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Math Help - System of Equations

  1. #1
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    System of Equations

    I have a system of equations that I think should be rather simple to solve, but I can't seem to get to the answer:

    f(r) = A Sin(j*r)
    g(r) = B e^(k*r) + C e^(-k*r)
    h(r) = D e^(-l*r)

    I have 4 boundary conditions:

    f(r)=g(r) at r=a
    f'(r)=g'(r) at r=a
    g(r)=h(r) at r=a+p
    g'(r) =h'(r) at r=a+p

    j, k, and l are functions of E that I want to find the roots of. p and a are constants, but I want to be able to solve these equations for any value of a and p

    With this information I should be able to eliminate A, B, C, and D and be able to calculate the values of E which give the zeros.

    I can simplify the system to:

    -j*((( B e^(k*a) + C e^(-k*a))/( B e^(k*a) + C e^(-k*a)))*(cot j*a) = l*(( B e^(k*(a+p)) + C e^(-k*(a+p))/( B e^(k*(a+p)) + C e^(-k*(a+p))

    but I can't seem to get rid of B and C. When I look at it written out it looks like I should easily be able to remove B and C, but it never works. I feel like there is some rational or exponential rule that I can't remember that would help me out.

    Thanks a lot and sorry if the notation is confusing.
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  2. #2
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    Two Questions:

    1. What is the underlying DE you're trying to solve?

    2. Do you know how j, k, and l depend on E?
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  3. #3
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    I'm not entirely sure how to answer 1, but I'll give it a shot; This problem is the particle in a sphere from quantum mechanics, so I think the answer is the Schroedinger equation: Hψ=Eψ or more explicity: iħdēψ(r)/drē + V(r) = Eψ(r). The solutions are described piecewise according to the equations above where a is the radius of the sphere and p is the length of an insulating shell. The wavefunction decays at a faster rate in the third, infinite layer. The general solution is slightly more complicated, in radial coordinates, and involves spherical harmonics, but I'm only interested in the answer for L=0, so this one dimensional solution is sufficient.

    and 2) j, k, l = sqrt((2*m*(E-V))/ħē)). E is the energy of the system, V is the potential - different for each layer of the sphere, thus j, k, and l, m is the mass of the particle, and ħ is the reduced planck constant. I simplified the equations by using a constant to lump all these together. I don't ever differentiate wrt E, so I thought that was appropriate. I do need the roots in E though.
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  4. #4
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    So are you using the

    V_{\text{eff}}(r)=V(r)+\dfrac{\hbar^{2}l(l+1)}{2m_  {0}r^{2}}?

    Thus, your Schrodinger equation becomes

    -\dfrac{\hbar^{2}}{2m_{0}}\,\dfrac{d^{2}\psi(r)}{dr  ^{2}}+V_{\text{eff}}(r)\,\psi(r)=E\,\psi(r).

    And, of course, you need continuity of the wave function and its first derivative. What is your potential energy function?

    I'm asking these questions because I'm not sure you've solved the DE correctly, so I'm just checking. Hope you don't mind.
    Last edited by Ackbeet; February 22nd 2011 at 01:08 PM. Reason: Psi over u.
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  5. #5
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    I don't mind at all. I took the solutions from a publication about 7 years old. I just assumed that they were correct. In my experience, the solutions are usually linear combinations of sines, cosines, and exponential decays. The exact solutions also have an E cos (jr) term in f(r) and an F e^(l*r) in h(r), but based on other conditions E and F must be 0.

    It seemed simpler to solve the system with 4 eqns and 4 unknowns than the 6 eqns and 6 unknowns especially considering I know two of the unknowns and the equations that describe them. Is that not correct?

    I'm just using a constant value for the potential step - a finite square well.

    Thanks for your help.
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  6. #6
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    So your potential looks like this, then:

    V(r)=\begin{cases}0,\quad&0\le r<a\\<br />
-V_{\text{mid}},\quad&a\le r<a+p\\<br />
0,\quad&a+p\le r\end{cases}.

    Is that correct? Here V_{\text{mid}}>0.

    Another question: are you looking at bound states (-V_{\text{mid}}<E<0) or at scattering states (0<E)?
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  7. #7
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    That potential is almost correct. For a+p\leq r \rightarrow V = V2 and V2>V1 (or mid as you have scripted)>0.

    These should all be bound states.
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