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- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/04%3A_Bra-ket_Formalism/4.08%3A_Exercise_ProblemsIn a certain basis, the Hamiltonian of a two-level system is described by the matrix \[\mathrm{H}=\left(\begin{array}{cc} E_{1} & 0 \\ 0 & E_{2} \end{array}\right), \quad \text { with } E_{1} \neq E_{...In a certain basis, the Hamiltonian of a two-level system is described by the matrix \[\mathrm{H}=\left(\begin{array}{cc} E_{1} & 0 \\ 0 & E_{2} \end{array}\right), \quad \text { with } E_{1} \neq E_{2},\] while the operator of some observable \(A\) of this system, by the matrix \[\mathrm{A}=\left(\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right) \text {. }\] For the system’s state with the energy definitely equal to \(E_{1}\), find the possible results of measurements of the observable \(A\)…
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/02%3A_1D_Wave_Mechanics/2.05%3A_Motion_in_Soft_PotentialsAs a result, we get the following (still approximate!) result:\[\begin{gathered} \frac{d \Phi_{1}}{d x}=\frac{i}{2} \frac{d^{2} \Phi_{0}}{d x^{2}} / \frac{d \Phi_{0}}{d x}=\frac{i}{2} \frac{d}{d x}\le...As a result, we get the following (still approximate!) result:\[\begin{gathered} \frac{d \Phi_{1}}{d x}=\frac{i}{2} \frac{d^{2} \Phi_{0}}{d x^{2}} / \frac{d \Phi_{0}}{d x}=\frac{i}{2} \frac{d}{d x}\left(\ln \frac{d \Phi_{0}}{d x}\right)=\frac{i}{2} \frac{d}{d x}[\ln k(x)]=i \frac{d}{d x}\left[\ln k^{1 / 2}(x)\right], \\ \left.i \Phi\right|_{\mathrm{WKB}} \equiv i \Phi_{0}+i \Phi_{1}=\pm i \int^{x} k\left(x^{\prime}\right) d x^{\prime}+\ln \frac{1}{k^{1 / 2}(x)}, \\ \psi_{\mathrm{WKB}}(x)=\frac{…
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/03%3A_Higher_Dimensionality_Effects/3.04%3A_Energy_Bands_in_Higher_DimensionsThe key notion of the band theory in \(d\) dimensions is the reciprocal lattice in the wave-vector (q) space, formed as \[\mathbf{Q}=\sum_{j=1}^{d} l_{j} \mathbf{b}_{j},\] with integer \(l_{j}\), and ...The key notion of the band theory in \(d\) dimensions is the reciprocal lattice in the wave-vector (q) space, formed as \[\mathbf{Q}=\sum_{j=1}^{d} l_{j} \mathbf{b}_{j},\] with integer \(l_{j}\), and vectors \(\mathbf{b}_{j}\) selected in such a way that the following natural generalization of Eq. (104) is valid for any pair of points of the direct and reciprocal lattices: \[e^{i \mathbf{Q} \cdot \mathbf{R}}=1 .\] One way to describe the physical sense of the lattice \(\mathbf{Q}\) is to say th…
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/02%3A_1D_Wave_Mechanics/2.10%3A_Harmonic_Oscillator-_Brute_Force_ApproachNow let us use Eq. (1.23) for a similar calculation of the expectation value of the system’s Hamiltonian in the trial state: \({ }^{87}\) \[\begin{aligned} \langle H\rangle_{\text {trial }} &=\int \ps...Now let us use Eq. (1.23) for a similar calculation of the expectation value of the system’s Hamiltonian in the trial state: \({ }^{87}\) \[\begin{aligned} \langle H\rangle_{\text {trial }} &=\int \psi_{\text {trial }}^{*} \hat{H} \psi_{\text {trial }} d^{3} x \equiv \sum_{n, n^{\prime}} \int \alpha_{n}^{*} \psi_{n}^{*} \hat{H} \alpha_{n^{\prime}} \psi_{n^{\prime}} d^{3} x \equiv \sum_{n, n^{\prime}} \alpha_{n^{\prime}} \alpha_{n}^{*} E_{n^{\prime}} \int \psi_{n}^{*} \psi_{n^{\prime}} d^{3} x \…
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/09%3A_Elements_of_Relativistic_Quantum_Mechanics/9.06%3A_Diracs_TheoryThe particular form of the \(2 \times 2\) matrices corresponding to the operators \(\hat{\boldsymbol{\sigma}}\) and \(\hat{I}\) in Eq. (98a) depends on the basis selected for the spin state representa...The particular form of the \(2 \times 2\) matrices corresponding to the operators \(\hat{\boldsymbol{\sigma}}\) and \(\hat{I}\) in Eq. (98a) depends on the basis selected for the spin state representation; for example, in the standard \(z\)-basis, in which the Cartesian components of \(\hat{\boldsymbol{\sigma}}\) are represented by the Pauli matrices (4.105), the \(4 \times 4\) matrix form of Eq. (98a) is \[\alpha_{x}=\left(\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ …
