Useful physics/math references

Trig Identities (top)

$\sin (2\theta )=2\sin (\theta )\cos (\theta )$

$\cos (2\theta )={\cos }^2(\theta )-{\sin }^2(\theta )=2{\cos }^2(\theta )-1=1-2{\sin }^2(\theta )$

$\tan (2\theta )=\frac{2\tan (\theta )}{1-{\tan }^2(\theta )}$

${\sin }^2(\theta )+{\cos }^2(\theta )=1$

${\tan }^2(\theta )+1={\sec }^2(\theta )$

${\cot }^2(\theta )+1={\csc }^2(\theta )$

$\sin (x)=\frac{i}{2}(e^{-ix}-e^{ix})$

$\cos (x)=\frac{1}{2}(e^{-ix}+e^{ix})$

${\sin }^2(x)=\frac{-1}{4}(e^{-2ix}+e^{2ix})+\frac{1}{2}$

${\cos }^2(x)=\frac{1}{4}(e^{-2ix}+e^{2ix})+\frac{1}{2}$

Hyperbolic Trig Identities (top)

$\sinh (x)=\frac{e^x+e^{-x}}{2}$

$\cosh (x)=\frac{e^x-e^{-x}}{2}$

$\cosh (x)-\sinh (x)=1$

Commutator Identities (top)

$[A,B]=AB-BA$

$[A,B]=-[B,A]$

$[A,B+C]=[A,B]+[A,C]$

$[A,BC]=[A,B]C+B[A,C]$

$[AB,C]=[A,C]B+A[B,C]$

Poisson Brackets (top)

$\{F,G\}={\mathrm{\Sigma }}_i(\frac{\partial F}{\partial q_i}\frac{\partial G}{\partial p_i}-\frac{\partial F}{\partial p_i}\frac{\partial G}{\partial q_i})$

$\{F,G\}=-\{G,F\}$

$\{aF+bG,H\}=a\{F,H\}+b\{G,H\}$

$\{FG,H\}=F[G,H]+[F,H]G$

$\{q_i,q_j\}=0;p_i,p_j=0;q_i,p_j={\delta }_{ij}$

Determinant Identities (top)

Levi-Cevita Symbol (top)

${ϵ}_{ijk}=1$ if $ijk=123,231,312;-1$ if $ijk=321,213,132$

$a\times b={ϵ}_{ijk}{a}_j{b}_k$

$L_i={ϵ}_{ijk}{x}_j{p}_k$

${ϵ}_{ijk}{ϵ}_{ilm}={\delta }_{jl}{\delta }_{km}-{\delta }_{jm}{\delta }_{kl}$

Coordinate Transformations (top)

Spherical:

$x=r\sin (\theta )\cos (\varphi )$

$y=r\sin (\theta )\sin (\varphi )$

$z=r\cos (\theta )$

Lagrangian:

${\dot{x}}^2+{\dot{y}}^2+{\dot{z}}^2={\dot{r}}^2+r^2{\dot{\theta }}^2+r^2{\sin }^2(\theta ){\dot{\varphi }}^2$

Taylor Expansion (top)

$f(x=a)=f(a)+\frac{f^{\prime }(a)}{1!}(x-a)+\frac{f^{″}(a)}{2!}(x-a{)}^2+\frac{f^{‴}(a)}{3!}(x-a{)}^3+...$

$\sin (x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+...$

$\cos (x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+...$

$\sinh (x)=x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+...$

$\cosh (x)=1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+...$

Approximations (top)

$\sqrt{1+x}\approx 1+\frac{x}{2}-\frac{x^2}{8}-...$

$e^{ix}\approx 1+ix-\frac{x^2}{2!}-i\frac{x^3}{3!}+\frac{x^4}{4!}+...$

$\sin (x)\approx x;\cos (x)\approx 1-\frac{x^2}{2};\tan (x)\approx x$

Complex numbers, conjugates, hermitian... (top)

$(a+bi{)}^{\ast }=(a-bi)$

Pauli Matrices (top)

$S_x=\frac{\hslash }{2}\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

$S_y=\frac{\hslash }{2}\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]$

$S_z=\frac{\hslash }{2}\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$

$[S_i,S_j]=i{ϵ}_{ijk}\hslash S_k$

$S_i,S_j=\frac{{\hslash }^2}{2}{\delta }_{ij}$

Matrix Exponentials (top)

$\exp (X^T)=\exp (X{)}^T$

$\exp (X^{\ast })=\exp X^{\ast }$

Jacobi's Formula: $\det (e^A)=e^{\mathrm{tr}(A)}$

Distribution

Gaussian distribution

Poisson distribution

Boltzmann Formula

Different pictures

Interaction picture transformation, for a given $H$, you would get

$U(t)=e^{-iHt};a(t)=U(t)a{U}^{†}(t))$

Heisenberg picture transformation:

$\frac{d{a}_I}{dt}=\frac{i}{\hslash }[H_I,a_I]$

Baker-Campbell-Hausdoff Formula (top)

