Suppose the potential function is a step function. The equation that gets satisfied is

When the potential function is a step function, the equations that describe the system’s behavior depend on the context in which the step function is applied. However, in quantum mechanics, a common scenario where a step potential is considered is to describe particles encountering a sudden change in potential energy, often used as a simplified model for barriers or wells.

In such a context, the Schrödinger equation is the foundational equation that gets satisfied. The Schrödinger equation describes how the quantum state of a physical system changes over time. For a particle of mass (m) encountering a step potential, the equation is typically split into regions, one for each side of the step.

The step potential can be described as follows:

[ V(x) = begin{cases}

V_0 & text{for } x > 0 \

0 & text{for } x < 0

end{cases} ]

Where (V_0) is the height of the step.

For each region, the time-independent Schrödinger equation applies:

[ -frac{hbar^2}{2m}frac{d^2psi(x)}{dx^2} + V(x)psi(x) = Epsi(x) ]

Here, (psi(x)) is the wave function of the particle, (E) is the energy of the particle, (V(x)) is the potential energy as a function of position (