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)

\( \{F, G\} = \Sigma_i \left( \frac{\partial F}{\partial q_i} \frac{\partial G}{\partial p_i} - \frac{\partial F}{\partial p_i} \frac{\partial G}{\partial q_i} \right) \)

\( \{F, G\} = -\{G, F\} \)

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

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

\( \{q_i, q_j\} = 0; \quad {p_i, p_j} = 0; \quad {q_i, p_j} = \delta_{ij} \)

Determinant Identities (top)

Levi-Cevita Symbol (top)

\( \epsilon_{ijk} = 1 \mathrm{if} ijk = 123, 231, 312; \quad -1 \mathrm{ if } ijk = 321, 213, 132 \)

\( a \times b = \epsilon_{ijk} a_j b_k \)

\( L_i = \epsilon_{ijk} x_j p_k \)

\( \epsilon_{ijk} \epsilon_{ilm} = \delta_{jl} \delta_{km} - \delta{jm} \delta_{kl} \)

Coordinate Transformations (top)

Spherical:

\( x = r \sin(\theta) \cos(\phi) \)

\( y = r \sin(\theta) \sin(\phi) \)

\( z = r \cos(\theta) \)

Lagrangian:

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

Taylor Expansion (top)

\( f(x=a) = f(a) + \frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{2!} (x-a)^2 + \frac{f'''(a)}{3!} (x-a)^3 + ... \)

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

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

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

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

Approximations (top)

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

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

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

Pauli Matrices (top)

\( S_x = \frac{\hbar}{2} \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \)

\( S_y = \frac{\hbar}{2} \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} \)

\( S_z = \frac{\hbar}{2} \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \)

\( [S_i, S_j] = i \epsilon_{ijk} \hbar S_k \)

\( {S_i, S_j} = \frac{\hbar^2}{2} \delta_{ij} \)

Matrix Exponentials (top)

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

\( \mathrm{exp}(X^*) = \mathrm{exp}{X}^* \)

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

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:

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

Quantum Harmonic Oscillator (top)

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

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

\( x = \sqrt{\frac{\hbar}{2 m \omega} } (a^\dagger + a); \quad p = i \sqrt{\frac{m \hbar \omega}{2}} (a^\dagger - a) \)

\( a|n\rangle = \sqrt{n} |n - 1\rangle; \quad a^\dagger |n \rangle = \sqrt{n+1} |n+1\rangle \)

Particle in a Box

\( V = 0, 0 \leq x \leq \)

\( E_n = \frac{n^2 \pi^2 \hbar^2}{2 m L^2} \)

\( \psi_n = \sqrt{\frac{2}{L}} \sin(\frac{n \pi x}{L}) \)

Spherical Harmonics (Central Potentials) (top)

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

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

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

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

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

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

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

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

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

Angular Momentum (top)

\( L^2 |l, m\rangle = \hbar l (l+1) |l, m \rangle \)

\( L_z |l, m\rangle = \hbar m |l, m \rangle \)

\( L_+ |l, m\rangle = \hbar \sqrt{ l (l+1) + m(m+1)} |l, m+1 \rangle \)

\( L_- |l, m\rangle = \hbar \sqrt{ l (l+1) + m(m-1)} |l, m-1 \rangle \)

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

Spin Additions (top)

Wigner-Eckert Theorem (top)

Perturbation Theory (top)

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

First order:

\( \Delta E_n^{(1)} = \lambda \langle n | V | n \rangle \)

Second order:

\( \Delta E_n^{(2)} = \lambda \Sigma_{i \neq n} \frac{|\langle i | V | n \rangle|^2}{E_n^{(0)} - E_i^{(0)}} \)

Degenerate perturbations:

\( W_{ij} = \langle i | V | j \rangle \)

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

Hamilton's Equations (top)

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

\( \frac{\partial H}{\partial p_i} = \dot{q}_i \)