Chapter 37

Write a plane-wave function \(\psi(\vec{r}, t)\) for a nonrelativistic free particle of mass \(m\) moving in three dimensions with momentum p, including correct time dependence as required by the Schrödinger equation. What is the probability density associated with this wave?

Suppose a quantum particle is in a stationary state (energy eigenstate) with a wave function \(\psi(x, t) .\) The calculation of \(\langle x\rangle,\) the expectation value of the particle's position, is shown in the text. Calculate \(d\langle x\rangle / d t(\operatorname{not}\langle d x / d t\rangle)\).

Although quantum systems are frequently characterized by their stationary states or energy eigenstates, a quantum particle is not required to be in such a state unless its energy has been measured. The actual state of the particle is determined by its initial conditions. Suppose a particle of mass \(m\) in a one-dimensional potential well with infinite walls (a "box") of width \(a\) is actually in a state with wave function $$ \Psi(x, t)=\frac{1}{\sqrt{2}}\left[\Psi_{1}(x, t)+\Psi_{2}(x, t)\right], $$ where \(\Psi_{1}\) denotes the stationary state with quantum number \(n=1\) and \(\Psi_{2}\) denotes the state with \(n=2 .\) Calculate the probability density distribution for the position \(x\) of the particle in this state.

Particle-antiparticle pairs are occasionally created out of empty space. Looking at energy-time uncertainty, how long would such particles be expected to exist if they are: a) an electron/positron pair? b) a proton/antiproton pair?

A positron and an electron annihilate, producing two 2.0 -MeV gamma rays moving in opposite directions. Calculate the kinetic energy of the electron when the kinetic energy of the positron is twice that of the electron.

Electrons from a scanning tunneling microscope encounter a potential barrier that has a height of \(U=4.0 \mathrm{eV}\) above their total energy. By what factor does the tunneling current change if the tip moves a net distance of \(0.10 \mathrm{nm}\) farther from the surface?

An electron is confined in a three-dimensional cubic space of \(L^{3}\) with infinite potentials. a) Write down the normalized solution of the wave function in the ground state. b) How many energy states are available up to the second excited state from the ground state? (Take the electron spin into account.)

The neutrons in a parallel beam, each having kinetic energy \(1 / 40 \mathrm{eV}\) (which is approximately corresponding to “room temperature"), are directed through two slits \(0.50 \mathrm{~mm}\) apart. How far apart will the interference peaks be on a screen \(1.5 \mathrm{~m}\) away?

Find the ground state energy (in units of eV) of an electron in a one- dimensional quantum box, if the box is of length \(L=0.100 \mathrm{nm}\).

An approximate one-dimensional quantum well can be formed by surrounding a layer of GaAs with layers of \(\mathrm{Al}_{x} \mathrm{Ga}_{1-x}\) As. The GaAs layers can be fabricated in thicknesses that are integral multiples of the single-layer thickness, \(0.28 \mathrm{nm}\). Some electrons in the GaAs layer behave as if they were trapped in a box. For simplicity, treat the box as an infinite one-dimensional well and ignore the interactions between the electrons and the Ga and As atoms (such interactions are often accounted for by replacing the actual electron mass with an effective electron mass). Calculate the energy of the ground state in this well for these cases: a) 2 GaAs layers b) 5 GaAs layers