Finite Quantum Wells

For each of the potential wells shown in the Figure above, make a qualitative sketch of the wave functions whose allowed energies are indicated. For each energy state, identify the classically allowed and forbidden regions. Discuss the important qualitative features of each state.

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For an infinite quantum well the wave function of the n-th level has the shape:

$Psi(x) =sqrt{2/L}*sin(k_n*x)$   where $k_n =(n*pi)/L$

This means two facts:
– The amplitude of the wave function is inversely proportional to the width of the well

– The standing wave has (n-1) nodes inside the well (that is in the allowed energy region $E_n > V(x)$

The basic shape of the wave functions described above (for infinite wells) are kept inside finite wells with a few differences.

For the given figures we assume the walls are finite in the regions where the energy levels are drawn outside the wells (that is both right walls are finite for all levels, and also the left wall in the second figure for energy level 5 – bottom level).  See the figures.

This means there will be a penetration of the wave functions inside the walls, regions where they will decrease exponentially towards zero value.

The allowed classical regions for each wave is inside the well (where $E_n > V(x)$) The classical forbidden regions are the regions in the walls ($En < V(x)$).  In QM all regions are allowed with the difference that inside the well (between the walls) the wave function is wave like (sinusoidal or equivalent $A*exp(i*k_n*x)$) and in the walls the wave function decreases exponentially with penetration depth ($A*exp(-k_n’*x)$).