Show that the unitary basis transformation ã₁ = Σ(μ|v), leaves the total particle number N = Σaa unchanged.
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- Consider a non-relativistic particle moving in a potential U(r). Can either the phase or the group velocity of the particle exceed the speed of light? What happens to the phase velocity vp at the turning point in the potential U(x), where the particle gets reflected and the group velocity vanishes, v - 0?In spherical polar coordinates, the angular momentum operators, 1² and Îz, can be written, Ә β = -ħ² 1 ə sin Ꮎ ᎧᎾ 1 2² sin² 0 00² дф él sin)+ + and (a) îψ(θ, φ) = λιψ(θ, φ) = Î₂ = = -ih Apply these operators to the unnormalized eigenfunction, (0, 4) = sin²0 e-²ip, and determine the eigenvalues of that are associated with each operator. (b) Î₂¥(0, 0) = 1₂4(0,4)Consider the real part of the spherical harmonic Y1, + 1. At which values of do angular nodes occur?
- The difference of the scalar potential squared and the modulus of the vector potential squared, Φ2 - |A|2, is Lorentz invariant (a Lorentz scalar). why the statement true?Please obtain the same result as in the book.Consider Euclidean space in D = 2. Show by direct computation that the Riemann curvature tensor, vanishes in: rest on image