Problem 1) Wave functions and probobility current densities Consid...

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Problem 1) Wave functions and probobility current densities Consider an electron that is confined a region of length L in one dimension. (This could be the idealization of a wire, but the realization is not the problem.) Inside the wire the potential vanishes. a) Show that wave functions of the form fulfil the free Schroedinger equation (for real k) and determine those k for which the probability density vanishes at the ends of the wire, i.e, at x = 0 and x = L. Calculate the probability current density associated with these wave functions b) Now assume that the wire forms a closed loop by identifying the endpoints x = 0 and x = L. Check that the wave function fulfils this boundary conditions and calculate the probability current density in the states PR. c) Give a brief discussion comparing the results in a) and b Problem 2) Variational Principle Consider the one-dimensional potential v(x) = C1x1 Where C > Ois a constant. Determine the energy of the ground state using the variational principle for wave functions of the form 4a(x)=A(c)e-us. Problem 3) Stationary States with Complex Energy? Assume that a stationary state has the complex energy E = c + thy. Show that this would lead to a contradiction with the conservation of probability. Problem 4) Wave function dynomics in the harmonic oscillator potential Consider an electron in a one-dimensional harmonic-oscillator potential v(x) = At = its state is given by the wave function *(x,t = 0) = A[and 91 are the normalized wave functions of the oscillator ground state and its first excited states with energies E0 = have and E1 = thwo. a) Determine the normalization constant A, so that $(x, t = 0) is normalized. b) Calculate the expectation value of the harmonic oscillator Hamiltonian in the states $(x,t = 0). How does the result compare to E0 and E1? c) Calculate *(x,t) and the corresponding probability density 14 (x,t)|2 for all times t 0. d) Calculate the expectation value (x)(t) in the state *(x,t). Hints for Problem 4: The Hermite polynomials are orthogonal in the sense that for n = m. You can assume the value of the integral Loudd=

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