Problem 1
An electron is moving as a free particle in the -\(x\)-direction with momentum that has magnitude 4.50 \(\times\) 10\(^{-24}\) kg \(\bullet\) m/s. What is the one- dimensional time-dependent wave function of the electron?
Problem 2
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.
Problem 4
A particle is described by a wave function \(\psi(x) = Ae^{-\alpha x^2}\), where A and \(\alpha\) are real, positive constants. If the value of \(\alpha\) is increased, what effect does this have on (a) the particle's uncertainty in position and (b) the particle's uncertainty in momentum? Explain your answers.
Problem 5
Consider a wave function given by \(\psi(x) = A \space sin \space kx\), where \(k = 2\pi/\lambda\) and \(A\) is a real constant. (a) For what values of \(x\) is there the highest probability of finding the particle described by this wave function? Explain. (b) For which values of \(x\) is the probability \(zero\)? Explain.
Problem 6
Compute \(\mid \Psi \mid ^2 for \space \Psi = \psi \space sin \space \omega t\), where \(\psi\) is time independent and \(\omega\) is a real constant. Is this a wave function for a stationary state? Why or why not?
Problem 9
(a) Find the lowest energy level for a particle in a box if the particle is a billiard ball (\(m\) = 0.20 kg) and the box has a width of 1.3 m, the size of a billiard table. (Assume that the billiard ball slides without friction rather than rolls; that is, ignore the \(rotational\) kinetic energy.) (b) Since the energy in part (a) is all kinetic, to what speed does this correspond? How much time would it take at this speed for the ball to move from one side of the table to the other? (c) What is the difference in energy between the \(n\) = 2 and \(n\) = 1 levels? (d) Are quantum-mechanical effects important for the game of billiards?
Problem 10
A proton is in a box of width \(L\). What must the width of the box be for the groundlevel energy to be 5.0 MeV, a typical value for the energy with which the particles in a nucleus are bound? Compare your result to the size of a nucleus-that is, on the order of 10\(^{-14}\) m.
Problem 12
When a hydrogen atom undergoes a transition from the \(n\) = 2 to the \(n\) = 1 level, a photon with \(\lambda\) = 122 nm is emitted. (a) If the atom is modeled as an electron in a one-dimensional box, what is the width of the box in order for the \(n\) = 2 to \(n\) = 1 transition to correspond to emission of a photon of this energy? (b) For a box with the width calculated in part (a), what is the ground-state energy? How does this correspond to the ground-state energy of a hydrogen atom? (c) Do you think a one-dimensional box is a good model for a hydrogen atom? Explain. (\(Hint\): Compare the spacing between adjacent energy levels as a function of n.)
Problem 14
An electron in a one-dimensional box has ground-state energy 2.00 eV. What is the wavelength of the photon absorbed when the electron makes a transition to the second excited state?
Problem 15
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.
