0$, the entropy is zero so the all particles collpase into one microstate.

Ideal Gas Law : The equation of state for non-interacting monoatomic particles of gas

$$
PV = Nk_B T
$$

Van der Waals’ Improvement of Ideal Gas : The equation of state adding particle interaction $b$ and size $a$ to the ideal gas.

Reversible Process : A quasistatic process which at first the system slowly does work into the system by slowly changing volume,

$$
\delta W = P d V
$$

In consequence,

$$
\delta Q = T \delta S
$$

The take away is that the entropy in the system changes at minimal

$$
\Delta S = \int  \frac{\delta Q}{T}
$$

Quasistatic Process : A process that change very slowly that on every step it’s in thermodynamic equilibrium

Adiabatic Process : A process that does not transfer heat

$$
\delta Q = 0
$$

Isentropic Process : A process that is both reversible and adiabatic so that entropy does not change

$$
\delta S = 0
$$

Isentropic Ideal Gas : The ideal gas in an isentropic process follows

$$
\begin{gather*}
PV^\gamma = \text{constant}\\
\gamma = \frac{C_P}{C_V}
\end{gather*}
$$

Iso-Processes : Any problem with iso-something process often ask for work. To do this,

1. Solve for work $\delta  W = P d V$ in terms of $P$
2. Use $P$ from the equation of state.

Free Expansion of Ideal Gas : An adiabatic process at which the ideal gas eventually occupy the whole volume

$$
PV = \text{const}
$$

Additionally since $PV$ doesn't change, adiabatic ideal gas is also isothermal.

Fundamental Thermodynamic Identity : The differential form of the second law of thermodyanmic,

$$
dU = TdS - PdV
$$

State Variables : The state variable can be dtermined from the thermodyanmic identity if you know the internal energy

$$
T = \frac{\partial U}{\partial S}\bigg\rvert_V
$$

$$
P = -\frac{\partial U}{\partial V}\bigg\rvert_T
$$

Heat Capacity : A material constant that is the amount of heat to change the temperature of the material.

$$
\begin{align*}
C_V = \left(\frac{\partial Q}{\partial T}\right)_V \\
C_P = \left(\frac{\partial Q}{\partial T}\right)_P
\end{align*}
$$

Heat Capcity from Equipartition Theorem : The equipartition theorem gives the internal energy as a function of temperature. This is useful to calculate heat capacity at constant volume

$$
C_V = \frac{\partial U}{\partial T}
$$

Ideal Gas Heat Capacity : $$ \begin{gather*} C_P - C_V = Nk_B\\ \big\Downarrow\\ \gamma = \frac{D+2}{D} \end{gather*} $$

Specific Heat : The heat capacity per mass $$ c = \frac{C}{M} $$

Useful for determining energy

$$
Q = mc\Delta T
$$

Efficiency of Heat Engine : $$ e = \frac{W}{Q_\text{in}} = 1 - \abs{\frac{Q_\text{out}}{Q_\text{in}}} $$

Carnot Efficiency : $$ e = 1 - \frac{T_\text{out}}{T_\text{in}} $$

Carnot Cycle : Two isothermal process interleaved by two isentropic process. The internal energy at the beginning and end of the cycle is the same thus the work is

$$
\Delta W = \Delta T \Delta S
$$

Ideal Gas Hamiltonian : $$ H = \frac{p^2}{2m} $$

Ideal Gas Partition Function : $$ \ln Z_N \propto N\ln\left(VT^{3/2}\right) $$

Ideal Gas Internal Energy : $$ U = \avg{E} = \frac{3}{2}Nk_BT $$

Ideal Gas RMS Velocity : $$ v_\text{rms} = \sqrt{\frac{(\dim \vec x) k_B T}{m}} $$

