$$i\hbar \frac{d}{dt} \|\psi (t) \rangle = H \|\psi (t) \rangle$$
$$H \|\psi \rangle = E \|\psi \rangle$$
$$\Omega_H (t) = e^{iHt/\hbar} \Omega e^{-iHt/\hbar}$$ $$\frac{d \Omega_H}{dt} = - \frac{i}{\hbar}[\Omega_h (t),H]$$
$$P = -i\hbar \nabla$$ $$[X,P]=i\hbar$$ $$\Delta \Omega = \sqrt{\langle \psi \| (\Omega-\langle \Omega \rangle)^2 \| \psi \rangle} = \sqrt{\langle \Omega^2 \rangle - \langle \Omega \rangle^2}$$
If
$$\frac{\partial H}{\partial t} = 0,$$
then:
$$\|\psi (t) \rangle = e^{-iEt/\hbar}\|\psi(0)\rangle = \sum_{E} e^{-iEt/\hbar}\|E \rangle \langle E\|\psi(0)\rangle$$
Probability Current:
$$\vec{J}=\frac{\hbar}{2mi}(\psi^*\nabla \psi-\psi \nabla \psi^*)$$
Bosons: $\|ab\rangle + \|ba\rangle$
Ferminons: $\|ab\rangle - \|ba\rangle$
$$\psi = A e^{ikx/\hbar} + B e^{-ikx/\hbar}$$ $$E = \frac{\hbar^2 k^2}{2m}$$
$\psi_n = \sqrt{2/L} \sin(n \pi x/L),$, for n even
$\psi_n = \sqrt{2/L} \cos(n \pi x/L)$ for n odd
$$E_n = \frac{\hbar^2 n^2 \pi^2}{2 m L^2}$$
$$[a,a^\dagger]=1$$
$$H = \hbar \omega (a^\dagger a + 1/2)$$
$$E_n=\hbar \omega (n+1/2)$$
$$a\|\lambda \rangle = \sqrt{\lambda} \| \lambda-1\rangle$$
$$a^\dagger\|\lambda \rangle = \sqrt{\lambda+1} \| \lambda+1\rangle$$
$a^\dagger a = N,$ The number operator
$$a a^\dagger = N + 1$$
$$X = \sqrt{\frac{\hbar}{2 m \omega}}(a+a^\dagger)$$
$$P = i \sqrt{\frac{m \omega \hbar}{2}}(a^\dagger - a)$$
$$E_n = \frac{Ry}{n^2}$$
$$Ry = \frac{m e^4}{2 \hbar^2}$$
$$a_0 = \frac{\hbar^2}{m e^2}$$
$$\psi = R_{nl} (r) Y_l^m (\theta,\phi)$$
$$R_{nl} \sim e^{-r/na_0} \left(\frac{r}{na_0}\right)^l L_{n-l-1}^{2l+1} !! \left(\frac{2r}{na_0} \right)$$
$$\psi_{100} = \sqrt{\frac{1}{\pi i a_0^3}} e^{-r/a_0} $$
$$(\frac{-\hbar^2 \partial^2}{2m \partial r^2}+V(r)+\frac{\hbar^2 l(l+1)}{2 m r^2})u_l=E_l u_l = H$$
Conserved Hermition Operators are generators of symetries
$$U = e^{-i \epsilon G/\hbar}$$
time $\rightarrow H(x)=H^*(x)$
translation -> constant P
rotation -> constant L
parity $(\Pi) \rightarrow \Pi \|x\rangle = \|-x\rangle, \Pi \|P\rangle = \|-P\rangle$
$[H,G]=0$ if G is conserved!!!
$$[J_i,J_j]=i\hbar\epsilon_{ijk}J_k$$
$$J^2\|\psi\rangle \ \hbar^2 j(j+1)\|\psi \rangle$$
$$J_z\|\psi\rangle = \hbar m \|\psi \rangle $$
$$J_+\|jm\rangle-\hbar\sqrt{j(j+1)-m(m+1)}\|j,m+1\rangle$$
$$J_-\|jm\rangle-\hbar\sqrt{j(j+1)-m(m-1)}\|j,m-1\rangle$$
$$J_x = \frac{1}{2}(J_+ + J_-)$$ $$J_y = \frac{1}{2i}(J_+ - J_-)$$
$$\vec{M}=g \gamma \vec{s},$$ where $$g\approx2, \gamma <1$$
$$\vec{s}=\frac{\hbar}{2}\vec{\sigma}$$
$$U = \frac{ge}{2m}\vec{s} \cdot \vec{B}=g \mu_B \frac{\vec{\sigma}}{2} \cdot \vec{B}$$
$$S_z = \frac{\hbar}{2} \sigma_z$$
$$\sigma_i \sigma_j = \delta_{ij}+ \epsilon_{ijk} i \sigma_k ?????????$$
$$TR(\sigma_i = 0)$$ $$\sigma_i,\sigma_j = 2i \epsilon_{ijk} \sigma_k $$
$$\|\pm\rangle_x=\frac{1}{\sqrt{2}}(\|+\rangle \pm \|-\rangle)$$ $$\|\pm\rangle_y=\frac{1}{\sqrt{2}}(\|+\rangle \pm i\|-\rangle)$$
$$\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$
$$\sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} $$
$$\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$
$$\sigma_0 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$
$$ \|JM \rangle = \sum_{m_1 m_2} \|j_1 m_1 j_2 m_2 \rangle \langle j_1 m_1 j_2 m_2 \| JM \rangle$$
$$j_1 \otimes j_2 = \sum_{j=\|j_1-j_2\|}^{j_1+j_2} \oplus J$$
Use:
$$J_z\|(j_1+j_2) (j_1+j_2)\rangle = (J_{z1} + J_{z2}) \|j_1 j_2,j_2 j_2 \rangle$$
to build states down from the top M level, which is easy and non-degenerate.
$$E[\psi] = \frac{\langle \psi \| E \| \psi \rangle}{\langle \psi \| \psi \rangle}$$
Use Gaussian or Exponential (depending on parity) trial wavefunctions and minimize the parameter.
