PX284 - B1 - the system

• statistical mechanics is concerned with understanding and predicting the properties of a 'system'
• eg: a box of gas, ions in a solid, electrons in a star
• there exists a 'reservoir' with which systems can exchange energy or particles
• typical systems:
  • $1l$ of N${}_2$ gas $\sim {10}^{22}$ molecules
  • $1g$ of NaCl $\sim {10}^{22}$ molecules
  • $1c{m}^3$ of Cu $\sim {10}^{22}$ electrons
  • the sun $\sim {10}^{57}$ electrons
• isolated systems: not connected to a reservoir, therefore the energy $(E)$, the number of particles $(N)$, and the volume $(V)$ are fixed

model system

• the simple harmonic oscillator
• potential energy:

• solution:

• $\boxed{{\displaystyle 2}}$ indicates a system in state, $n=2$
• there can be $N>1$ simple harmonic oscillators
• consider $N=4:$

• the energy of the system:

• the last term is fixed at $\frac{N\hslash \omega }{2}$, which is called the zero-point energy (ZPE), the energy of the ground state
• ZPE is an important result of quantum mechanics that may sometimes be ignored for convenience