Particle in a Ring. Wells

1. The wave function of a particle in a ring is $psi(phi,t)=frac{1}{sqrt{2pi}}frac{1}{sqrt{2}}e^{-iphi}e^{ihbar t/2I}-frac{1}{sqrt{2pi}}frac{1}{sqrt{2}}e^{-iphi}e^{i2hbar t/2I}$. Please find the expectation value of the energy.

$psi=frac{1}{sqrt{2pi}}(C_1phi_1+C_2phi_2)$ is a mix of elementary states.

For first state $C_1=1/sqrt{2}$ and $phi_1=exp(-iphi)exp(-ihbar t/2I)$

For second state $C_1=-isqrt{2}$ and $phi_2=exp(iphiexp(-2ihbar t/2I)$

By comparying the elementary states  with the general type $phi=exp(imphi)exp(-iomega t)$

we obtain two energies in the spectrum

$E_1=hbaromega_1=hbar^2/2I$ and $e_2=hbaromega_2=2hbar^2/I=4hbar^2/2I$

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 $psisim e^{-x/d}$.

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 $E_n=hbaromega_n$.