Quantum Mechanics

Operator Properties

$$H^{\dagger} = (H^{*})^{-1} = H$$ (93)

$$U^{-1}U = UU^{-1} = \mathbb{1}$$ (94)

$$O^{\dagger}O = \mathbb{1}$$ (95)

$$\begin{bmatrix} a & b\\ c & d \end{bmatrix}^{-1} = \frac{1}{det} \begin{bmatrix} d & -b\\ -c & a \end{bmatrix}$$

Pauli Spin Matrices:

$$\sigma_x = \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}$$ (96)

$$\sigma_y = \begin{bmatrix} 0 & -i\\ i & 0 \end{bmatrix}$$ (97)

$$\sigma_z = \begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix}$$ (98)

$$\mathbb{1} = \delta_{ij}$$ (99)

$$\sigma_i\sigma_j = i\sigma_k$$ (100)

$$\sigma_i^2 = \mathbb{1}$$ (101)

$$\mathbf{\sigma} = \sigma_x\hat{x} + \sigma_y\hat{y} + \sigma_z \hat{z}$$ (102)

Density Matrix:

$$\rho = \sum_i w_i \left\vert a^i\right\rangle \left\langle a^i\right\vert$$ (103)

$$Tr(\rho)=1$$ (104)

$$Tr(\rho A) = \langle A\rangle$$ (105)

$$\rho (\rho^2 = \rho)$$ (106)

Raising and Lowering operators (SHO)

$$a^{\pm} = \big(\frac{m\omega}{2\hbar}\big)^{1/2} x \mp i \big(\frac{1}{2\hbar\omega m}\big)^{1/2} p$$ (107)

$$x = c_1 (a + a^{\dagger}); p = c_2 (a - a^{\dagger} )$$ (108)

$$a \left\vert n\right\rangle = \sqrt{n}\left\vert n-1\right\rangle$$ (109)

$$a^{\dagger}\left\vert n\right\rangle = \sqrt{n+1} \left\vert n+1\right\rangle$$ (110)

$$a^{\dagger}a\left\vert n\right\rangle = n\left\vert n\right\rangle$$ (111)

Bose-Einstein Distribution

$$\mathcal{Z} = \sum^{\infty}_{n=0}(e^{\mu\beta} e^{-E\beta})^N$$ (112)

$$\rightarrow \boxed{\frac{1}{1 - e^{(E - \mu)\beta}} = \mathcal{Z}}$$ (113)

$$n = \frac{1}{\beta}\frac{\partial^2ln(\mathcal{Z})}{\partial\mu^2} \rightarrow$$ (114)

$$\langle n\rangle = \big[\exp{(E-\mu)\beta} - 1\big]^{-1}$$

Fermi-Dirac Distribution

$$\mathcal{Z} = e^{0\mu \beta- 0\beta} + e^{\mu \beta - \epsilon \beta}$$ (115)

$$= 1 + e^{(\mu - \epsilon) \beta}$$

$$N = kT\frac{\partial}{\partial \mu}ln\mathcal{Z}$$ (116)

$$\boxed{= \frac{1}{e^{(\epsilon - \mu})\beta}+1}$$

Angular Momentum

$$[J_i,J_j] = i\hbar \epsilon_{ijk}J_k$$ (117)

$$[\mathbf{J}^2,J_k] = 0$$ (118)

$$J_{\pm} = J_x \pm iJ_y [J_+,J_-] = 2\hbar J_z$$ (119)

$$[J_z,J_{\pm}]=\pm\hbar J_{\pm}$$ (120)

$$J^2\left\vert j,m\right\rangle = j(j+1)\hbar^2\left\vert j,m\right\rangle$$ (121)

$$J_z\left\vert j,m\right\rangle = m\hbar\left\vert j,m\right\rangle$$ (122)

$$J_{\pm} \left\vert j,m\right\rangle = \sqrt{(j\mp m)(j \pm m + 1)}\hbar \left\vert j, m\pm 1\right\rangle$$ (123)

Time independent Perturbation theory
Non-degenerate case:

$$H_0 + H', E = E_0 + \lambda E_1 + \lambda^2 E_2...$$ (124)

$$\left\vert\psi\right\rangle = \left\vert n_0\right\rangle + \lambda \left\vert n_1\right\rangle + \lambda^2 \left\vert n_2\right\rangle ...$$ Collect like-terms:

$$E_1 = \left\langle n_0\right\vert H' \left\vert n_0\right\rangle$$ (125)

$$\left\vert n_1\right\rangle = \sum_{m=1_, m\neq n}^{\infty} \frac... ...H'\left\vert n_0\right\rangle }{E_n^{(0)}-E_m^{(0)}}\left\vert m_0\right\rangle$$ (126)

$$E2 =\sum_{m=1_, m\neq n}^{\infty} \frac{\left\vert\left\langle m_0\right\vert H'\left\vert n_0\right\rangle \right\vert^2}{E_n^{(0)}-E_m^{(0)}}$$ (127) Etc.

