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## Quantization of Energy in Mechanical Oscillators (15 minutes)

- Before beginning this lecture, ask the students to Draw Some Bound States. This will help put students into the mindset of energies and bound states.
- The most common bound state is the single mass on a spring, as seen below.

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- We can also extend this bound state premise to a system with multiple masses and springs as well. Looking at the first normal mode for a five-mass system (if students have already performed the Monoatomic Chain Lab, refer to this), if we keep the envelope function the same shape but increase the amplitude, the potential energy for the system changes with very similar behavior to the single mass system.

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In essence, each normal mode for a multi-particle oscillator can be thought of as a collective mass bound by an effective potential.

- How do we determine the energy for the multi-particle oscillator? Since only certain normal modes are allowed due to the boundary conditions of the system, we can expect that only certain energies are allowed as well. The total energy stored in a normal mode is

$$U_{mode} \, = \, \left(n_{phonon}\, + \, \frac{1}{2}\right)\hbar \omega_{mode} \, \, , $$

where $\hbar \omega_{mode}$ is the phonon energy and $n_{phonon}$ is called the phonon number and increases with temperature and in whole steps (i.e. $n_{phonon}=0,1,2…$).