Ideal Gas Entropy : $$ S = Nk_B \ln \frac{VT^{3/2}}{N} + \text{constants} $$

Speed of Sound in Ideal Gas : $$ \begin{align*} c_s &= \sqrt{\gamma\frac{P}{\rho}}\\ &= \sqrt{\gamma \frac{k_B T}{m}} \end{align*} $$

Occupation Number Distribution : The distribution of occupation number for an energy level $N_i$ is different depending on if the particles are bosons or fermions.

$$
F = \avg{N_i} = \frac{1}{\exp[\beta (E_i-\mu)] + 1} \qquad
\begin{cases}
	+ & \text{Fermion} \\
	- & \text{Boson}
\end{cases}
$$

The total average occupancy number sums across all possible energy level and all possible ways a particle can be at the energy level (i.e., the degeneracy)

$$
\avg{N} = \sum g(E_i)\avg{N_i}
$$

Continuous Occupancy Number Distribution : The continuous distribution changes the degeneracy function to be the density of state function

$$
\begin{gather*}
\rho = \frac{dg}{dE}\\
\avg{N} = \int \rho(E)F(E)~dE
\end{gather*}
$$

Fermi Energy : $$ E_F \propto \left(\frac{N}{V}\right)^{2/3}m^{-1} $$

Special Relativity#

Coordinate Tranform : For the person the S frame, the person sees the S’ frame is moving away at velocity $v$. $$ \begin{align*} t = \gamma (t' + \frac{v}{c^2}x)\\ x = \gamma (x' + vt')\\ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \end{align*} $$

Simultaneity : Two events occuring at the same time for one frame, occurs at different times for the other

Time Dilation : Measuring time with fixed position $x'$

$$
\Delta t = \gamma \Delta t'
$$

Length Contraction : Measure two points of an object with fixed time $t$ $$ L = \frac{L'}{\gamma} $$

Velocity Addition : $$ \frac{v+u}{1 + \frac{vu}{c^2}} $$

Position Four-Vector : $$ x^\mu = (ct, x, y, z) $$

Energy-Momentum Four-Vector : $$ p^\mu = (E/c, p_x, p_y, p_z) $$

Lorentz Transform : $$ \begin{pmatrix} \gamma & -\gamma \beta & 0 & 0\\ -\gamma \beta & \gamma & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{pmatrix} $$

Four-Vector Dot Products : The four-vector dot product is given by

$$
a \cdot b = a^0b^0 - a^1b^1 - a^2b^2 - a^3b^3
$$

Note that all four-vector dot product is Lorentz invariant

Energy-Momentum Formula : From expanding $p^2$ and taking the rest frame (the dot product is invariant) $$ \begin{gather*} p^2 = m^2c^2\\ E^2 = \abs{\vec p}^2 c^2 + m^2c^4 \end{gather*} $$

Relativistic Doppler Shift : The doppler shift only depends on the relative velocity between the source and the observer $$ \begin{gather*} \beta = \frac{v}{c}\\ \frac{\lambda'}{\lambda} = \sqrt{\frac{1 + \beta}{1 - \beta}} \end{gather*} $$

Pythagorean Triples and Relativity : The pythagorean triples are useful for relativity to solve for $\gamma$ given $\beta$,

$$
(A,B,C) \to \left(\beta_1, \beta_2, 1 \right) \to (\gamma_2^{-1}, \gamma_1^{-1}, 1)
$$

A common one is $(3, 4, 5)$ such that,

$$
(\beta_1 = \tfrac{3}{5}, \beta_2=\tfrac{4}{5}, 1) \to (\gamma_2=\tfrac{5}{3}, \gamma_1=\tfrac{5}{4}, 1)
$$

Statistics#

Error : Error often mean the sample variance of some data sample $S = \set{X_1=x_1, X_2=x_2, \ldots, X_n=x_n}$. The error is whats known as the unbiased estimator of the underlying variance of the distribution $X_i$

$$
\sigma_S^2 = \frac{1}{n-1} \sum_{i=1}^n(x_i - \bar x)^2
$$

The unbiased estimator changes the normalization factor from $n \rightarrow n-1$. This effect disappears as the sample size reaches the population size $n \rightarrow N$.

