Wavefunctions and energies for CHEM10212 Quantum Mechanics

particle in a box wavefunction

$$\psi\left(x\right)=\sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$$

particle in a box energy

$$E=\frac{n^2h^2}{8mL^2}$$

particle on a ring wavefunction

$$\psi\left(\phi\right)=\frac{1}{\sqrt{2\pi}}e^{i\left(ml\right)\phi}$$

particle on a ring energy

$$E=\frac{\left(ml\right)^2\left(hbar\right)^2}{2I}$$

particle on a sphere energy

$$E=\frac{\left(hbar\right)^2}{2I}l\left(l+1_{}\right)$$

rigid rotor energy

$$E=BJ\left(J+1\right)$$

B

$$B=\frac{h}{8\pi^2Ic}$$

transition frequency rigid rotor

2B(J+1)

harmonic oscillator energy

$$E=\left(v+\frac12\right)\left(hbar\right)\omega$$

omega

$$\omega=\sqrt{\frac{k}{\mu}}$$