Chapter 36

Alpha particles are accelerated through a potential difference of $20.0\mathrm{kV}$. What is their de Broglie wavelength?

Consider an electron whose de Broglie wavelength is equal to the wavelength of green light (about $550\mathrm{nm}$ ). a) Treating the electron nonrelativistically, what is its speed? b) Does your calculation confirm that a nonrelativistic treatment is sufficient? c) Calculate the kinetic energy of the electron in $\mathrm{eV}$.

After you told him about de Broglie's hypothesis that particles of momentum $p$ have wave characteristics with wavelength $\lambda =h/p$, your 60.0 -kg roommate starts thinking of his fate as a wave and asks you if he could be diffracted when passing through the 90.0 -cm-wide doorway of your dorm room. a) What is the maximum speed at which your roommate can pass through the doorway in order to be significantly diffracted? b) If it takes one step to pass through the doorstep, how long should it take your roommate to make that step (assume the length of his step is $0.75\mathrm{m}$ ) in order for him to be diffracted? c) What is the answer to your roommate's question? Hint: Assume that significant diffraction occurs when the width of the diffraction aperture is less that 10.0 times the wavelength of the wave being diffracted.

Consider de Broglie waves for a Newtonian particle of mass $m,$ momentum $p=mv,$ and energy $E=p^2/(2m),$ that is, waves with wavelength $\lambda =h/p$ and frequency $f=E/h$. a) Calculate the dispersion relation $\omega =\omega (k)$ for these waves. b) Calculate the phase and group velocities of these waves. Which of these corresponds to the classical velocity of the particle?

Now consider de Broglie waves for a (relativistic) particle of mass $m$, momentum $p=mv\gamma$, and total energy $E=m{c}^2\gamma$, with $\gamma ={[1-(v/c{)}^2]}^{-1/2}$. The waves have wavelength $\lambda =h/p$ and frequency $f=E/h$ as before, but with the relativistic momentum and energy. a) Calculate the dispersion relation for these waves. b) Calculate the phase and group velocities of these waves. Now which corresponds to the classical velocity of the particle?

A 50.0 -kg particle has a de Broglie wavelength of $20.0\mathrm{cm}.$ a) How fast is the particle moving? b) What is the smallest speed uncertainty of the particle if its position uncertainty is $20.0\mathrm{cm}?$

Suppose that Fuzzy, a quantum-mechanical duck, lives in a world in which Planck's constant $\hslash =1.00\mathrm{J}$ s. Fuzzy has a mass of $0.500\mathrm{kg}$ and initially is known to be within a $0.750-\mathrm{m}-$ wide pond. What is the minimum uncertainty in Fuzzy's speed? Assuming that this uncertainty prevails for $5.00\mathrm{s}$, how far away could Fuzzy be from the pond after 5.00 s?

Consider a quantum state of energy $E$, which can be occupied by any number $n$ of some bosonic particles, including $n=0$. At absolute temperature $T$, the probability of finding $n$ particles in the state is given by $P_n=N\exp (-nE/k_{\mathrm{B}}T)$, where $k_{\mathrm{B}}$ is Boltzmann's constant and the normalization factor $N$ is determined by the requirement that all the probabilities sum to unity. Calculate the mean or expected value of $n$, that is, the occupancy, of this state, given this probability distribution.

Consider a system made up of $N$ particles. The average energy per particle is given by $⟨E⟩=(\sum E_i{e}^{-E_i/k_{B}T})/Z$ where $Z$ is the partition function defined in equation $36.29.$ If this is a two-state system with $E_1=0$ and $E_2=E$ and $g_1=$ $g_2=1,$ calculate the heat capacity of the system, defined as $N(d⟨E⟩/dT)$ and approximate its behavior at very high and very low temperatures (that is, $k_{\mathrm{B}}T\gg 1$ and $k_{\mathrm{B}}T\ll 1$ ).

Given that the work function of tungsten is $4.55\mathrm{eV}$ what is the stopping potential in an experiment using tungsten cathodes at $360\mathrm{nm}?$