Finite Square Well Potential Equation, Derivation & Rearrangements

Core idea

Overview

The potential is zero inside the active region and finite outside, which is the simplest model that still produces tunneling and bound states.

When to use: Use this when the wavefunction must be matched across a finite barrier or finite well.

Why it matters: The tunneling picture explains why wavefunctions oscillate in allowed regions and decay exponentially in forbidden regions.

Free formulas

Rearrangements

Solve for $_{s} t a t u s$

b l oc k e d_{n} o n_{i} n v er t ib l e

Solve for reason

T h ee q u a t i o n d e f in es a p i ece w i se p o t e n t ia l f u n c t i o nV (x) ma pp in g p os i t i o n t oe n er g y v a l u es . I t i s a d e f ini t i o n r a t h er t hana s t an d a r d a l g e b r ai ce q u a l i t y t ha t c anb er e a r r an g e d t oso l v e f or in t er na l p a r am e t er s l ik e L or V ina g e n er i cse n se, a s t h ese a r e f i x e d p a r am e t er so f t h e p o t e n t ia l p r o f i l e .

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

Contains advanced layout notation (matrix/cases)

Why it behaves this way

Intuition

Imagine a flat-bottomed trench of width L dug into a level landscape. Inside the trench (from 0 to L), the floor is perfectly flat and set at a baseline height of zero. Outside the trench, the ground level abruptly jumps up to a constant height V. This creates a 'well' where a particle can be trapped; unlike a well with infinitely high walls, a particle here has a mathematical probability of existing slightly inside the walls even if it doesn't have enough energy to climb out.

V(x)

Potential Energy Profile

The 'map' of potential energy at every point along the x-axis, showing where the particle is free to move and where it encounters a barrier.

V

Barrier Height

The constant energy level of the regions outside the well. It represents the 'strength' or 'height' of the walls that the particle must overcome to escape.

x

Position

The horizontal coordinate representing the particle's location in one-dimensional space.

L

Well Width

The distance between the two walls, defining the size of the 'free' region where the potential energy is zero.

Signs and relationships

(0, L): This interval defines the interior of the well. Within these bounds, the potential is set to 0, meaning no external forces act on the particle here.
? (Union symbol): This indicates that the potential V applies to both the region to the far left (negative infinity to 0) and the region to the far right (L to infinity) simultaneously.

Free study cues

Insight

Canonical usage

The finite square well potential defines regions of constant potential energy, which are typically expressed in units of energy (e.g., Joules, electron-volts) or as a potential difference (e.g., Volts).

Common confusion

Students may confuse the potential energy V with electric potential (Volts) if not explicitly stated, or incorrectly assume the potential is dimensionless.

Dimension note

While the potential energy V and length L have physical units, the solutions to the Schrödinger equation for this potential often involve dimensionless parameters derived from these quantities and fundamental constants

Unit systems

$V$ J | eV | V · Represents the height of the potential barrier or the depth of the potential well. Commonly expressed in electron-volts (eV) in atomic and solid-state physics.

$L$ m | nm · Represents the width of the finite potential well or barrier. Often expressed in nanometers (nm) for atomic-scale systems.

One free problem

Practice Problem

In the region (0, L), what is the general form of the wavefunction if the total energy E is greater than zero but less than V?

contextbound state region

Solve for: $V(x)

Hint: Consider the sign of (E - V) inside the well where the potential is zero.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In setting up a finite well before solving for bound-state energies, Finite Square Well Potential is used to calculate $V(x) from the measured values. The result matters because it helps size components, compare operating conditions, or check a design margin.

Study smarter

Tips

Inside the classically allowed region you get sine and cosine solutions.
In the forbidden region the physically acceptable branch is exponential decay.
Barrier width matters exponentially, not linearly.
The same piecewise potential can describe either a well or a barrier depending on the energy reference.

Avoid these traps

Common Mistakes

Using an oscillatory solution where the energy is below the barrier.
Forgetting to match both the wavefunction and its derivative at the boundaries.
Underestimating how quickly the tunneling signal drops with barrier width.
Assuming an infinite wall when the model is explicitly finite.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Use this when the wavefunction must be matched across a finite barrier or finite well.

The tunneling picture explains why wavefunctions oscillate in allowed regions and decay exponentially in forbidden regions.

Using an oscillatory solution where the energy is below the barrier. Forgetting to match both the wavefunction and its derivative at the boundaries. Underestimating how quickly the tunneling signal drops with barrier width. Assuming an infinite wall when the model is explicitly finite.

In setting up a finite well before solving for bound-state energies, Finite Square Well Potential is used to calculate $V(x) from the measured values. The result matters because it helps size components, compare operating conditions, or check a design margin.

Inside the classically allowed region you get sine and cosine solutions. In the forbidden region the physically acceptable branch is exponential decay. Barrier width matters exponentially, not linearly. The same piecewise potential can describe either a well or a barrier depending on the energy reference.

References

Sources

Engineering LibreTexts, finite square well and tunneling-barrier notes, accessed 2026-04-09
Peverati, The Live Textbook of Physical Chemistry 2, quantum weirdness/tunneling section, accessed 2026-04-09
Engineering LibreTexts, field enhanced emission and tunnelling effects, accessed 2026-04-09
Griffiths, D. J. (2018). Introduction to Quantum Mechanics.
Liboff, R. L. (2002). Introductory Quantum Mechanics.
Griffiths, David J. (2018). Introduction to Quantum Mechanics (3rd ed.). Cambridge University Press.
Landau, L. D., & Lifshitz, E. M. (1977). Quantum Mechanics: Non-Relativistic Theory (Vol. 3, 3rd ed.). Pergamon Press.
NIST CODATA Value of the Planck constant

Finite Square Well Potential