10.4.1: Energy Systems

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Paul Penfield, Jr.
Massachusetts Institute of Technology via MIT OpenCourseWare

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An object that stores, transmits, or converts energy must have possible states. Such an object typically might consist of a large number (say Avogadro’s number \(N_A = 6.02 × 10^{23}\)) of similar or identical particles and therefore a huge number of stationary states. The Schrödinger equation cannot be solved in such circumstances. Interactions with the environment would occur often in order to transfer energy to and from the environment. It is impossible to know whether the system is in a stationary state, and even if it is known, unpredictable interactions with the environment make such knowledge irrelevant rapidly.

The most that can be done with such systems is to deal with the probabilities \(p_j\) of occupancy of the various stationary states

\(p_j = |a_j |^2 \tag{10.13}\)

The expected value of the energy \(E\) would then be

\(E = \displaystyle \sum_{j} e_j p_j \tag{10.14}\)

This model is set up in a way that is perfectly suited for the use of the Principle of Maximum Entropy to estimate the occupation probability distribution \(p_j\). This topic will be pursued in Chapter 11 of these notes.