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4.8: Exercise Problems
https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/04%3A_Bra-ket_Formalism/4.08%3A_Exercise_Problems
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_{...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 $A$ …
8.2: Singlets, Triplets, and the Exchange Interaction
https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/08%3A_Multiparticle_Systems/8.02%3A_Singlets_Triplets_and_the_Exchange_Interaction
Indeed, in the $1^{\text {st }}$ approximation of the perturbation theory, the total energy $E_{\mathrm{e}}$ of the system may be expressed as \(\varepsilon_{100}+\varepsilon_{n l m}+E_{\text {int...Indeed, in the $1^{\text {st }}$ approximation of the perturbation theory, the total energy $E_{\mathrm{e}}$ of the system may be expressed as $\varepsilon_{100}+\varepsilon_{n l m}+E_{\text {int }}{ }^{(1)}$ , with \[E_{\text {int }}^{(1)}=\left\langle U_{\text {int }}\right\rangle=\int d^{3} r_{1} \int d^{3} r_{2} \psi_{\mathrm{e}}^{*}\left(\mathbf{r}_{1}, \mathbf{r}_{2}\right) U_{\text {int }}\left(\mathbf{r}_{1}, \mathbf{r}_{2}\right) \psi_{\mathrm{e}}\left(\mathbf{r}_{1}, \mathbf{r}_{…
2.5: Motion in Soft Potentials
https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/02%3A_1D_Wave_Mechanics/2.05%3A_Motion_in_Soft_Potentials
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}\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{…
3.4: Energy Bands in Higher Dimensions
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_Dimensions
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 ...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…
2.10: Harmonic Oscillator- Brute Force Approach
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_Approach
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 \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 \…
9.6: Dirac’s Theory
https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/09%3A_Elements_of_Relativistic_Quantum_Mechanics/9.06%3A_Diracs_Theory
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 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 \\ …
3.7: Atoms
https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/03%3A_Higher_Dimensionality_Effects/3.07%3A_Atoms
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 $l \leq n-1 .$ ...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…
5.4: Revisiting Harmonic Oscillator
https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/05%3A_Some_Exactly_Solvable_Problems/5.04%3A_Revisiting_Harmonic_Oscillator
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\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…
1.6: Time Evolution
https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/01%3A_Introduction/1.06%3A_Time_Evolution
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}=\...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…
Back Matter
https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/zz%3A_Back_Matter
5.2: The Ehrenfest Theorem
https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_(Likharev)/05%3A_Some_Exactly_Solvable_Problems/5.02%3A_The_Ehrenfest_Theorem
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...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}…

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