Final answer:
The de Broglie wavelength of neutrons at different temperatures can be calculated by converting average kinetic energy from temperature into momentum and then using the de Broglie equation λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum.
Step-by-step explanation:
The question concerns the calculation of de Broglie wavelengths for neutrons at different temperatures using principles from quantum mechanics and thermal physics. The de Broglie wavelength is a concept that describes the wave-like behavior of particles and is determined by the equation λ = h/p, where λ represents the wavelength, h is Planck's constant, and p is the momentum of the particle.
For particles in thermal equilibrium, their average kinetic energy can be given by the equipartition theorem from statistical mechanics, which states that each degree of freedom has an average energy of (1/2)kT, where k is the Boltzmann constant and T is the temperature. The momentum p of a particle can be related to its kinetic energy K by the formula p = √(2mK), where m is the mass of the particle. Therefore, to find the de Broglie wavelength of the neutrons at different temperatures, we would convert the average thermal kinetic energy at each temperature to an equivalent average momentum and then use the de Broglie equation to determine the wavelengths.