Question

Consider a one-dimensional harmonic oscillator system in equilibrium with a thermal heat bath at absolute temperature T. The oscillator is characterized by angular frequency w = √k/m, where k is the spring constant characterizing the oscillator's motion and m is its mass. The allowed (quantum mechanical) states of the oscillator are labeled n = 0, l, 2, ... with the energy of state n given by: En = hꞷ(n + 1/2). We showed in class that the internal energy of this system is

E(T) = hꞷ/2 cos (hꞷ / 2kBT).

a. Deduce a formula for Cv, the constant volume heat capacity of this system. Cv(T) = dE(T) / dT

Answer 1

The student is tasked with deriving the formula for Cv(T), the constant volume heat capacity for a quantum harmonic oscillator. The calculation involves taking the temperature derivative of the given internal energy expression.

Step-by-step explanation:

The question pertains to the calculation of the heat capacity at constant volume, Cv, for a one-dimensional quantum harmonic oscillator in thermal equilibrium at temperature T. The given internal energy E(T) is expressed as a function of temperature, and the task is to find Cv(T) as the derivative of E(T) with respect to T.

For a one-dimensional harmonic oscillator in equilibrium with a heat bath at temperature T, the allowed energy levels are quantized, denoted by En = hꞷ(n + 1/2), with n = 0, 1, 2, ... The oscillator's angular frequency w is determined by the spring constant k and the mass m, such that w = √(k/m). The relationship between internal energy and heat capacity is critical here, as heat capacity represents how much energy is needed to change the system's temperature.

The heat capacity at constant volume, Cv, for the oscillator can be calculated using the provided internal energy expression E(T) = hꞷ/2 cos(hꞷ / 2kBT). Taking the derivative of E(T) with respect to temperature T gives us Cv(T). The specific steps for this calculation involve differentiating the cosine function and applying the chain rule of calculus, considering that h, w, and kB are constants.