397 lines
17 KiB
TeX
397 lines
17 KiB
TeX
\def\sigmaxmatrix{\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}}
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\def\sigmaymatrix{\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}}
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\def\sigmazmatrix{\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}}
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\def\sigmaxbraket{\ket{0}\bra{1} + \ket{1}\bra{0}}
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\def\sigmaybraket{-i \ket{0}\bra{1} + i \ket{1}\bra{0}}
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\def\sigmazbraket{\ket{0}\bra{0} - \ket{1}\bra{1}}
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\Part[
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\eng{Quantum Mechanics}
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\ger{Quantenmechanik}
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]{qm}
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\Section[
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\eng{Basics}
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\ger{Basics}
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]{basics}
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\Subsection[
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\eng{Operators}
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\ger{Operatoren}
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]{op}
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\GER[row_vector]{Zeilenvektor}
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\GER[column_vector]{Spaltenvektor}
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\ENG[column_vector]{Column vector}
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\ENG[row_vector]{Row vector}
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\begin{formula}{dirac_notation}
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\desc{Dirac notation}{}{}
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\desc[german]{Dirac-Notation}{}{}
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\eq{
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\bra{x} \vspace*{1cm} \text{"Bra" \GT{row_vector}} \\
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\ket{x} \vspace*{1cm} \text{"Ket" \GT{column_vector}} \\
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\hat{A}\ket{\beta} = \ket{\alpha} \Rightarrow \bra{\alpha} = \bra{\beta} \hat{A}^\dagger
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}
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\end{formula}
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\begin{formula}{dagger}
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\desc{Dagger}{}{}
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\desc[german]{Dagger}{}{}
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\eq{
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\hat{A}^\dagger &= (\hat{A}^*)^\mathrm{T} \\
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(c \hat{A})^\dagger &= c^* \hat{A}^\dagger \\
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(\hat{A}\hat{B})^\dagger &= \hat{B}^\dagger \hat{A}^\dagger \\
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}
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\end{formula}
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\begin{formula}{adjoint_op}
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\desc{Adjoint operator}{}{}
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\desc[german]{Adjungierter operator}{}{}
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\eq{\braket{\alpha|\hat{A}^\dagger|\beta} = \braket{\beta|\hat{A}|\alpha}^*}
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\end{formula}
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\begin{formula}{hermitian_op}
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\desc{Hermitian operator}{}{}
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\desc[german]{Hermitescher operator}{}{}
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\eq{\hat{A} = \hat{A}^\dagger}
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\end{formula}
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\Subsection[
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\ger{Wahrscheinlichkeitstheorie}
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\eng{Probability theory}
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]{probability}
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\begin{formula}{conservation_of_probability}
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\desc{Continuity equation}{}{$\rho$ density of a conserved quantity $q$, $j$ flux density of $q$}
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\desc[german]{Kontinuitätsgleichung}{}{$\rho$ Dichte einer Erhaltungsgröße $q$, $j$ Fluß von $q$}
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\eq{\frac{\partial\rho(\vec{x}, t)}{\partial t} + \nabla \cdot \vec{j}(\vec{x},t) = 0}
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\end{formula}
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\begin{formula}{state_probability}
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\desc{State probability}{}{}
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\desc[german]{Zustandswahrscheinlichkeit}{}{}
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\eq{TODO}
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\end{formula}
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\begin{formula}{dispersion}
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\desc{Dispersion}{}{}
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\desc[german]{Dispersion}{}{}
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\eq{\Delta \hat{A} = \hat{A} - \braket{\hat{A}}}
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\end{formula}
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\begin{formula}{variance}
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\desc{Variance}{}{}
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\desc[german]{Varianz}{}{}
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\eq{\sigma^2 = \braket{(\Delta \hat{A})^2} = \braket{\hat{A}^2} - \braket{\hat{A}}^2}
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\end{formula}
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\begin{formula}{generalized_uncertainty}
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\desc{Generalized uncertainty principle}{}{}
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\desc[german]{Allgemeine Unschärferelation}{}{}
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% \eq{\braket{(\Delta \hat{A})^2} \braket{(\Delta \hat{B})^2} \ge \frac{1}{4} \braket{[\hat{A},\hat{B}]}^2}
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\eq{
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\sigma_A \sigma_B &\ge \frac{1}{4} \braket{[\hat{A},\hat{B}]}^2 \\
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\sigma_A \sigma_B &\ge \frac{1}{2} \abs{\braket{[\hat{A},\hat{B}]}}
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}
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\end{formula}
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\Subsubsection[
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\eng{Pauli matrices}
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\ger{Pauli-Matrizen}