$e^X{e}^Y=e^Z,\mathrm{where}Z=X+Y+\frac{1}{2}[X,Y]+\frac{1}{12}[X,[X,Y]]-\frac{1}{12}[Y,[X,Y]]+...$

Baker Hasudorff Lemma:

$e^{-G}A{e}^G=A+[G,A]+\frac{1}{2!}[G,[G,A]]+\frac{1}{3!}[G,[G,[G,A]]]+...$

Zassenhaus Formula

$e^{A+B}=e^A{e}^B{e}^{-\frac{1}{2}[A,B]}$

Quantum Harmonic Oscillator (top)

$E_n=(n+\frac{1}{2})\hslash \omega$

$H=\frac{p^2}{2m}+\frac{1}{2}m{\omega }^2{x}^2$

$a|n⟩=\sqrt{n}|n-1⟩;a^{†}|n⟩=\sqrt{n+1}|n+1⟩$

$[a,a^{†}]=1$

$a=\frac{1}{2}(\frac{x}{x_{zp}}+\frac{ip}{p_{zp}})$

$x_{zp}=\sqrt{\frac{\hslash }{2m\omega }},p_{zp}=\sqrt{\frac{2m\hslash }{\omega }}$

$x=\sqrt{\frac{\hslash }{2m\omega }}(a^{†}+a);p=i\sqrt{\frac{m\hslash \omega }{2}}(a^{†}-a)$

$[X,Y]=\frac{i}{2}$

Coherent states (top)

$D(\alpha )=e^{\alpha a^{†}-{\alpha }^{\ast }a}$

$D(\alpha )|0⟩=|\alpha ⟩$

$D(-\alpha )aD(\alpha )=a+\alpha$

$[a,D(\alpha )]=D\alpha$

$a|\alpha ⟩=\alpha |\alpha ⟩$

Particle in a Box

$V=0,0\le x\le$

$E_n=\frac{n^2{\pi }^2{\hslash }^2}{2m{L}^2}$

${\psi }_n=\sqrt{\frac{2}{L}}\sin (\frac{n\pi x}{L})$

Spherical Harmonics (Central Potentials) (top)

$Y_0^0=\frac{1}{2\sqrt{\pi }}$

$Y_1^1=-\sqrt{\frac{3}{8\pi }}{e}^{i\varphi }\sin (\theta )$

$Y_1^0=\frac{1}{2}\sqrt{\frac{3}{\pi }}\cos (\theta )$

$Y_1^{-1}=\sqrt{\frac{3}{8\pi }}{e}^{-i\varphi }\sin (\theta )$

$Y_2^{-2}=\frac{1}{4}\sqrt{\frac{15}{2\pi }}{e}^{-2i\varphi }{\sin }^2(\theta )$

$Y_2^{-1}=\frac{1}{2}\sqrt{\frac{15}{2\pi }}{e}^{-i\varphi }\sin (\theta )\cos (\theta )$

$Y_2^0=\frac{1}{4}\sqrt{\frac{5}{\pi }}(3{\cos }^2(\theta )-1)$

$Y_2^1=-\frac{1}{2}\sqrt{\frac{15}{2\pi }}{e}^{i\varphi }\sin (\theta )\cos (\theta )$

$Y_2^2=\frac{1}{4}\sqrt{\frac{15}{2\pi }}{e}^{2i\varphi }{\sin }^2(\theta )$

Angular Momentum (top)

$L^2|l,m⟩=\hslash l(l+1)|l,m⟩$

$L_z|l,m⟩=\hslash m|l,m⟩$

$L_{+}|l,m⟩=\hslash \sqrt{l(l+1)+m(m+1)}|l,m+1⟩$

$L_{-}|l,m⟩=\hslash \sqrt{l(l+1)+m(m-1)}|l,m-1⟩$

$L_x=\frac{1}{2}(L_{+}+L_{-});L_y=\frac{1}{2i}(L_{+}-L_{-})$

Spin Additions (top)

Wigner-Eckert Theorem (top)

Perturbation Theory (top)

Non-degenerate perturbations: $H^{\prime }=H^{(0)}+\lambda V$

First order:

$\mathrm{\Delta }{E}_n^{(1)}=\lambda ⟨n|V|n⟩$

Second order:

$\mathrm{\Delta }{E}_n^{(2)}=\lambda {\mathrm{\Sigma }}_{i\ne n}\frac{|⟨i|V|n⟩{|}^2}{E_n^{(0)}-E_i^{(0)}}$

Degenerate perturbations:

$W_{ij}=⟨i|V|j⟩$

Solve for eigenvalues of W to find the lifted degeneracies for energy and eigenvectors for new "good" states.

Hamilton's Equations (top)

$\frac{\partial H}{\partial q_i}=-{\dot{p}}_i$

$\frac{\partial H}{\partial p_i}={\dot{q}}_i$

More advanced qubit formula references!