Chapter 4, The Principles of Wave Mechanics Video Solutions, Quantum mechanics

Problem 1

Show that the addition of an imaginary part to the potential in the quantal wave equation describes the presence of sources or sinks of probability. (Work out the appropriate continuity equation.)
Solve the wave equation for a potential of the form $V=-V_0(1+i \zeta)$, where $V_0$ and $\zeta$ are positive constants. If $\zeta \ll 1$, show that there are stationary state solutions that represent plane waves with exponentially attenuated amplitude, describing absorption of the waves. Calculate the absorption coefficient.

Dr. Rajveer Singh

Numerade Educator

01:18

Problem 2

If a particle of mass $m$ is constrained to move in the $x y$ plane on a circular orbit of radius $\rho$ around the origin 0 , but is otherwise free, determine the energy eigenvalues and the eigenfunctions.

Suzanne W.

Numerade Educator

Problem 3

A particle of mass $m$ and charge $q$ is constrained to move in the $x y$ plane on a circular orbit of radius $\rho$ around the origin 0 , as in Problem 2, and a magnetic field, represented by the vector potential $\mathbf{A}=\Phi \hat{\mathbf{k}} \times \mathbf{r} /\left[2 \pi(\hat{\mathbf{k}} \times \mathbf{r})^2\right]$, is imposed.
(a) Show that the magnetic field approximates that of a long thin solenoid with flux $\Phi$ placed on the $z$ axis.
(b) Determine the energy spectrum in the presence of the field and show that it coincides with the spectrum for $\Phi=0$ if the flux assumes certain quantized values. Note that the energy levels depend on the strength of a magnetic field $\mathbf{B}$ which differs from zero only in a region into which the particle cannot penetrate (the AharonovBohm effect).

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