Difference Equation: Explained & Derived - QM1

[Domnu]

In the following link: http://electron6.phys.utk.edu/QM1/modules/m2/square_potentials.htm what does the difference equation mean, and how is it gotten from the previous step?

 

[George Jones]

Rearranged slightly, the equation above it is

$$\frac{\partial^2 \phi_\epsilon \left( x \right)}{\partial x} = \frac{2m}{\hbar^2} \left( V_\epsilon \left( x \right) - E \right) \phi_\epsilon \left( x \right).$$

What happens when you integrate both sides of this equation from $x = x_1 - \epsilon$ to $x = x_1 + \epsilon$?

 

[denian]

The difference equation in the provided link refers to the Schrödinger equation, which is a fundamental equation in quantum mechanics. It describes the behavior of a quantum system, such as an electron, in a given potential energy field.

The difference equation is derived from the previous step, which is the time-independent Schrödinger equation. This equation describes the energy states of a quantum system in a potential energy field. The difference equation is obtained by discretizing the time-independent Schrödinger equation, which means approximating it by dividing the continuous time interval into smaller discrete steps. This allows us to solve the equation numerically and obtain the energy eigenvalues and eigenfunctions of the system.

In simpler terms, the difference equation is a mathematical representation of the Schrödinger equation that allows us to calculate the energy levels and wavefunctions of a quantum system in a potential energy field. It is an important tool in understanding and predicting the behavior of quantum systems, and it is used extensively in various fields such as physics, chemistry, and engineering.