35. Special Relativity

An electron is in a box of width $3.0\times {10}^{-10}$ m. What are the de Broglie wavelength and the magnitude of the momentum of the electron if it is in (a) the $n=1$ level; (b) the $n=2$ level; (c) the $n=3$ level? In each case how does the wavelength compare to the width of the box?

What wavelength photon does a hydrogen atom emit in a 200→199 transition?

Identify the unknown isotope $X$ in the following decays. ${}^7\text{Be}+{\text{e}}^{-}\to \text{X}+\nu$

$10.0$-g marble is gently placed on a horizontal tabletop that is $1.75$ m wide.

(a) What is the maximum uncertainty in the horizontal position of the marble?

(b) According to the Heisenberg uncertainty principle, what is the minimum uncertainty in the horizontal velocity of the marble?

(c) In light of your answer to part (b), what is the longest time the marble could remain on the table? Compare this time to the age of the universe, which is approximately $14$ billion years. (Hint: Can you know that the horizontal velocity of the marble is exactly zero?)

An electron with initial kinetic energy $5.0$ eV encoun­ters a barrier with height $U_0$ and width $0.60$ nm. What is the transmission coefficient if (a) $U_0=7.0$ eV; (b) $U_0=9.0$ eV; (c) $U_0=13.0$ eV?

An excited state of an atom has a 25 ns lifetime. What is the probability that an excited atom will emit a photon during a 0.50 ns interval?

An electron is moving with a speed of $8.00\times {10}^6$ m/s. What is the speed of a proton that has the same de Broglie wavelength as this electron?

A 2.0-μm-diameter water droplet is moving with a speed of 1.0 μm/s in a 20-μm-long box. Estimate the particle’s quantum number.

The allowed energies of a simple atom are 0.00 eV, 4.00 eV, and 6.00 eV. An electron traveling with a speed of 1.30×10⁶ m/s collides with the atom. Can the electron excite the atom to the n = 2 stationary state? The n = 3 stationary state? Explain.

A particle confined in a rigid one-dimensional box of length 10 fm has an energy level E_n = 32.9 MeV and an adjacent energy level E_n+1 = 51.4 MeV. Determine the values of n and n+1.

A particle is described by the wave function where L = 2.0 mm. Calculate the probability of finding the particle within 1.0 mm of the origin.

(II) What is the maximum sum-of-the-angles for a triangle on a sphere?

What is the minimum uncertainty in position, in nm, of an electron whose velocity is known to be between 3×10⁵ m/s and 4 ×10⁵ m/s? Give your answer to one significant figure.

Identify the isotope that is 11 times as heavy as ¹²C and has 18 times as many protons as ⁶Li . Give your answer in the form ᴬS, where S is the symbol for the element. See Appendix C: Atomic and Nuclear Data.

For those that are not stable, identify both the decay mode and the daughter nucleus.