Consider a freely moving quantum particle with mass m and speed v. Its energy is $E=K+U=\frac{1}{2} m v^{2}+0$. Determine the phase speed of the quantum wave representing the particle and show that it is different from the speed at which the particle transports mass and energy.

The average lifetime of a muon is about 2 $\mu \mathrm { s }$. Estimate the minimum uncertainty in the energy of a muon.

A proton is confined in a uranium nucleus of radius $7.2 \times 10^{-15}$ m. Determine the proton’s minimum kinetic energy according to the uncertainty principle if the proton is confined to a one-dimensional box that has length equal to the nuclear diameter.

Use the uncertainty principle to show that if an electron were confined inside an atomic nucleus of diameter on the order of $10^{-14} \mathrm{m},$ it would have to be moving relativistically, whereas a proton confined to the same nucleus can be moving nonrelativistically.

Show that the equation $\sigma(n)=k$ has at most a finite number of solutions when k is a positive integer.

The contribution of a single electron or hole to the electric conductivity of a semiconductor can be expressed by an important property called the mobility. The mobility, $\mu$, is defined as the particle drift speed per unit electric field or, in terms of a formula, $\mu=v_{\mathrm{d}} / E$. Note that mobility describes the ease with which charge carriers drift in an electric field and that the mobility of an electron, $\mu_{n}$, may be different from the mobility of a hole, $\mu_{n}$, in the same material. (a) Show that the current density may be written in terms of mobility as $J=n e \mu_{n} E$, where $\mu_{n}=e \tau / m_{\mathrm{e}}$. (b) Show that when both electrons and holes are present, the conductivity may be expressed in terms of $\mu_{p}$ and $\mu_{n}$ as $\sigma=n e \mu_{n}+p e \mu_{p}$, where n is the electron concentration and p is the hole concentration. (c) If a germanium sample has $\mu_{n}=3900 \mathrm{cm}^{2} / \mathrm{V} \cdot \mathrm{s}$, calculate the drift speed of an electron when a field of 100 V/cm is applied. (d) A pure (intrinsic) sample of germanium has $\mu_{n}=3900 \mathrm{cm}^{2} / \mathrm{V} \cdot \mathrm{s}$ and $\mu_{p}=1900 \mathrm{cm}^{2} / \mathrm{V} \cdot \mathrm{s}$. If the hole concentration is equal to $3.0 \times 10^{13} \mathrm{cm}^{-3}$, calculate the resistivity of the sample.

Show that if k > 1 is an integer, then the equation $\tau(n)=k$ has infinitely many solutions.

A photon having energy $E_{0}=0.880$ MeV is scattered by a free electron initially at rest such that the scattering angle of the scattered electron is equal to that of the scattered photon. Determine the scattering angle of the photon and the electron.

Which statements are true? Explain why or why not. Peroxisomes are found in only a few specialized types of eukaryotic cell.

Two light sources are used in a photoelectric experiment to determine the work function for a particular metal surface. When green light from a mercury lamp ($\lambda = 546.1$ nm) is used, a stopping potential of 0.376 V reduces the photocurrent to zero. Based on this measurement, what is the work function for this metal?

A 0.001 60-nm photon scatters from a free electron. For what (photon) scattering angle does the recoiling electron have kinetic energy equal to the energy of the scattered photon?

Underline each adjective clause and draw an arrow to the word it modifies. My mom will take anyone who wants to go.

Why, according to Lahiri, would she never try to make pulao?

Summarize the need for alternative energy sources.