Problem 1.18 Very roughly speaking, quantum mechanics is relevant when the de Broglie wavelength of the particle in question (h/p) is greater than the characteristic size of the system (d). In thermal equilibrium at (Kelvin) temperature T, the average kinetic energy of a particle is p 3 2m (where kg is Boltzmann's constant), so the typical de Broglie wavelength is (1.45) V3mkBT The purpose of this problem is to determine which systems will have to be treated quantum mechanically, and which can safely be described classically. (a) Solids. The lattice spacing in a typical solid is around d = 0.3 nm. Find the temperature below which the unbound² electrons in a solid are quantum mechanical. Below what temperature are the nuclei in a solid quantum mechanical? (Use silicon as an example.) Moral: The free electrons in a solid are always quantum mechanical; the nuclei are generally not quantum mechanical. The same goes for liquids (for which the interatomic spacing is roughly the same), with the exception of helium below 4 K. (b) Gases. For what temperatures are the atoms in an ideal gas at pressure P quantum mechanical? Hint: Use the ideal gas law (PV = NkgT) to deduce the interatomic spacing. Answer: T < (1/kg) (h²/3m)³/³ p2/5. Obviously (for the gas to show quantum behavior) we want m to be as small as possible, and P as large as possible. Put in the numbers for helium at atmospheric pressure. Is hydrogen in outer space (where the interatomic spacing is about 1 cm and the temperature is 3 K) quantum mechanical? (Assume it's monatomic hydrogen, not H2.)
Problem 1.18 Very roughly speaking, quantum mechanics is relevant when the de Broglie wavelength of the particle in question (h/p) is greater than the characteristic size of the system (d). In thermal equilibrium at (Kelvin) temperature T, the average kinetic energy of a particle is p 3 2m (where kg is Boltzmann's constant), so the typical de Broglie wavelength is (1.45) V3mkBT The purpose of this problem is to determine which systems will have to be treated quantum mechanically, and which can safely be described classically. (a) Solids. The lattice spacing in a typical solid is around d = 0.3 nm. Find the temperature below which the unbound² electrons in a solid are quantum mechanical. Below what temperature are the nuclei in a solid quantum mechanical? (Use silicon as an example.) Moral: The free electrons in a solid are always quantum mechanical; the nuclei are generally not quantum mechanical. The same goes for liquids (for which the interatomic spacing is roughly the same), with the exception of helium below 4 K. (b) Gases. For what temperatures are the atoms in an ideal gas at pressure P quantum mechanical? Hint: Use the ideal gas law (PV = NkgT) to deduce the interatomic spacing. Answer: T < (1/kg) (h²/3m)³/³ p2/5. Obviously (for the gas to show quantum behavior) we want m to be as small as possible, and P as large as possible. Put in the numbers for helium at atmospheric pressure. Is hydrogen in outer space (where the interatomic spacing is about 1 cm and the temperature is 3 K) quantum mechanical? (Assume it's monatomic hydrogen, not H2.)
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Transcribed Image Text:Problem 1.18 Very roughly speaking, quantum mechanics is relevant when the de
Broglie wavelength of the particle in question (h/p) is greater than the
characteristic size of the system (d). In thermal equilibrium at (Kelvin)
temperature T, the average kinetic energy of a particle is
p
3
2m
(where kg is Boltzmann's constant), so the typical de Broglie wavelength is
(1.45)
V3mkBT
The purpose of this problem is to determine which systems will have to be
treated quantum mechanically, and which can safely be described classically.
(a) Solids. The lattice spacing in a typical solid is around d = 0.3 nm. Find
the temperature below which the unbound² electrons in a solid are
quantum mechanical. Below what temperature are the nuclei in a solid
quantum mechanical? (Use silicon as an example.)
Moral: The free electrons in a solid are always quantum mechanical; the
nuclei are generally not quantum mechanical. The same goes for liquids
(for which the interatomic spacing is roughly the same), with the
exception of helium below 4 K.
(b) Gases. For what temperatures are the atoms in an ideal gas at pressure P
quantum mechanical? Hint: Use the ideal gas law (PV = NkgT) to
deduce the interatomic spacing.
Answer: T < (1/kg) (h²/3m)³/³ p2/5. Obviously (for the gas to show
quantum behavior) we want m to be as small as possible, and P as large as
possible. Put in the numbers for helium at atmospheric pressure. Is
hydrogen in outer space (where the interatomic spacing is about 1 cm and
the temperature is 3 K) quantum mechanical? (Assume it's monatomic
hydrogen, not H2.)
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