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- A real wave function is defined on the half-axis: [0≤x≤00) as y(x) = A(x/xo)e-x/xo where xo is a given constant with the dimension of length. a) Plot this function in the dimensionless variables and find the constant A. b) Present the normalized wave function in the dimensional variables. Hint: introduce the dimensionless variables = x/xo and Y(5) = Y(5)/A.The amplitude of a scattered wave is given by 1 S(0) = (21 + 1)exp[id] sin 3i P(cos 0), l=0 where e is the angle of scattering, I is the angular momentum, ik is the incident momentum, and & is the phase shift produced by the central potential that is doing the scattering. The total cross section is oiot = S IS(O)²a2. Show that 47 Otot = FL(21+ 1)sin² § . %3D0The normalised wavefunction representing a particle enclosed in a box is given by: а. 2 sin-e - iot y (x,t) =, a a b. 2 IX - iot y (x,t) =, sin -e a a O c. y (x,t) =, С. 2 sin -e nx - iot a a d. 2 sin -e NAX - iot y (x,t) =, a a
- sin kx 2) Consider the wavefunction is given y(x) =2N For what value of N х will it be normalized? Hint: a useful integral is (sin dt 3) Prove that(a) Write the wave function w(x)= wo•e'kx in the form w(x) function 4(x, t) that corresponds to w(x) written in this form. = a + ib, where a and b are real quantities. (Assume that wo is real.) (b) Write the time-dependent waveFind the Fourier series of the function f(x) = x sin x on the interval [−π, π). Useit to prove that (1/1*3) -(1/3*5)+(1/5*7)-(1/7*9) +.........=(π-2)/4
- Find the Fourier series of f(x) on the given interval. -п хе(-1,0), f(x) = x € [0, 1).In the region 0 < x < a, a particle is described by the wave function y₁(x) = -b(x² - a²). In the region a≤ x ≤w, its wave function is y2(x) = (x-d)² - c. For x≥w, ¥3(x) = 0. (a) By applying the continuity conditions at x= a, find c and d in terms of a and b. (b) Find w in terms of a and b.Find the phase velocity and group velocity for the following two wave-forms (x and t are measured in SI units) (i) 1(x,t) = cos(4t - 2x) + cos(8t- 4.r). (ii) 2(x,t) Does these waveforms satisfy the wave equation %3D = cos(8t- 6a) + cos (4t – 4.x). %3D v² Ət? where v is a constant independent of frequency. Explain.