## Transcribed Text

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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