# Scaling when solving Schrodinger equation numerically

• dilloncyh
In summary, the conversation discusses solving the Schrodinger equation numerically for simple examples like harmonic potential and square well. The program sometimes gives strange results due to the extremely small values of the constants involved. The speaker sets hbar^2/m = 1 and expresses the length scale in terms of Bohr's radius and all energy (E and V) in terms of eV. They need to find the eigenvalue E that satisfies the equation and then convert all values back to sensible units. The units are commonly defined by distance, mass, and time scales, and in atomic physics, atomic units are used to simplify the calculations.
dilloncyh
I guess this is just a maths problem about algebra. I'm learning to solve Schrodinger equation numerically, and right now I'm just dealing with the simplest examples like harmonic potential, square well, etc. The problem is that sometimes my program gives some strange results and I suspect it is deal to the extremely small values of the constants involved. So let's say I set hbar^2/m = 1, express the length scale in terms of Bohr's radius (5.3e-11 becomes 1) and express all the energy (E and V) in terms of eV. I need to first find the eigenvalue E that satisfies the equation, then I can use the value of E to find the initial conditions which enable me to find solution everywhere. But how do I convert E and all the values of wave function at different points back to some sensible units? I just do not know what the units of the results I get mean.

thanks

How did you "express all energy in terms of eV"?
(It looks a bit like you have over-specified your units.)

Note: If ##\hbar^2/m = 1## then you particle at rest has energy: ##mc^2 = \hbar^2c^2 = 197.33\text{keV}##.

Units are commonly defined by the distance, mass, and time scales.

If you set c=1 then the mass of your particle is the unit of energy. If you put m=1 then your energies will all be in "particle mass-energy units".
If your particle is an electron, say, then multiplying by 511 will get the energy converted to keV.

Doing this has consequences:
If you also set unit-distance to, say, the bohr radius, then the unit of time becomes the duration for light to travel the bohr radius.
That's pretty small - but quantum stuff can happen on small time scales and you are more interested in stationary states anyway.

## 1. How does scaling affect the accuracy of solving the Schrodinger equation numerically?

Scaling can significantly impact the accuracy of numerical solutions to the Schrodinger equation. If the scaling is not chosen appropriately, it can lead to errors in the results. This is because the scaling affects the size of the numerical grid used to solve the equation, and if the grid is too coarse or too fine, it can result in inaccurate solutions.

## 2. What is the purpose of scaling when solving the Schrodinger equation numerically?

The purpose of scaling is to transform the original Schrodinger equation into a dimensionless form, making it easier to solve numerically. This is achieved by choosing appropriate units and scaling parameters that eliminate any unnecessary physical constants from the equation.

## 3. How do you determine the appropriate scaling parameters for solving the Schrodinger equation numerically?

The appropriate scaling parameters can be determined by analyzing the physical system and understanding the dominant terms in the Schrodinger equation. These parameters should be chosen such that the resulting dimensionless equation is well-behaved and can be solved accurately using numerical methods.

## 4. What are the common methods for scaling when solving the Schrodinger equation numerically?

The most commonly used methods for scaling when solving the Schrodinger equation numerically are the dimensionless scaling method, the natural units scaling method, and the atomic units scaling method. Each of these methods involves choosing appropriate units and scaling parameters to eliminate physical constants from the equation.

## 5. Can scaling be applied to all types of systems when solving the Schrodinger equation numerically?

Yes, scaling can be applied to all types of systems when solving the Schrodinger equation numerically. However, the choice of scaling parameters may vary depending on the specific physical system being studied. It is important to carefully analyze the system and choose appropriate scaling parameters for accurate results.

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