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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_Problems
      In a certain basis, the Hamiltonian of a two-level system is described by the matrix \[\mathrm{H}=\left(E100E2\right), \quad \text { with } E_{1} \neq E_{...In a certain basis, the Hamiltonian of a two-level system is described by the matrix H=(E100E2), with E1E2, while the operator of some observable A of this system, by the matrix A=(1111) For the system’s state with the energy definitely equal to E1, find the possible results of measurements of the observable A
    • https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_(Likharev)/06%3A_Electromagnetism
      This chapter discusses two major effects that arise when electric and magnetic fields are changing in time: the “electromagnetic induction” of an additional electric field by changing magnetic field, ...This chapter discusses two major effects that arise when electric and magnetic fields are changing in time: the “electromagnetic induction” of an additional electric field by changing magnetic field, and the reciprocal effect of the “displacement currents”- actually, the induction of an additional magnetic field by changing electric field.
    • https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Essential_Graduate_Physics_-_Classical_Mechanics_(Likharev)/01%3A_Review_of_Fundamentals/1.02%3A_Kinematics-_Basic_Notions
      If the frames move versus each other by translation only (no mutual rotation!), similar relations are valid for the velocity and the acceleration as well: \[\begin{aligned} &\left.\mathbf{v}\right|_{\...If the frames move versus each other by translation only (no mutual rotation!), similar relations are valid for the velocity and the acceleration as well: v|in 0=v|in 0+v0|in 0a|in 0=a|in 0+a0|in 0 Note that in the case…
    • https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Essential_Graduate_Physics_-_Classical_Mechanics_(Likharev)/07%3A_Deformations_and_Elasticity/7.05%3A_Rod_Bending
      Now proceeding in the same fashion to Eq. (76), we get φy=ρgA2EIy(zl)33+ const =ρgA6EIy[(zl)3+l3], where th...Now proceeding in the same fashion to Eq. (76), we get φy=ρgA2EIy(zl)33+ const =ρgA6EIy[(zl)3+l3], where the integration constant is selected to satisfy the clamping condition at the left end of the rod: φy =0 at z=0. (Note that this is different from the support condition illustrated on the lower panel of Figure 9 b, which allows the angle at z=0 to be different f…
    • https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Essential_Graduate_Physics_-_Statistical_Mechanics_(Likharev)/05%3A_Fluctuations/5.07%3A_The_Fokker-Planck_equation
      where jq (which was called jw in the last section) is the probability current density in the coordinate space, and q (which was denoted as in that sec...where jq (which was called jw in the last section) is the probability current density in the coordinate space, and q (which was denoted as in that section) is the usual vector operator in the space, while jp is the current density in the momentum space, and p is the similar vector operator in that space:
    • https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_(Likharev)/02%3A_Charges_and_Conductors/2.09%3A_Variable_Separation__Polar_Coordinates
      15a  (E0>0), the surface charge is positive on the right-hand side of the cylinder and negative on its left-hand side, thus creating a field directed from the right to the left, whi...15a  (E0>0), the surface charge is positive on the right-hand side of the cylinder and negative on its left-hand side, thus creating a field directed from the right to the left, which exactly compensates the external field inside the conductor, where the net field is zero. (Please take one more look at the schematic Fig.
    • https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Essential_Graduate_Physics_-_Classical_Mechanics_(Likharev)/02%3A_Lagrangian_Analytical_Mechanics/2.05%3A_Exercise_Problems
      (i) At not very high frequencies (whose quantum ω is lower than the binding energy 2Δ of the Cooper pairs), the Josephson effect in a sufficiently small junction may be descri...(i) At not very high frequencies (whose quantum ω is lower than the binding energy 2Δ of the Cooper pairs), the Josephson effect in a sufficiently small junction may be described by the following coupling energy: U(φ)=EJcosφ+ const  where the constant EJ describes the coupling strength, while the variable φ (called the Josephson phase difference) is connected to the voltage V across the junction by the famous …
    • 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 1st  approximation of the perturbation theory, the total energy Ee of the system may be expressed as \(\varepsilon_{100}+\varepsilon_{n l m}+E_{\text {int...Indeed, in the 1st  approximation of the perturbation theory, the total energy Ee of the system may be expressed as ε100+εnlm+Eint (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}_{…
    • https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_(Likharev)/02%3A_Charges_and_Conductors/2.03%3A_Exercise_Problems
      Calculate the mutual capacitance between the terminals of the semi-infinite lumped-capacitor circuit shown in the figure on the right, and the law of decay of the applied voltage along the system. Use...Calculate the mutual capacitance between the terminals of the semi-infinite lumped-capacitor circuit shown in the figure on the right, and the law of decay of the applied voltage along the system. Use the method of images to find the Green’s function of the system shown in the figure on the right, where the bulge on the conducting plane has the shape of a semi-sphere of radius  R.
    • 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{…
    • 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 Q=dj=1ljbj, with integer lj, and ...The key notion of the band theory in d dimensions is the reciprocal lattice in the wave-vector (q) space, formed as Q=dj=1ljbj, with integer lj, and vectors bj 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: eiQR=1. One way to describe the physical sense of the lattice Q is to say th…

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