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=====Lecture: Reviewing Several Energy Eigenvalues (10 minutes)=====
  * Review the different eigenvalues covered in the previous quantum paradigms
  * Particle in a box
  * Harmonic oscillator
  * Rotating molecules






=====Lecture notes from Dr. Roundy's 2014 course website:=====
I'm going to quickly review and introduce the energy eigenvalues for some simple quantum mechanical problems. For each of the following, I will sketch out the potential, then sketch the wavefunctions and the spacing of the energy levels.

**Particle in a box** {{  courses:lecture:eelec:particle-in-box.png?300|Particle in a Box}}

The first problem you handled was a particle in an infinite square well potential: 
$$\mathcal{H} = \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2}$$
$$E_n = \frac{\hbar^2 \pi^2 n^2}{2m L^2}$$
where $n≥1$. We could solve the same problem in three dimensions, and the three coordinates would separate (i.e. we could use separation of variables), and we would have:
$$\mathcal{H} = \frac{-\hbar^2}{2m}\nabla^2$$
$$= -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} - \frac{\hbar^2}{2m}\frac{\partial^2}{\partial y^2} - \frac{\hbar^2}{2m}\frac{\partial^2}{\partial z^2}$$
$$ E_{n_xn_yn_z} = \frac{\hbar^2 \pi^2 \left(n_x^2 + n_y^2 + n_z^2\right)}{L^2}$$

**Rigid rotor**

The next moderately simple problem is the rigid rotator. In this case, the only energy in the Hamiltonian is the angular kinetic energy:
$$\mathcal{H} = \frac{-\hbar^2}{2I}L^2$$
$$E_{lm} = \frac{\hbar^2 l(l+1)}{2I}$$

**Simple harmonic oscillator** {{  courses:lecture:eelec:simple-harmonic-oscillator.png?300|Simple Harmonic Oscillator}}

Finally, we have a simple problem that hasn't yet come up in the paradigms, which is the simple harmonic oscillator. In this case we have both kinetic //and// potential energy:
$$\mathcal{H} = \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + \frac{m\omega_0^2}{2}x^2$$
$$E_{n} = \left(n + \frac12\right)\hbar \omega_0$$
Of course, you also studied the hydrogen atom, but its solution is less general than those we've listed here. Any diatomic molecule behaves like a rigid rotator and a simple harmonic oscillator, and like a particle in a box, too!








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