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]{pauli_matrices}
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\begin{formula}{pauli_matrices}
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\desc{Pauli matrices}{}{}
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\desc[german]{Pauli Matrizen}{}{}
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\eqAlignedAt{2}{
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\sigma_x &= \sigmaxmatrix &&= \sigmaxbraket \label{eq:pauli_x} \\
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\sigma_y &= \sigmaymatrix &&= \sigmaybraket \label{eq:pauli_y} \\
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\sigma_z &= \sigmazmatrix &&= \sigmazbraket \label{eq:pauli_z}
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}
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\end{formula}
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% $\sigma_x$ NOT
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% $\sigma_y$ PHASE
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% $\sigma_z$ Sign
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\Subsection[
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\eng{Commutator}
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\ger{Kommutator}
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]{commutator}
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\begin{formula}{commutator}
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\desc{Commutator}{}{}
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\desc[german]{Kommutator}{}{}
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\eq{[a,b] = ab - ba}
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\end{formula}
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\begin{formula}{anticommutator}
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\desc{Anticommutator}{}{}
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\desc[german]{Antikommmutator}{}{}
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\eq{\{a,b\} = ab + ba}
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\end{formula}
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\begin{formula}{commutation_relations}\
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\desc{Commutation relations}{}{}
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\desc[german]{Kommutatorrelationen}{}{}
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\eq{[a, bc] = \{a, b\}c - b\{a,c\}}
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\end{formula}
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\begin{formula}{jacobi_identity}
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\desc{Jacobi identity}{}{}
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\desc[german]{Jakobi-Identität}{}{}
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\eq{[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0}
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\end{formula}
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\begin{formula}{canon_comm_relation}
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\desc{Canonical commutation relation}{}{$x$, $p$ canonical conjugates}
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\desc[german]{Kanonische Vertauschungsrelationen}{}{$x$, $p$ kanonische konjugierte}
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\eq{
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[x_i, x_j] &= 0 \\
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[p_i, p_j] &= 0 \\
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[x_i, p_j] &= i \hbar \delta_{ij}
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}
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\end{formula}
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\Subsection[
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\eng{Schrödinger equation}
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\ger{Schrödingergleichung}
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]{schroedinger_equation}
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\begin{formula}{energy_operator}
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\desc{Energy operator}{}{}
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\desc[german]{Energieoperator}{}{}
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\eq{E = i\hbar \frac{\partial}{\partial t}}
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\end{formula}
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\begin{formula}{momentum_operator}
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\desc{Momentum operator}{}{}
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\desc[german]{Impulsoperator}{}{}
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\eq{\vec{p} = -i\hbar \vec{\nabla_x}}
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\end{formula}
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\begin{formula}{space_operator}
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\desc{Space operator}{}{}
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\desc[german]{Ortsoperator}{}{}
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\eq{\vec{x} = i\hbar \vec{\nabla_p}}
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\end{formula}
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\begin{formula}{stationary_schroedinger_equation}
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\desc{Stationary Schrödingerequation}{}{}
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\desc[german]{Stationäre Schrödingergleichung}{}{}
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\eq{\hat{H}\ket{\psi} = E\ket{\psi}}
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\end{formula}
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\begin{formula}{schroedinger_equation}
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\desc{Schrödinger equation}{}{}
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\desc[german]{Schrödingergleichung}{}{}
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\eq{i\hbar\frac{\partial}{\partial t}\psi(x, t) = (- \frac{\hbar^2}{2m} \vec{\nabla}^2 + \vec{V}(x)) \psi(x)}
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\end{formula}
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The time evolution of the Hamiltonian is given by:
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\begin{formula}{time_evolution_op}
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\desc{Time evolution operator}{}{$U$ unitary}
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\desc[german]{Zeitentwicklungsoperator}{}{$U$ unitär}
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\eq{\ket{\psi(t)} = \hat{U}(t, t_0) \ket{\psi(t_0)}}
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\end{formula}
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\Subsubsection[
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\eng{Schrödinger- and Heisenberg-pictures}
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\ger{Schrödinger- und Heisenberg-Bild}
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]{s_h_pictures}
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\eng[s_h_pictures_desc]{
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In the \textbf{Schrödinger picture}, the time dependecy is in the states
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while in the \textbf{Heisenberg picture} the observables (operators) are time dependent.