Bound States: $\int_{x_1}^{x_2} p(x) \, dx = (n+1/2)\pi \hbar$
Tunneling Amplitudes: $\psi(x_e)=\psi(x_o) e^{\frac{i}{\hbar}\int_{x_0}^{x_e}i\sqrt{2m(V(x)-E)}\,dx}$
Remember to square (or even think of probability current) for tunneling probability!!
$$H = H^0 + H^1$$
First Order: $E_n^1= \langle n^0 \| H^1 \| n^0 \rangle$
$$\|n^1 \rangle = \sum_{m \not= n} \|m^0 \rangle \frac{\langle m^0 \| H^1 \| n^0 \rangle}{E_n^0-E_m^0}$$
Second Order: $E_n^2 = \sum_{m \not= n} \frac{\|\langle m^0 \| H^1 \| n^0 \rangle \|^2}{E_n^0-E_m^0}$
For degeneracy, diagonalize on the subspace of the degenerate states to do second order.
$$ E_n = E_n^0 + \langle n^0 \| H^1 \| n^0 \rangle + \sum_{m \not= n} \frac{\|\langle m^0 \| H^1 \| n^0 \rangle \|^2}{E_n-E_m^0} $$
$$H = H^0 + H^1(t)$$
$$i \hbar \frac{d}{dt} \| \psi (t) \rangle = H \| \psi (t) \rangle $$
If we use diagonalized states in $H^0 \rightarrow \| n^0 \rangle$, then $d_n (t)$ denotes the coefficient of that state:
$$\| \psi (t) \rangle = \sum_n d_n (t) \, e^{-i E_n^0 t/\hbar} \| n^0 \rangle $$
$$i \hbar \dot d_f = \sum_n \langle f_0 \| H^1(t) \| n^0 \rangle e^{i \omega_{fn} t} d_n$$
$$\omega_{fn} = \frac{E_f^0 - E_n^0}{\hbar}$$ $$P_{n \rightarrow f} = \| d_f \|^2$$
Bosons: by the number in each state
$$\| n_1, n_2,\ldots, n_m \rangle = \sqrt{\frac{n_1! n_2!\cdots n_m!}{N!}}[\mathrm{permutations}^\dagger]$$
Fermions: only occupied states are listed
$$\|1,2,\ldots,m\rangle = \sqrt{\frac{1}{m!}}[\mathrm{permutations}^\dagger]$$
$^\dagger$Sum over distinct permutations of states (see identical particles Bosons/Fermions above). Remember order is important with Fermions!
Creation/Annihilation Operators add one or take one particle away from only the state they are dealing with.
Commutation relations for Bosons, Anti-commutation relations for Fermions:
$$[a_i,a_j]=\delta_{ij}$$
$$[a_i^\dagger,a_j^\dagger]=0$$
Building Boson states and Fermion states, respectively: $\|n_1,n_2,\ldots,n_m \rangle = \frac{(a_1^\dagger)^{n_1}}{\sqrt{n_1!}} \cdots \frac{(a_m^\dagger)^{n_m}}{\sqrt{n_m!}} \| 0 \rangle $
$\|1,2,\ldots,m \rangle = a_1^\dagger a_2^\dagger \cdots a_m^\dagger \|0 \rangle $ Where $\|0\rangle$ is the Vacuum State
One and two body operators look like: $\Omega = \sum_{\alpha,\beta} \langle \alpha \| \Omega_{single} \| \beta \rangle a_\alpha^\dagger a_\beta$
$$U = \frac{1}{2} \sum_{\alpha,\beta,\gamma,\delta} \langle \alpha \beta \| U_{12} \| \gamma \delta \rangle a_\alpha^\dagger a_\beta^\dagger a_\delta a_\gamma $$
$$Y_0^0 = 1/ \sqrt{4 \pi}$$
$$Y_1^{0} = \sqrt{3/4 \pi} \cos(\theta)$$
$$Y_1^{\pm 1} = \mp \sqrt{3/8 \pi} \sin(\theta) e^{\pm i \phi}$$
$$Y_2^{0} = \sqrt{5/16 \pi} (3\cos^2(\theta) -1)$$
$$Y_2^{\pm 1} = \mp \sqrt{15/8 \pi} \sin(\theta) \cos(\theta) e^{\pm i \phi}$$
$$Y_2^{\pm 2} = \sqrt{15/32 \pi} \sin^2(\theta) e^{\pm 2 i \phi}$$
Almost all of this is taken from Dr. Stanton\'s class lectures. A few parts are from R. Shankar's Book Principles of QM.