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/03%3A_Higher_Dimensionality_Effects/3.07%3A_AtomsBecause of this reason, \(n\) is usually called the principal quantum number, and the above relation between it and the "more subordinate" orbital quantum number \(l\) is rewritten as \[l \leq n-1 .\]...Because of this reason, \(n\) is usually called the principal quantum number, and the above relation between it and the "more subordinate" orbital quantum number \(l\) is rewritten as \[l \leq n-1 .\] Together with the inequality (162), this gives us the following, very important hierarchy of the three quantum numbers involved in the Bohr atom problem: \[1 \leq n \leq \infty \quad \Rightarrow \quad 0 \leq l \leq n-1 \quad \Rightarrow \quad-l \leq m \leq+l\] Taking into account the \((2 l+1)\)-d…
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/05%3A_Some_Exactly_Solvable_Problems/5.04%3A_Revisiting_Harmonic_OscillatorTaking all phases \(\varphi_{n}\) and \(\varphi_{n}\) ’ equal to zero for simplicity, we may spell out Eqs. (82) as \(^{32}\) \[\hat{a}^{\dagger}|n\rangle=(n+1)^{1 / 2}|n+1\rangle, \quad \hat{a}|n\ran...Taking all phases \(\varphi_{n}\) and \(\varphi_{n}\) ’ equal to zero for simplicity, we may spell out Eqs. (82) as \(^{32}\) \[\hat{a}^{\dagger}|n\rangle=(n+1)^{1 / 2}|n+1\rangle, \quad \hat{a}|n\rangle=n^{1 / 2}|n-1\rangle .\] Now we can use these formulas to calculate, for example, the matrix elements of the operator \(\hat{x}\) in the Fock state basis: \[\begin{aligned} \left\langle n^{\prime}|\hat{x}| n\right\rangle & \equiv x_{0}\left\langle n^{\prime}|\hat{\xi}| n\right\rangle=\frac{x_{0…
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/01%3A_Introduction/1.06%3A_Time_Evolution7, in which externally applied voltage \(V\) fixes the difference \(\phi_{1}-\phi_{2}\) between the electrochemical potentials of two superconductors, Eq. (72) takes the form \[\frac{d \varphi}{d t}=\...7, in which externally applied voltage \(V\) fixes the difference \(\phi_{1}-\phi_{2}\) between the electrochemical potentials of two superconductors, Eq. (72) takes the form \[\frac{d \varphi}{d t}=\frac{2 e}{\hbar} V .\] If the link between the superconductors is weak enough, the electric current \(I\) of the Cooper pairs (called the supercurrent) through the link may be approximately described by the following simple relation, \[I=I_{\mathrm{c}} \sin \varphi,\] where \(I_{\mathrm{c}}\) is so…
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/zz%3A_Back_Matter
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/05%3A_Some_Exactly_Solvable_Problems/5.02%3A_The_Ehrenfest_TheoremTo derive them, for the simplest case of 1D orbital motion, let us calculate the following commutator: \[\left[\hat{x}, \hat{p}_{x}^{2}\right] \equiv \hat{x} \hat{p}_{x} \hat{p}_{x}-\hat{p}_{x} \hat{p...To derive them, for the simplest case of 1D orbital motion, let us calculate the following commutator: \[\left[\hat{x}, \hat{p}_{x}^{2}\right] \equiv \hat{x} \hat{p}_{x} \hat{p}_{x}-\hat{p}_{x} \hat{p}_{x} \hat{x} \text {. }\] Let us apply the commutation relation (4.238) in the following form: \[\hat{x} \hat{p}_{x}=\hat{p}_{x} \hat{x}+i \hbar \hat{I},\] to the first term of the right-hand side of Eq. (24) twice, with the goal to move the coordinate operator to the rightmost position: \[\hat{x}…
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/03%3A_Higher_Dimensionality_Effects/3.02%3A_Landau_Levels_and_the_Quantum_Hall_EffectNow, expanding the eigenfunction into the Fourier integral in the \(y\)-direction: \[\psi(x, y)=\int X_{k}(x) \exp \left\{i k\left(y-y_{0}\right)\right\} d k,\] we see that for each component of this ...Now, expanding the eigenfunction into the Fourier integral in the \(y\)-direction: \[\psi(x, y)=\int X_{k}(x) \exp \left\{i k\left(y-y_{0}\right)\right\} d k,\] we see that for each component of this integral, Eq. (41) yields a specific equation \[-\frac{\hbar^{2}}{2 m}\left\{\mathbf{n}_{x} \frac{d}{d x}+i \mathbf{n}_{y}\left[k-\frac{q}{\hbar} \mathscr{B}\left(x-x_{0}\right)\right]\right\}^{2} X_{k}=E X_{k} .\] Since the two vectors inside the curly brackets are mutually perpendicular, its squa…
