Solving using schrodinger equation techniques

In summary, the person is asking for help with a problem they have and asks if anyone has had the same problem and solved it. They also mention that the problem may be examined as a 2D schrodinger equation, though the eigenvalue for k1 is known. They ask for help with finding the solutions and mention that a better program is COMSOL multiphysics.
  • #1
amnon_willi
1
0
Hello,
I have problem I wish to solve, and I wonder if anyone already delt with it when solving the schrodinger 2D equation.

say E(x,y) is a scalar field function that complies with

( [tex]\frac{d}{dx}[/tex]2+[tex]\frac{d}{dy}[/tex]2 ) *E(x,y)+k(x,y)*E(x,y)=k1*E(x,y)


where k(x,y)={k2 for x2+y2<R2 and 0 otherwise}, i.e. a tube potential.
All is known but E(x,y).
I think it can be examined as a 2D schrodinger equation, even thow the eigenvalue k1 is known.

How can I get to start finding the solutions of this equation?
Can I expect to know how many are there? - one, two, many?
 
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  • #2
At least in 3D, a rotationally invariant system is best dealt with in spherical coordinates, expanding in eigenfunctions of angular momentum and then solving the radial equation, which will be an ODE. In 2D, the "spherical harmonics" are just the functions [itex]e^{im \phi}[/itex] where m is an integer. So try writing [itex]f_m(r,\phi) = u_m(r) e^{i m \phi} [/itex], which should give you an ordinary differential equation for [itex]u_m(r)[/itex]. Then the general solution will be a linear combination of the [itex]f_m[/itex], ie, you should expect a linearly independent solution for each integer m.

This is a little messier in 2D than in 3D, and the equation will have a first derivative term that doesn't appear in the ordinary schrodinger equation. In fact, the solutions [itex]u_m(r,\phi)[/itex] are probably going to be Bessel functions.
 
  • #3
Numerically you could use pdetool within MATLAB (type: pdetool at the MATLAB prompt), where you could solve this eigen value problem in 2D (Its not that you know or could specify k1, you will get it from the numeric solution). Other better program is COMSOL multiphysics.

This problem looks like cylinder symmetry, but depend on weather k(x,y)=f(r) or not. In that case you could obtain an effective 1D eigen value problem in Psi(r), using the angular quantum quantum number m as specified in the other post here.
 

1. What is the Schrodinger equation and how is it used in solving scientific problems?

The Schrodinger equation is a mathematical equation that describes the behavior of quantum particles. It is used in quantum mechanics to calculate the probability of finding a particle in a specific location at a specific time. This equation is essential for understanding the behavior of atoms, molecules, and other quantum systems.

2. What are some common techniques used to solve problems using the Schrodinger equation?

Some common techniques used to solve problems using the Schrodinger equation include analytical methods, such as separation of variables and perturbation theory, and numerical methods, such as the finite difference method and the variational method.

3. How does the Schrodinger equation differ from classical equations of motion?

The Schrodinger equation differs from classical equations of motion in that it describes the behavior of quantum particles, which exhibit wave-like properties and do not follow the same rules as classical particles. The Schrodinger equation takes into account the probabilistic nature of quantum mechanics, whereas classical equations of motion are deterministic.

4. Can the Schrodinger equation be applied to all types of particles?

Yes, the Schrodinger equation can be applied to all types of particles, including electrons, protons, neutrons, and even larger particles such as atoms and molecules. It is a fundamental equation in quantum mechanics and is used to study the behavior of all quantum systems.

5. Are there any limitations or drawbacks to using the Schrodinger equation?

One limitation of the Schrodinger equation is that it does not take into account the effects of relativity. This is why it is not applicable to particles traveling at very high speeds, such as particles in a particle accelerator. Additionally, the Schrodinger equation can only be used for systems with a finite number of particles, so it is not suitable for describing systems with a large number of particles, such as solids and liquids.

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