91Ó°ÊÓ

A free particle moving in one dimension has wave function $$\Psi(x, t) = A[e^{i(kx-vt)} - e^{i(2kx-4vt)}]$$ where \(k\) and \(\omega\) are positive real constants. (a) At \(t\) = 0 what are the two smallest positive values of \(x\) for which the probability function \(\mid \Psi(x, t) \mid ^2\) is a maximum? (b) Repeat part (a) for time \(t = 2\pi/\omega\). (c) Calculate \(v_{av}\) as the distance the maxima have moved divided by the elapsed time. Compare your result to the expression \(v_{av} = (\omega_2 - \omega_1)/(k_2 - k_1)\) from Example 40.1.

Consider a particle moving in one dimension, which we shall call the \(x\)-axis. (a) What does it mean for the wave function of this particle to be \(normalized\)? (b) Is the wave function \(\psi(x) = e^{ax}\) , where a is a positive real number, normalized? Could this be a valid wave function? (c) If the particle described by the wave function \(\psi(x) = Ae^{-bx}\), where \(A\) and \(b\) are positive real numbers, is confined to the range \(x \geq 0\), determine A (including its units) so that the wave function is normalized.

When an electron in a one-dimensional box makes a transition from the \(n\) = 1 energy level to the \(n\) = 2 level, it absorbs a photon of wavelength 426 nm. What is the wavelength of that photon when the electron undergoes a transition (a) from the \(n\) = 2 to the \(n\) = 3 energy level and (b) from the n = 1 to the \(n\) = 3 energy level? (c) What is the width \(L\) of the box?