Exam 38: Particles Behaving As Waves

Light shines through atomic hydrogen gas. It is seen that the gas absorbs light readily at a wavelength of 91.63 nm. What is the value of n of the level to which the hydrogen is being excited by the absorption of light of this wavelength? Assume that the most of the atoms in the gas are in the lowest level. (h = 6.626 × 10^-34 J ∙ s, c = 3.00 × 10⁸ m/s, 1 eV = 1.60 × 10^-19 J, the Rydberg constant is R = 1.097 × 10⁷ m^-1)

A perfectly black body at 100°C emits light of intensity I that has the strongest intensity near wavelength λ. The temperature of this body is now increased to 200°C. (a) In terms of I, what is the intensity at which this hotter body radiates? (b) In terms of λ, near what wavelength does light radiated from this hotter body have the strongest intensity?

The Bohr radius of the hydrogen atom is 0.529 × 10^-10 m. What is the radius of the n = 2 state?

A perfectly black sphere 18.0 cm in diameter is held at a temperature of 215°C. (σ = 5.670 × 10-8 W/m² ∙ K⁴, Wien displacement law constant is 2.90 × 10^-3 m ∙ K, h = 6.626 × 10^-34 J ∙ s, c = 3.00 × 10⁸ m/s) (a) Near what wavelength does this sphere radiate most strongly? (b) If all the radiated energy were at the wavelength found in part (a), how many photons would the sphere emit each second?

If the accuracy in measuring the position of a particle increases, the accuracy in measuring its velocity will

A nonrelativistic electron is confined to a length of 500 pm on the x-axis. What is the kinetic energy of the electron if its speed is equal to the minimum uncertainty possible in its speed? ( = 1.055 × 10^-34 J ∙ s, mel = 9.11 × 10^-31 kg, 1 eV = 1.60 × 10^-19 J)

A measurement of an electron's speed is 2.0 × 10⁶ m/s and has an uncertainty of 10%. What is the minimum uncertainty in its position? ( = 1.055 × 10^-34 J ∙ s, mel = 9.11 × 10^-31 kg)

A nonrelativistic proton is confined to a length of 2.0 pm on the x-axis. What is the kinetic energy of the proton if its speed is equal to the minimum uncertainty possible in its speed? (1 eV = 1.60 × 10^-19 J, = 1.055 × 10^-34 J ∙ s, mproton = 1.67 × 10^-27 kg)

An electron inside a hydrogen atom is confined to within a space of 0.110 nm. What is the minimum uncertainty in the electron's velocity? ( = 1.055 × 10^-34 J ∙ s, mel = 9.11 × 10^-31 kg)

What is the orbital radius of the n = 3 excited state in the Bohr model of the hydrogen atom? The ground-state radius of the hydrogen atom is 0.529 × 10^-10 m.

A certain particle's energy is measured by a detector to within 1.0 × 10-18. What is the minimum uncertainty we can have in its arrival time at the detector? ( = 1.055 × 10^-34 J ∙ s)

You need 14 W of infrared laser light power with wavelength 1270 nm to bore a hole in a diamond. How many downward atomic transitions per second must occur in the laser if all of them result in light directed onto the diamond? (c = 3.00 × 10⁸ m/s, h = 6.626 × 10^-34 J ∙ s)

Light of wavelength 105 nm falls on a metal surface for which the work function is 5.00 eV. What is the minimum de Broglie wavelength of the photoelectrons emitted from this metal? (h = 6.626 × 10^-34 J ∙ s = 4.14 × 10^-15 eV ∙ s, c = 3.00 × 10⁸ m/s, mel = 9.11 × 10^-31 kg, 1 eV = 1.60 × 10^-19 J)

A single slit is illuminated at normal incidence with a parallel beam of light having a wavelength of 411 nm. The entire central band of the diffraction pattern is observed at ±90°. The illumination is now replaced by a nonrelativistic beam of electrons, each having a kinetic energy of 980 eV. When this beam hits the slit at normal incidence, at what angle will the first minimum of the electron diffraction pattern occur? (h = 6.626 × 10^-34 J ∙ s, mel = 9.11 × 10^-31 kg, 1 eV = 1.60 × 10^-19 J)

In the vicinity of what frequency does an object with a temperature of 1000 K radiate the largest amount of power? (c = 3.00 × 10⁸ m/s, Wien displacement law constant equals 2.90 × 10^-3 m ∙ K, σ = 5.670 × 10^-8 W/m² ∙ K⁴)

A laser produces a beam of 4000-nm light. A shutter allows a pulse of light, 30 ps in duration, to pass. Which of the following is closest to the uncertainty in the energy of a photon in the pulse? ( = 1.055 × 10^-34 J ∙ s = 6.59 × 10^-16 eV ∙ s)

The lifetime of an excited nuclear state is 1.0 ns. What is the minimum uncertainty in the energy of this state? ( = 1.055 × 10^-34 J ∙ s = 6.591 × 10^-16 eV ∙ s, 1 eV = 1.60 × 10^-19 J)

A nonrelativistic electron has a kinetic energy of 5.4 eV. What is the energy of a photon whose wavelength is the same as the de Broglie wavelength of the electron? (mel = 9.11 × 10^-31 kg, c = 3.00 × 10⁸ m/s, 1 eV = 1.60 × 10^-19 J)

An ultraviolet source produces a monochromatic beam of 200-nm light. A shutter allows a pulse to pass that is 10,000 wavelengths long. The uncertainty in the energy of a photon in this pulse is closest to which of the following? ( = 1.055 × 10^-34 J ∙ s = 6.59 × 10^-16 eV ∙ s, c = 3.00 × 10⁸ m/s)

A hydrogen atom is excited to the n = 10 stated. It then decays to the n = 4 state by emitting a photon which is detected in a photographic plate. What is the frequency of the detected photon? The lowest level energy state of hydrogen is -13.6 eV. (h = 6.626 × 10^-34 J ∙ s, 1 eV = 1.60 × 10^-19 J)