Let's consider a more real-life example -- an Einstein Solid. In an Einstein Solid, we have particles that are trapped in a quantum harmonic oscillator. Each particle has 3 oscillators they are part of, one for each dimension. The energy in an oscillator must be some integer multiple of a unit value, and so we can count the number of ways energy can be allocated.
Playlist: • An Introduction to Thermal Physics by...
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The textbook I am using is: Schroeder, Daniel V. "An Introduction to Thermal Physics" 1st ed., Addison Wesley Longman, 2005.