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I am trying to solve an equation of the form as follows

$\left(\begin{array}{cc} -\frac{\hbar^{2}}{2m}\frac{\delta^{2}}{\delta z^{2}}+\sin^{2}\left(z\right) & z\\ z & \frac{\hbar^{2}}{2m}\frac{\delta^{2}}{\delta z^{2}}-\sin^{2}\left(z\right) \end{array}\right)\left(\begin{array}{c} u_{n}(z)\\ v_{n}(z) \end{array}\right)==\epsilon_{n}\left(\begin{array}{c} u_{n}(z)\\ v_{n}(z) \end{array}\right)$

Assuming required no. of boundary conditions but with no knowledge of the eigenvalues, please tell how to solve such a system of equations in Mathematica. Here $n$ is an index for the eigenfunction and the corresponding eigenvalue. Take $z$ as real and assume that $z$ runs on a finite lattice of fixed size of length $L=Na$ where $N$ is the number of sites on the lattice and a is the lattice constant. If you take $a==1$, take $N$ of the order of $10000$. There is a specified value of the eigenfunctions at the boundary of the lattice which you can take zero for the simplest case. Please multiply everything by units such that dimensions are consistent- Here the derivative term is the kinetic energy term while sin2z term is the potential term.

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closed as off-topic by Jens, RunnyKine, Yves Klett, Kuba, MarcoB Apr 14 at 12:00

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question cannot be answered without additional information. Questions on problems in code must describe the specific problem and include valid code to reproduce it. Any data used for programming examples should be embedded in the question or code to generate the (fake) data must be included." – Jens, RunnyKine, Yves Klett, Kuba, MarcoB
If this question can be reworded to fit the rules in the help center, please edit the question.

Is this a Mathematica related question? If so do you have any Mathematica code. It would save the time to type the equation.... – user21 Jun 20 '13 at 7:03
I want to know how to solve this in Mathematica. There are algorithms to solve the coupled eigenvalue differential equations but I have not been able to find a Mathematica command(s) that directly utilizes them for solving such a set of equations. And that or a combination of commands to help solve this, is what I am looking for. – cleanplay Jun 20 '13 at 10:57
What kind of solution is expected? A closed form may be not attainable. Where seems to be no a ready procedure to find eigenvalues numerically, so probably an implementation of an algorithm you're mentioning would be needed. – Andrew Jun 20 '13 at 11:41
Your units are dimensionally inconsistent. The derivative term has units of $s^{-1}$, but $\sin^2(z)$ is unitless. Also, what are the boundary conditions? They affect the eigenvalues of the system. Also, as this appears to be a Hamiltonian, is $z$ real? If not, shouldn't the matrix be Hermitian? – rcollyer Jun 20 '13 at 13:24
@rcollyer I somtimes find in my own research that $2=1$ :D – sebhofer Jun 20 '13 at 15:26

Sorry, I formulate my comments here, because there is not enough space in the comment section.

1) The dimension question cannot be ignored. The original Schrödinger equation should look like

$-\frac{h^2}{2m} \left(\frac{d}{\text{dx}}\right)^2 u(x)+A \sin ^2\left(\frac{2 \pi x}{L}\right) u(x)+B x v(x)=a u(x)$

Letting $x \to q z$ and $u \to p U, v \to p V$ the equation becomes

$-\frac{h^2 p}{2m q^2} \left(\frac{d}{\text{dz}}\right)^2 U(z)+A p \sin ^2\left(\frac{2 \pi q z}{L}\right) U(z)+\left(B p q z\right) V(z)=\left(a p\right)U(z)$

Now we can set two factors equal to 1 letting $q =\left (\frac{h^2}{2 m B} \right)^\frac {1}{3}$ and $p =\left (\frac{2m}{h^2 B^2} \right)^\frac {1}{3}$ resulting in

$- \left(\frac{d}{\text{dz}}\right)^2 U(z)+S \sin ^2\left(R z\right) U(z)+V(z)=T U(z)$

an equation with two parameters S and R and eigenvalue T. The second equation has a similar form.

2) The system seems to describe the motion of a charged particle in a periodic potential in a constant electric field. (I would be helpful if you describe the phsical problem). The equations are hence a mixture between Mathieu- and Airy differential equations.

3) Both are tricky when it comes to Eigenvalues. Airy is easier, it corresponds to the motion of a charged particle in a constant electric field; the eigenvalues are related to the zeroes of the Airy-function. Mathieu (coming from the Sin^2-terms) is much more complicated. It leads, roughly speaking, to energy "bands". You might wish to consult a book on quantum mechanics.

4) It's always a good idea to simplify the equations. Here it means neglecting first the z-Term -> Mathieu, and then the sin^2-term -> Airy, and study the two areas separately.

5) Once you have clearly formulated your physical problem you can go and solve the system in MMA numerically. I have done it with simple boundary conditions like {u,v}[0]==0 and {u',v'}[0] = {1,1} and varying a and b. No problem but of no use without a proper question.

Best regards, Wolfgang

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