"Nikolaus Jewell" <nik.jewell@gmail.com> wrote in message
<g85h4q$1bs$1@fred.mathworks.com>...
> I am currently programming a model of a finite quantum
well and have run
> into a couple of problems. After applying boundary
conditions, I get two
> equations:
>
> tan(theta) = sqrt(theta_knot^2/theta^2  1)
> cot(theta) = sqrt(theta_knot^2/theta^2 1)
>
> The parameters theta and theta_knot have the width of
well, barrier potential,
> and mass of the electron embedded in them.
>
> I tried solving this in two ways:
>
> 1) I arranged loops to solve using the "solve" command for
transcendental
> equation but the cotangent argument was causing a problem.
>
> 2) My other attempt consisted of incrementing theta
between odd intervals of
> pi for tangent and even intervals of pi for cotangent
until upon substitution
> into the above equations they did equal within 0.00000001.
This approach is
> currently giving me some problems. Am on I doing this
correctly or is there a
> better way. A friend suggested using NewtonRaphson method
but I am
> unsure how to apply this because as far as I am aware that
method is for
> finding zeroes and I am interested in the intersections of
these equations so
> that I can derive the possible energy levels. Thanks in
advance.
>
> Nik
>
>
>
Hi, It does not seem likely that you get only trigonometric
functions, because the wave functions in the barrier must
decay, something like exp(kappa*x)
,kappa=sqrt(2*m*(VE)/hbar^2) and in the well
exp(i*k*x),exp(i*k*x), etc. Fitting Psi and derivatives at
x=0 and L gives you a linear problem with 4 unknown
coefficcients. Taking the determinant=0 you get a
oscillating expression weighted with cosh and sinh
contributions as I remember (simplify by using simple if you
have symbolic toolbox).
You should NOT try to arrange things like: cos+k^2sin=0 into
tan(x(E))=1/k^2 because tan is much more difficult to solve
than the original:
f(E)=cos+k^2sin=0 (now more complicated in the finite wall
problem). Just use plot(Evec,fvec) to see graphically where
your zeros are and you get the eigenvalues there.
Good luck,
Per
