##  [Potential Step](/potential-step-0) 

 Definition

A canonical model concept defining a standard Hamiltonian or potential used to illustrate and solve quantum behavior. It specifies idealized conditions that allow analytic solutions or controlled approximations for spectra and dynamics. It does not capture all real-world effects and typically omits interactions, dissipation, or complex geometry unless explicitly added. It provides reference solutions that calibrate intuition and benchmark numerical methods and experimental interpretation. The concept is generally stable, though extensions and solution techniques evolve over time.



 

 

 

 

 

 





## Principle

Principle

Continuity of the wavefunction and its derivative (or current continuity) across the discontinuity determines reflection and transmission amplitudes; for particle energy above the step there is partial transmission and reflection, while for energy below the step the transmitted part is an evanescent decaying wave in the classically forbidden region.

 

 

 

 

 





## Demonstration

Demonstration

A particle of energy E incident from the left on a step of height V0: for E>V0 transmitted waves propagate in the right region with reduced wavenumber and reflection coefficient R<1; for E

 

 

 

 

## Misapplication

Misapplication

Treating the step as if classical (full reflection for EV0) ignores quantum reflection and evanescent penetration; or using step formulas where the potential actually varies smoothly can misestimate reflection coefficients and phase shifts.

 

 

 

 

 





## Consequence

Consequence

Predictable partial reflection and transmission with energy‑dependent coefficients, phase shifts on transmission and reflection, and nonzero probability density in classically forbidden regions — key ingredients in build­ing more complex scattering, barrier, and resonance phenomena.

 

 

 

 

## Reversal

Reversal

Replacing the abrupt discontinuity by a smooth ramp or extended barrier changes scattering: adiabatic variations reduce reflection and modify phase accumulation, while extended barriers introduce resonant transmission and multiple‑reflection interference not captured by a single step.

 

 

 

 

 





## Boundary

Boundary

Model assumes a single sharp, one-dimensional discontinuity, nonrelativistic particle and stationary scattering; excludes smooth spatially varying potentials, inelastic channels, many‑body interactions and time‑dependent step movements unless explicitly modeled.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Closely related to the finite square well and potential barrier problems; the step isolates the simplest scattering discontinuity but can be confused with a finite barrier (two steps) or with smooth interfaces where WKB or numerical methods are required.

 

 

 

 

 





## Synthesis

Synthesis

The potential step is the elementary scattering element: an abrupt change in potential where continuity conditions yield reflection and transmission amplitudes that depend on energy, produce evanescent penetration below the step and partial transmission above it, and serve as building blocks for understanding tunneling and resonance in more complex potentials.