Problem 4.28. Thermodynamic properties of a system of harmonic oscillators (a) Show that for one oscillator 1 f = hw + kT In(1 – e¯Bhw), (4.130) Bhw s = klBhw -Bhu In(1 1 (4.131) - e 1 (4.132) eßhw Equation (4.132) is Planck's formula for the mean energy of an oscillator at temperature T. The heat capacity is discussed in Problem 4.50.

: In class we did a problem where we saw that Planck’s law, which is given byI(λ, T) = 2πhc2λ5(ehc/λkbT − 1),at high wavelengths reduces to the classical predictionI(λ, T) = 2πckBTλ4.We did this by using the MacLaurin series for an exponential.1 + x +x22! +x33! +x44! + ...When we did this problem in class, we reasoned that,when the wavelength (λ) is large, the term hc/λkbTis small enough that any term (hc/λkbT)2 or (hc/λkbT)3 or any higher power is neglible. That’s how weshowed that, in the limit of large λ, Planck’s law reduces to the classical prediction. Now assume that wewant to make a slightly better approximation. We still assume that λ is large and therefore (hc/λkbT)3 and(hc/λkbT)4 and all higher powers are negligible, but now we want to work at the level of precision where(hc/λkbT)2is not negligible. What does Planck’s law reduce to in this case?

3.1 Write down the equation for the Planck function B, as a function of the frequency v and define the various parameters.

An electron has de Broglie wavelength 2.78×10-10 m . Part A Determine the magnitude of the electron's momentum pe. Express your answer in kilogram meters per second to three significant figures. ▸ View Available Hint(s) VG ΑΣΦ 24 Pe 2.29 10 . Submit Previous Answers ? × Incorrect; Try Again; One attempt remaining kg. m/s