Particle in a Ring. Wells

1. The wave function of a particle in a ring is $\psi(\phi,t)=\frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{2}}e^{-i\phi}e^{i\hbar t/2I}-\frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{2}}e^{-i\phi}e^{i2\hbar t/2I}$. Please find the expectation value of the energy.

$\psi=\frac{1}{\sqrt{2\pi}}(C_1\phi_1+C_2\phi_2)$ is a mix of elementary states.
For first state $C_1=1/\sqrt{2}$ and $\phi_1=\exp(-i\phi)\exp(-i\hbar t/2I)$
For second state $C_1=-i\sqrt{2}$ and $\phi_2=\exp(i\phi\exp(-2i\hbar t/2I)$

By comparying the elementary states  with the general type $\phi=\exp(im\phi)\exp(-i\omega t)$
we obtain two energies in the spectrum
$E_1=\hbar\omega_1=\hbar^2/2I$ and $e_2=\hbar\omega_2=2\hbar^2/I=4\hbar^2/2I$
For the coefficients we have $|C_1|^2+|C_2|^2=1$  so that the expectation energy is

Rank for the penetration distance $d$ of the wavefunction for the energy levels in the figure. The penetration distance is defined from $\psi\sim e^{-x/d}$.

Three Wells

The decay of the wavefunction in the well wall is of the type $\psi \sim \exp (-kx)$ so that $d=1/k=\frac{\hbar}{\sqrt{2m(U-E)}}$ and thus the penetration depth depends only on difference \sqrt{U-E}.
For wells in figures a) and b) the energy difference is the same $U-E=5 eV$. therefore the penetration depth is the same in a) and b) cases. For case c) $U-E=16-10=6 eV >5eV$ so that the penetration depth is in this case smaller than in the first two cases (a and b).

Specify the features of the wavefunctions for the indicated energies.

One Well

-The energy level n has  n-1 nodes inside the well.
-The amplitude of a wave inside a well is proportional to the $\sqrt{1/L}$ where $L$ is the width of the well. So lower energy levels will have a bigger amplitude in the case given.
-The penetration depth is inversely proportional to the difference $\sqrt{U-E}$ so that the lowest energy level will have a smaller penetration depth into the wall.
-All wavefunctions have a a zero at the left wall (it is infinite).
The frequency of the wavefunction is increasing with the energy level, according to the relation $E_n=\hbar\omega_n$.


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