Schrodinger equation in cylindrical coordinates.

In summary, the conversation discusses writing the Schrodinger equation in cylindrical coordinates with a total potential consisting of Woods-Saxon, spin-orbit, and Coulomb potentials. The Schrodinger equation can be written in a specific form and the person is looking for ideas on how to write the total potential explicitly, including the use of the cylindrical coordinate variable ##r^2 = \rho^2+z^2##.
  • #1
BRN
108
10
Hi guys!
For nuclear case, I need to write an Schrodinger equation in cylindrical coordinates with an total potential formed by Woods-Saxon potential, spin-orbit potential and the Coulomb potential.

Schrodinger equation can be written in this form:

$$[-\frac{\hbar^2}{2m}(\frac{\partial ^2}{\partial r^2}+\frac{1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial \varphi^2 }+\frac{\partial^2}{\partial z^2})+V_{tot}]\Psi (r, \varphi, z)=E \Psi (r, \varphi, z)$$


How can I write explicitly ##V_{tot}##?

Someone could give me some ideas on how to write Woods-Saxon, spin-orbit, Coulomb potentials in cylindrical coordinates?

Thanks!
 
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  • #2
[itex] r^2 = \rho^2+z^2[/itex] get the idea?
 

Related to Schrodinger equation in cylindrical coordinates.

1. What is the Schrodinger equation in cylindrical coordinates?

The Schrodinger equation in cylindrical coordinates is a mathematical formula that describes the behavior of a quantum mechanical system in three dimensions, specifically in cylindrical coordinates (r, θ, z). It is a partial differential equation that relates the wave function, which represents the state of a quantum system, to its energy.

2. How is the Schrodinger equation in cylindrical coordinates different from the one in Cartesian coordinates?

The Schrodinger equation in cylindrical coordinates is different from the one in Cartesian coordinates because it takes into account the cylindrical symmetry of the system. This means that the equation includes terms that account for the radial and angular dependence of the wave function, in addition to the usual time and spatial dependence. In contrast, the Schrodinger equation in Cartesian coordinates only considers the spatial dependence of the wave function.

3. What are the advantages of using the Schrodinger equation in cylindrical coordinates?

Using the Schrodinger equation in cylindrical coordinates can simplify the mathematical description of certain physical systems, especially those with cylindrical symmetry. This can make it easier to solve the equation and obtain analytical solutions. Additionally, the cylindrical coordinates are often more suited to describe the behavior of particles moving in circular or cylindrical potentials, such as in atomic and molecular systems.

4. Can the Schrodinger equation in cylindrical coordinates be applied to any quantum system?

Yes, the Schrodinger equation in cylindrical coordinates can be applied to any quantum system. However, it is most commonly used for systems with cylindrical symmetry, such as atoms, molecules, and particles moving in circular potentials. For systems without cylindrical symmetry, it may be more appropriate to use the Schrodinger equation in Cartesian coordinates.

5. What are some real-life applications of the Schrodinger equation in cylindrical coordinates?

The Schrodinger equation in cylindrical coordinates has many applications in quantum mechanics and has led to important discoveries in physics, chemistry, and materials science. Some examples include using it to study the behavior of electrons in cylindrical nanowires, predicting the energy levels of atoms in magnetic fields, and calculating the wave function of particles in a vortex. It is also used in developing quantum technologies such as quantum computing and quantum cryptography.

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