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}
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\ger[s_h_pictures_desc]{Im Schrödinger-Bild sind die Zustände zeitabhänig, im Heisenberg-Bild
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sind die Observablen (Operatoren) zeitabhänig
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}
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\gt{s_h_pictures_desc}\\
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\begin{formula}{schroediner_time_evolution}
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\desc{Schrödinger time evolution}{}{}
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\desc[german]{Schrödinger Zeitentwicklug}{}{}
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\eq{
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\ket{\psi(t)_\textrm{S}} = \hat{U}(t,t_0)\ket{\psi(t_0)}
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}
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\end{formula}
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\begin{formula}{heisenberg_time_evolution}
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\desc{Heisenberg time evolution}{}{\textrm{H} and \textrm{S} being the Heisenberg and Schrödinger picture, respectively}
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\desc[german]{Heisenberg Zeitentwicklung}{}{mit \textrm{H} und \textrm{S} dem Heisenberg- und Schrödinger-Bild}
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\eq{
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\ket{\psi_\mathrm{H}} = \ket{\psi_\mathrm{S}(t_0)} \\
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A_\textrm{H} = U^\dagger(t,t_0)A_\textrm{S}U(t,t_0) \\
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\diff{\hat{A}_\textrm{H}}{t} = \frac{1}{i\hbar}[\hat{A}_\textrm{H}, \hat{H}_\textrm{H}] + \Big(\diffp{\hat{A}_\textrm{S}}{t}\Big)_\textrm{H}
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}
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\end{formula}
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\Subsubsection[
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\ger{Korrespondenzprinzip}
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\eng{Correspondence principle}
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]{correspondence_principle}
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\begin{ttext}{desc}
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\ger{Die klassischen Bewegungsgleichungen lassen sich als Grenzfall (große Quantenzahlen) aus der Quantenmechanik ableiten.}
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\eng{The classical mechanics can be derived from quantum mechanics in the limit of large quantum numbers.}
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\end{ttext}
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\Subsubsection[
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\eng{Ehrenfest theorem}
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\ger{Ehrenfest-Theorem}
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]{ehrenfest_theorem}
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\GT{see_also} \ref{sec:qm:basics:schroedinger_equation:correspondence_principle}
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\begin{formula}{ehrenfest_theorem}
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\desc{Ehrenfesttheorem}{applies to both pictures}{}
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\desc[german]{Ehrenfest-Theorem}{gilt für beide Bilder}{}
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\eq{
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\diff{}{t} \braket{\hat{A}} = \frac{1}{i\hbar}\braket{[\hat{A},\hat{H}]} + \Braket{\diffp{\hat{A}}{t}}
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}
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\end{formula}
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\begin{formula}{ehrenfest_theorem_x}
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\desc{}{Example for $x$}{}
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\desc[german]{}{Beispiel für $x$}{}
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\eq{m\diff[2]{}{t}\braket{x} = -\braket{\nabla V(x)} = \braket{F(x)}}
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\end{formula}
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% \eq{Time evolution}{\hat{H}\ket{\psi} = E\ket{\psi}}{sg_time}
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\Section[
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\eng{Pertubation theory}
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\ger{Störungstheorie}
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]{qm_pertubation}
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\eng[desc]{The following holds true if the pertubation $\hat{H_1}$ is sufficently small and the $E^{(0)}_n$ levels are not degenerate.}
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\ger[desc]{Die folgenden Gleichungen gelten wenn $\hat{H_1}$ ausreichend klein ist und die $E_n^{(0)}$ Niveaus nicht entartet sind.}
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\gt{desc}
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\begin{formula}{pertubation_hamiltonian}
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\desc{Hamiltonian}{}{}
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\desc[german]{Hamiltonian}{}{}
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\eq{\hat{H} = \hat{H_0} + \lambda \hat{H_1}}
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\end{formula}
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\begin{formula}{pertubation_series}
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\desc{Power series}{}{}
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\desc[german]{Potenzreihe}{}{}
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\eq{
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E_n &= E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + ... \\
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\ket{\psi_n} &= \ket{\psi_n^{(0)}} + \lambda \ket{\psi_n^{(1)}} + \lambda^2 \ket{\psi_n^{(2)}} + ...
