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Villhelm
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Homework Statement
Solve for the allowed energy values E of a finite square quantum well of depth U0 = 25eV, width a = 0.5nm that contains an electron of mass m (I'm presuming that m = 9.11*10^-31kg, the question doesn't indicate a specific value to use).
I'm defining the interior potential to be 0eV (so that the walls are +25eV).
I'm also defining the well as extending over x = [0,a]
Homework Equations
Time-independent schrodinger equation in 1 dimension + some basic DE stuff.
The Attempt at a Solution
Starting from the TISE, I end up with the following:
2 * sqrt(E*U0 - E^2) / (2E - U0)
= tan(sqrt(2*m*E)*a/hbar)
Upto there I'd (apparently) done things correctly. However, when I estimated the value(s) of E which satisfy the equation numerically I did not get the results that were expected. I did the estimation using openoffice spreadsheet. The tan function is using radians.
My first thought was to see if my value assumed for the electron mass was wrong, so I tried multiples in the range m' = [0.5m,2m] just to see what happened to the results and noticed that the higher the value of m', the closer the values of E that solved the equation came to those I was indicated to have gotten. I tried m'=10m and found that it pushes the values of E too far, so I started iterating through some values until I ended up settling down to m'=6.28m which struck me as being 2*pi*m.
The RHS of the equation now looks like:
RHS = tan(sqrt(2*(2*pi)*m*E)*a/hbar)
and results in a reproduction of the given energy values (there are five, approximately at {1.123, 4.461, 9.905, 17.162, 24.782}eV).
I don't understand why this works, however, especially when the 2*pi is inside the square root.
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