1.1 Consider a traveling sinusoidal wave of the form, E, =E,cos(wt-kz +4,). Show that this wave satisfies the Maxwell's wave equation.
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- Problem 2.13 A particle in the harmonic oscillator potential starts out in the state ¥ (x. 0) = A[3¥o(x)+ 4¼1(x)]. (a) Find A. (b) Construct ¥ (x, t) and |¥(x. t)P. (c) Find (x) and (p). Don't get too excited if they oscillate at the classical frequency; what would it have been had I specified ¥2(x), instead of Vi(x)? Check that Ehrenfest's theorem (Equation 1.38) holds for this wave function. (d) If you measured the energy of this particle, what values might you get, and with what probabilities?Consider a transverse periodic (sinusoidal) wave passing through a very long string of mass density 0.250 kg/m. The wave function for this wave is found to be: y (x,t) = (0.125 m) cos [(1.10 rad/m) x - (15.0 rad/s) t] From the equation find the following quantites; 1. Wave amplitude 2. Wave number 3. Angular frequency 4. Wavelength 5. Period 6. FrequencyThe Asymmetric Planar Waveguide. Examine the TE field in an asymmetric planar wave- guide consisting of a dielectric slab of width d and refractive index n1 placed on a substrate of lower refractive index n2 and covered with a medium of refractive index ng < n2 < n1, as illustrated in Fig. 9.2-10. (a) Determine an expression for the maximum inclination angle 0 of plane waves undergoing total internal reflection, and the corresponding numerical aperture NA of the waveguide. (b) Write an expression for the self-consistency condition, similar to (9.2-4). (c) Determine an approximate expression for the number of modes M (valid when M is very large). Figure 9.2-10 Asymmetric planar waveguide. n3 d n1
- Prove that the spherical wave given by: ?(?,?) = (A ei k (r-vt))/r is indeed a solution of the three-dimensional wave equation Hint: Write the wave equation in spherical coordinates (therefore, you should use the Laplacian written in terms of r, ?, ?, and insert the solution given above into the wave equation.NoneExplain the answer to the second and last equation in detail
- Please help with this QsA number N of plane waves are travelling parallel; the waves share a common wavevector k and common direction of electric field vector along unit vector û, but each wave has its own distinct amplitude Eo and phase difference &;. Therefore the ith wave (for 1 ≤ i ≤ N) has electric field given by E₁ = Eoiû Re [exp j(k-r - wt+d;)]. From this, prove that the total intensity Inet including interference is given by Ec|² 27⁰ where the complex amplitude E, is defined by Inet and 70 is the impedance of free space. N Ec = Eoi exp(joi), 7 i=11.3. Determine an orthogonal basis for the subspace of C (-1, 1] spanned by functions: {f(x) = x, f(x) = x³, f(x) = x³] using Gram-Schmidt process.