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}
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\end{formula}
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\begin{formula}{1o_energy}
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\desc{1. order energy shift}{}{}
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\desc[german]{Energieverschiebung 1. Ordnung}{}{}
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\eq{E_n^{(1)} = \Braket{\psi_n^{(0)}|\hat{H_1}|\psi_n^{(0)}}}
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\end{formula}
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\begin{formula}{1o_state}
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\desc{1. order states}{}{}
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\desc[german]{Zustände}{}{}
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\eq{\ket{\psi_n^{(1)}} = \sum_{k\neq n}\frac{\Braket{\psi_k^{(0)}|\hat{H_1}|\psi_n^{(0)}}}{E_n^{(0)} - E_k^{(0)}}\ket{\psi_k^{(0)}}}
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\end{formula}
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\begin{formula}{2o_energy}
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\desc{2. order energy shift}{}{}
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\desc[german]{Energieverschiebung 2. Ordnung}{}{}
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% \eq{E_n^{(1)} = \Braket{\psi_n^{(0)}|\hat{H_1}|\psi_n^{(0)}}}
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\eq{E_n^{(2)} = \sum_{k\neq n}\frac{\abs*{\Braket{\psi_k^{(0)}|\hat{H_1}|\psi_n^{(0)}}}^2}{E_n^{(0)} - E_k^{(0)}}}
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\end{formula}
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% \begin{formula}{qm:pertubation:}
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% \desc{1. order states}{}{}
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% \desc[german]{Zustände}{}{}
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% \eq{\ket{\psi_n^{(1)}} = \sum_{k\neq n}\frac{\Braket{\psi_k^{(0)}|\hat{H_1}|\psi_n^{(0)}}}{E_n^{(0)} - E_k^{(0)}}\ket{\psi_k^{(0)}}}
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% \end{formula}
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\Section[
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\eng{Harmonic oscillator}
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\ger{Harmonischer Oszillator}
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]{qm_hosc}
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\begin{formula}{hamiltonian}
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\desc{Hamiltonian}{}{}
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\desc[german]{Hamiltonian}{}{}
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\eq{
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H&=\frac{p^2}{2m}+\frac{1}{2} m\omega^2 x^2\\
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&=\frac{1}{2} \hbar\omega+\omega a^\dagger a
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}
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\end{formula}
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\begin{formula}{hosc_spectrum}
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\desc{Energy spectrum}{}{}
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\desc[german]{Energiespektrum}{}{}
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\eq{E_n = \hbar\omega \Big(\frac{1}{2} + n\Big)}
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\end{formula}
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\GT{see_also} \ref{sec:qm:hosc:c_a_ops}
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\Subsection[
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\ger{Erzeugungs und Vernichtungsoperatoren}
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\eng{Creation and Annihilation operators}
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]{c_a_ops}
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\begin{formula}{c_a_ops_def}
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\desc{Particle number operator/occupation number operator}{}{$\ket{n}$ = Fock states, $\hat{a}$ = Annihilation operator, $\hat{a}^\dagger$ = Creation operator}
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\desc[german]{Teilchenzahloperator/Besetzungszahloperator}{}{$\ket{n}$ = Fock-Zustände, $\hat{a}$ = Vernichtungsoperator, $\hat{a}^\dagger$ = Erzeugungsoperator}
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\eq{
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\hat{N} &:= a^\dagger a \\
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\hat{N}\ket{n} &= n \ket{N}
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}
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\end{formula}
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\begin{formula}{c_a_commutator}
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\desc{Commutator}{}{}
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\desc[german]{Kommutator}{}{}
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\eq{
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[\hat{a},\hat{a}^\dagger] &= 1 \\
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[N, \hat{a}] &= -\hat{a} \\