Lippman-Schwinger Equation
Integral form:

$$\left\vert\psi^+\right\rangle = \left\vert i\right\rangle + \frac{1}{E_i - H_0 + i \hbar \epsilon}V\left\vert\psi^+\right\rangle$$ (128) Iterate recursively (First order is the Born Approximation)

$$\psi^{\pm}(\mathbf{x},t) = \phi^{\pm}(\mathbf{x},t) - \frac{m}{2\pi\hbar^2}\int_{\infty}^{\infty}d\mathbf{x'}$$ (129)

$$\times \frac{e^{ \pm i k \left\vert\mathbf{x}-\mathbf{x'}\right\v... ...ft\vert\mathbf{x}-\mathbf{x'}\right\vert}V(\mathbf{x'}\psi^{\pm}(\mathbf{x'},t)$$

This is the integral form. To get the Born approximation, transform into the momentum basis, and investigate the regime where $\left\vert\mathbf{x}-\mathbf{x'}\right\vert\rightarrow r$ for large $r$...

Born Approximation (scattering)

$$f^{(1)}(k,k') = \frac{-m}{\hbar^2 2 \pi}\int_{\infty}^{\infty}e^{i(\mathbf{k}-\mathbf{k'})\cdot\mathbf{r}} V(\mathbf{r})d^3\mathbf{r}$$ (130) Spherically symmetric solution;

$$f^{(1)}(\theta) = \frac{-2m}{\hbar^2q} \int_0^{\infty} r dr V(r) sin(qr)$$ (131)

$$\frac{d\sigma}{d\Omega} = \left\vert(f^{(1)}(\theta)\right\vert^2$$ (132)

Fermi's Golden Rule

$$w_{ij} = \frac{2\pi}{\hbar}\left\vert\left\langle\psi_f\right\vert V\left\vert\psi_i\right\rangle \right\vert^2 \rho(E_i)$$ (133)

If you wait long enough, the total transition rates becomes constant (time independent).

Variational Method (Ground State):

$$E = \frac{\left\langle\alpha\right\vert H\left\vert\alpha\right\rangle }{\langle\alpha\vert\alpha\rangle }$$ (134)

$$\alpha$$

$$\frac{dE}{d\alpha} = 0\iff E = E_0;$$ (135)

$$\alpha$$

Adiabatic and Sudden approximations

$$\dot{c}_m(t) = -c_m(t)\left\langle m;t\right\vert\big[\frac{\partial}{\partial t} \left\vert m;t\right\rangle \big]$$ (136)

$$- \sum_n c_n(t) e^{i(\theta_n-\theta_m)} \frac{\left\langle m;t\right\vert H\left\vert n;t\right\rangle }{E_n-E_m}$$ In the adiabatic approximation, the second term disappears.

$$i\frac{\partial}{\partial s} U(t,t_0) = \frac{H}{\hbar/T}U(t,t_0)=\frac{H}{\hbar\Omega} U(t,t_0)$$ (137)

$$T\rightarrow 0, \hbar\Omega>>H$$

$$U(t,t_0)\rightarrow 1 T\rightarrow 0$$

Schrodinger, Hamiltonian,
Interaction Pictures:

	operators	wavefunc/ket
Schrodinger	Time independent	$\left\vert\psi(t)\right\rangle _s$
	A	Evolve in time
Heisenberg	Time Evolution	$e^{iHt} \left\vert\psi\right\rangle _s = \left\vert\psi\right\rangle _H$
	$U^{\dagger}AU = A_H$	No evolution
Interaction	$U^{\dagger}AU$	$\left\vert\psi\right\rangle _I = e^{iH_0t}\left\vert\psi(t)\right\rangle _s$
	$e^{iH_0t} A e^{-iH_0 t} = A_I$	evolution by
	evolution with un-	$V_I$
	perturbed hamiltonian