Errors are interpreted as Gaussian deviations from the true mean $x^*$ interpreted as the underlying true value.

$$
x^* = \bar x \pm \sigma_S
$$

Error Propogation : Two measurements of the distribution $X$ and $Y$ has an effective error governed by error progation. If $X$ and $Y$ are independent,

$$
\sigma = \sqrt{\sigma_X^2 + \sigma_Y^2}
$$

Given the measurements are multivariate (i.e., the data points depends on more than one variable $z(x_1,x_2,\ldots x_n)$), the distributions are then multivariate.

$$
\sigma_z^2 = \sum_{i=1}^n \left(\frac{\partial z}{\partial x_i}\right)^2 \sigma_{x_i}^2
$$

Other rules of propgation for two measurements $A$ and $B$ are as follow

$$
\begin{gather*}
\sigma(\alpha A) = \alpha \sigma_A\\
\sigma(AB) = AB\sqrt{\left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2}\\
\sigma(A/B) = \frac{A}{B}\sqrt{\left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2}
\end{gather*}
$$

Error of Poisson Process : The error (sample variance) in a sampling of a Poisson distribution is,

$$
\sigma_S = \sqrt{x}
$$

Where $x$ is the count of events occured within the sample period.

Poisson Waiting Time : The waiting time to observe a Poisson count or more mathematically the time between two Poisson count is distributed as the exponential part of Poisson.

$$
P(t) = \lambda e^{-\lambda t}
$$

Electronics#

Impedence : The impedence $Z$ is the phase and magnitude of a current given the source is AC of angular frequency $\omega$. The impedence ties into many classical E&M circuits.

$$
\begin{gather*}
\text{Ohm's Law:} & V = IZ\\
\text{Capcitor:} & Z = \frac{1}{i \omega C}\\
\text{Inudctor:} & Z = i\omega L \\
\text{Resistor:} & Z = R
\end{gather*}
$$

De-Morgan’s Law : NAND and NOR gates and be disected as their individual OR and AND gates respectively

$$
\overline{A \cdot B} = \overline A + \overline B\\
\overline{A + B} = \overline A \cdot \overline B
$$

Particle Interactions#

Nuclei : Facts about nuclei interactions

* Are bigger than electrons thus gets stopped faster upon collision of particle field
* Travel in straight lines because nuclei are heavier
* Interact mostly with atomic electrons

Photons : Facts about photon interactions

* Photoabsorption is when photon is absorbed by the atom causing an electron to be emitted with energy,

	$$
	E_\text{max} = E_\gamma - \phi
	$$

Where $\phi$ is the work function or more commonly the binding energy.

* Compton Scattering is when photons elastically collide with an electron and both particles scatter. The scattered photon has the Compton wavelength

	$$
	\lambda = \frac{h}{m_ec}
	$$

* At $E_\gamma \ge 2m_e c^2 \sim 10 \text{MeV}$, the nuclei's electric field can induce a photon to produce an electron-positron pair

Radioactive Decay : A substance of $N_0$ particles under radioactive decay, loses particles at the rate of exponential decay

$$
N = N_0 e^{-t / \tau}
$$

Where $\tau$ is the mean lifetime. Given multiple channels of decay the effective mean lifetime is

$$
\tau^{-1} = \sum_i\tau_i^{-1}
$$

Numbers to Remember#

Hydrogen Ground State Energy : $13.6$ eV

Electron Rest Mass Energy : $511$ eV

Rayleigh Criterion Circular Coefficient : $1.22$

Wein Displacement Constant : $3 \times 10^{-3}$

CMB Temperature : $2.7$ K

Astrophysics#

Redshift : The Doppler redshift is the factor that the observed wavelength expanded setting no expansion at $z=0$

$$
\frac{\lambda'}{\lambda} = 1 + z
$$