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[N, \hat{a}^\dagger] &= \hat{a}^\dagger
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}
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\end{formula}
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\begin{formula}{c_a_on_state}
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\desc{Application on states}{}{}
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\desc[german]{Anwendung auf Zustände}{}{}
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\eq{
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\hat{a} \ket{n} &= \sqrt{n}\ket{n-1} \\
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\hat{a}^\dagger \ket{n} &= \sqrt{n+1}\ket{n+1} \\
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\ket{n} &= \frac{1}{\sqrt{n!}} (\hat{a}^\dagger)^n \ket{0}
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}
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\end{formula}
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\Subsubsection[
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\eng{Harmonischer Oszillator}
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\ger{Harmonic Oscillator}
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]{hosc}
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\begin{formula}{c_a_ops}
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\desc{Harmonic oscillator}{}{}
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\desc[german]{Harmonischer Oszillator}{}{}
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\eq{
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% \tilde{X} &= \sqrt{\frac{m\omega}{\hbar}} \hat{x} &= \frac{1}{\sqrt{2}} (\hat{a} + \hat{a}^\dagger) \\
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% \tilde{P} &= \frac{1}{\sqrt{m\omega\hbar}} \hat{p} &= \frac{-i}{\sqrt{2}} (\hat{a} - \hat{a}^\dagger) \\
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\hat{x} &= \sqrt{\frac{\hbar}{2m\omega}} (\hat{a} + \hat{a}^\dagger) \\
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\hat{p} &= -i\sqrt{\frac{m\omega\hbar}{2}} (\hat{a} - \hat{a}^\dagger) \\
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\hat{H} &= \frac{\hat{p}^2}{2m} + \frac{m\omega^2 \hat{x}^2}{2} &= \hbar\omega\Big(a^\dagger a + \frac{1}{2}\Big) \\
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a &= \frac{1}{\sqrt{2}} (\tilde{X} + i\tilde{P}) \\
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a^\dagger &= \frac{1}{\sqrt{2}} (\tilde{X} - i\tilde{P})
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% \hat{a}^\dagger ? \sqrt{\frac{}{}}
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}
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\end{formula}
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% \eq{Heisenberg}{\frac{dA}{dt}=\frac{\partial A}{\partial t}+\frac{[A,H]}{i\hbar}}{heisenberg}
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% \begin{align}
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% \label{eq:k}
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% A=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x+\frac{ip}{\sqrt{2m\omega}} \\
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% A^\dagger=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x-\frac{ip}{\sqrt{2m\omega}} \\
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% HAu_E=(E-\hbar\omega)Au_E \\
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% u_n=\frac{1}{\sqrt{n!}}\left(\frac{A^\dagger}{\sqrt{\hbar}}\right)^nu_0 \\
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% u_0=\sqrt[4]{\frac{m\omega}{\pi\hbar}}\exp\left(-\frac{m\omega x^2}{2\hbar}\right) \\
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% E_n=( \frac{1}{2} +n)\hbar\omega
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% \end{equation}
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\Section{angular_momentum}
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\times
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\begin{formula}{bloch_waves}
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\desc{Bloch waves}{
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Solve the stat. SG in periodic potential with period
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$\vec{R}$: $V(\vec{r}) = V(\vec{r} + \vec{R})$\\
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}{
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$\vec{k}$ arbitrary, $u$ periodic function
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}
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\desc[german]{Blochwellen}{
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Lösen stat. SG im periodischen Potential mit Periode
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$\vec{R}$: $V(\vec{r}) = V(\vec{r} + \vec{R})$\\
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}{
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$\vec{k}$ beliebig, $u$ periodische Funktion
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}
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\eq{\psi_k(\vec{r}) = e^{i \vec{k}\cdot \vec{r}} \cdot u_{\vec{k}}(\vec{r})}
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\end{formula}
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