Chapter 39

Suppose that the uncertainty of position of an electron is equal to the radius of the $n$ = 1 Bohr orbit for hydrogen. Calculate the simultaneous minimum uncertainty of the corresponding momentum component, and compare this with the magnitude of the momentum of the electron in the $n$ = 1 Bohr orbit. Discuss your results.

(a) A particle with mass $m$ has kinetic energy equal to three times its rest energy. What is the de Broglie wavelength of this particle? ($Hint$: You must use the relativistic expressions for momentum and kinetic energy: $E^2=(p{c}^2)+(m{c}^2{)}^2$ and $K=E-m{c}^2$.) (b) Determine the numerical value of the kinetic energy (in MeV) and the wavelength (in meters) if the particle in part (a) is (i) an electron and (ii) a proton.

The radii of atomic nuclei are of the order of 5.0 $\times$ 10${}^{-15}$ m. (a) Estimate the minimum uncertainty in the momentum of an electron if it is confined within a nucleus. (b) Take this uncertainty in momentum to be an estimate of the magnitude of the momentum. Use the relativistic relationship between energy and momentum, Eq. (37.39), to obtain an estimate of the kinetic energy of an electron confined within a nucleus. (c) Compare the energy calculated in part (b) to the magnitude of the Coulomb potential energy of a proton and an electron separated by 5.0 $\times$ 10${}^{-15}$ m. On the basis of your result, could there be electrons within the nucleus? ($Note$: It is interesting to compare this result to that of Problem 39.72.)

The neutral pion (${\pi }^0$) is an unstable particle produced in high-energy particle collisions. Its mass is about 264 times that of the electron, and it exists for an average lifetime of 8.4 $\times$ 10${}^{-17}$ s before decaying into two gamma-ray photons. Using the relationship $E=m{c}^2$ between rest mass and energy, find the uncertainty in the mass of the particle and express it as a fraction of the mass.

If your wavelength were 1.0 m, you would undergo considerable diffraction in moving through a doorway. (a) What must your speed be for you to have this wavelength? (Assume that your mass is 60.0 kg.) (b) At the speed calculated in part (a), how many years would it take you to move 0.80 m (one step)? Will you notice diffraction effects as you walk through doorways?

A certain atom has an energy level 2.58 eV above the ground level. Once excited to this level, the atom remains in this level for 1.64 $\times$ 10${}^{-7}$ s (on average) before emitting a photon and returning to the ground level. (a) What is the energy of the photon (in electron volts)? What is its wavelength (in nanometers)? (b) What is the smallest possible uncertainty in energy of the photon? Give your answer in electron volts. (c) Show that $\mid \mathrm{\Delta }E/E\mid =\mid \mathrm{\Delta }\lambda /\lambda \mid if\mid \mathrm{\Delta }\lambda /\lambda \mid \ll$ 1. Use this to calculate the magnitude of the smallest possible uncertainty in the wavelength of the photon. Give your answer in nanometers.

A certain atom has an energy state 3.50 eV above the ground state. When excited to this state, the atom remains for 2.0 $\mu$s, on average, before it emits a photon and returns to the ground state. (a) What are the energy and wavelength of the photon? (b) What is the smallest possible uncertainty in energy of the photon?

To investigate the structure of extremely small objects, such as viruses, the wavelength of the probing wave should be about one-tenth the size of the object for sharp images. But as the wavelength gets shorter, the energy of a photon of light gets greater and could damage or destroy the object being studied. One alternative is to use electron matter waves instead of light. Viruses vary considerably in size, but 50 nm is not unusual. Suppose you want to study such a virus, using a wave of wavelength 5.00 nm. (a) If you use light of this wavelength, what would be the energy (in eV) of a single photon? (b) If you use an electron of this wavelength, what would be its kinetic energy (in eV)? Is it now clear why matter waves (such as in the electron microscope) are often preferable to electromagnetic waves for studying microscopic objects?

Consider a particle with mass m moving in a potential $U=\frac{1}{2}k{x}^2$, as in a mass-spring system. The total energy of the particle is $E=(p^2/2m)+12k{x}^2$. Assume that $p$ and $x$ are approximately related by the Heisenberg uncertainty principle, so $px\approx h$. (a) Calculate the minimum possible value of the energy $E$, and the value of $x$ that gives this minimum E. This lowest possible energy, which is not zero, is called the $zero-pointenergy$. (b) For the $x$ calculated in part (a), what is the ratio of the kinetic to the potential energy of the particle?

Imagine another universe in which the value of Planck's constant is 0.0663 J $\cdot$ s, but in which the physical laws and all other physical constants are the same as in our universe. In this universe, two physics students are playing catch. They are 12 m apart, and one throws a 0.25-kg ball directly toward the other with a speed of 6.0 m/s. (a) What is the uncertainty in the ball's horizontal momentum, in a direction perpendicular to that in which it is being thrown, if the student throwing the ball knows that it is located within a cube with volume 125 cm${}^3$ at the time she throws it? (b) By what horizontal distance could the ball miss the second student?