4. Consider a particle of mass , moving in one dimension confined to the positiv

Question

4. Consider a particle of mass , moving in one dimension confined to the positive x-axis by a potential energy of the form: V = infinity for x 0, where A is a positive constant. Use the uncertainty principle to estimate the ground state energy for such a particle. Use the Bohr-Sommerfeld quantization rule integral pdr = nh to determine the form of the energy levels associated with this potential. Your answer should provide a sketch of the classical trajectory in the x-p (phase space) plane. c) Now consider an exact analysis which involves solving the Schrodinger equation (SE). What are the appropriate length (l) and energy (u) scales associated with this problem? Your answers should be in terms of m, h, A. It may prove useful to utilize dimensional analysis. Write down the time independent SE in terms of the dimensionless parameters y = x/l and epsilon = E/u. d) Consider the limiting behavior when y >> 1. How does psi behave in this limit? Is there more than one possibility? If so, what principle may be used to choose between the alternatives? e) Suppose that psi ~ F for large y, where F(y) is the function you found in part (d). Write psi = FG. substitute this expression into the SE. and thus determine the equation satisfied by G(y). Assume a series solution for G of the form G = and use the equation for G to determine a recursion relation satisfied by the coefficients an. f) How does the series solution for G behave if it does not terminate? Utilize this fact to determine what vales of epsilon are permitted. What energies E follow from these values, and how do these compare with what you found in part (b)? Determine the normalized wave functions corresponding lo the ground and first excited state for this potential. Are these two states orthogonal? g) For the ground state evaluate the expectation values , , Are these expectation values consistent with the uncertainty principle? f) Consult your text on the various electron states that are possible in a hydrogen atom. What relation, if any, might this problem have to this physical system? The original Bohr model of the H-atom implies all electron states have non-zero angular momentum. Does this problem have any relation to the ''s-states'' of such an atom?

Explanation / Answer

Uncertainty: dp*dx ? h
For minimum uncertainty, dp*dx = h
Uncertainty, dx ? r
For minimum uncertainty, p ? dp ? h/r

Potential energy V(r) = - A/r
energy, E = p^2/(2*m) + V(r)
E = h^2/(2*m*r^2) - A/r
Ground state energy E is minimum with respect to r
dE/dr = 0,
dE/dr = -h^2/(m*r^3) + A/(r^2)
r = h^2/(m*A)

E = h^2/(2*m*r^2) - A/r = m*A^2/(2*h^4) - m/h^2

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