Quantum Tunneling Probability Calculator for a Rectangular Barrier

Introduction to quantum tunneling through a rectangular barrier

This rectangular-barrier quantum tunneling calculator estimates how likely a particle is to cross a one-dimensional barrier when its mass, energy, height, and width are known. The result is most useful when you want to see how quickly transmission changes as the barrier gets wider, the barrier gets taller, or the particle gets heavier.

Because tunneling probabilities can change by orders of magnitude for small shifts in width or mass, the most important part of using the calculator is entering a scenario that actually matches the system you want to study. A clean set of inputs makes the result easier to interpret and much easier to compare against a second barrier design.

The sections below explain what the rectangular-barrier model is doing, how to choose realistic values, how the transmission formula changes with the energy regime, and which assumptions matter most when you read the result.

What this rectangular-barrier calculator computes

The question behind this rectangular barrier calculator is whether a particle with a given mass and energy is expected to transmit through the barrier or be reflected by it. The output gives a probability for the specific barrier shape assumed here, so it is ideal for comparing how a thinner barrier, a taller barrier, or a different particle mass changes the answer.

If you are setting up a new scenario, state it in plain language first: “How transparent is this barrier to this particle?”, “What happens if the barrier is narrowed?”, or “Does the energy sit below or above V₀?” Framing the problem that way makes it easier to pick numbers that belong together and to spot input mistakes before they distort the probability.

How to use this quantum tunneling probability calculator

This rectangular-barrier tunneling calculator is easiest to use when the mass, energy, barrier height, and barrier width all describe the same physical setup. Enter one complete scenario, read the probability, and then change only one input at a time if you want to see which parameter is driving the change.

1. Enter Particle mass (multiples of electron mass m_e) with the unit shown beside the field.
2. Enter Particle energy (eV) with the unit shown beside the field.
3. Enter Barrier height V ₀ (eV) with the unit shown beside the field.
4. Enter Barrier width (nm) with the unit shown beside the field.
5. Run the calculation to refresh the tunneling probability.
6. Compare the output with a second barrier setup if you want to see how the result shifts.

If you are comparing two candidate barriers, keep a note of the values you used so the transmission estimate can be reproduced later without guessing which width or energy was entered. When you change only one input at a time, it becomes much easier to tell whether the barrier, the mass, or the particle energy is responsible for the new result.

Choosing mass, energy, height, and width for rectangular tunneling

Choosing values for this rectangular barrier calculator is mostly about unit consistency and physical plausibility. The model depends strongly on whether the particle energy sits below or above the barrier height, and the width matters enough that even small changes can dominate the result. Use the checklist below to keep the scenario internally consistent.

• Units: confirm the unit shown beside each field and keep the particle and barrier quantities in the same system before entering them.
• Ranges: if a value feels extreme, test a more modest case first and then expand outward to see how sharply the probability responds.
• Defaults: the prefilled numbers are only starting points; replace them with values from your own tunneling scenario before drawing conclusions.
• Consistency: make sure the particle energy, barrier height, and barrier width are describing the same barrier and not a mix of unrelated examples.

Typical inputs for a rectangular-barrier tunneling estimate usually fall into four categories:

• Particle mass (multiples of electron mass m_e): the mass of the particle whose wavefunction is trying to cross the barrier.
• Particle energy (eV): the incident kinetic energy in eV.
• Barrier height V₀ (eV): the potential energy step that defines how hard the barrier is to penetrate.
• Barrier width (nm): the physical thickness of the barrier in nanometers.

If a value is uncertain, it is often better to run a low and a high estimate. Tunneling probability can change dramatically with small changes in mass or width, so a bracketed range is more informative than a single overconfident number. The same is true when the barrier height is only approximately known: the output is often more sensitive to a modest shift in height than a casual estimate suggests.

How the rectangular-barrier transmission formula is evaluated

This rectangular-barrier calculator follows the standard one-dimensional tunneling workflow: convert the inputs into consistent units, evaluate the branch that matches the energy regime, and report the transmission probability. When the particle energy sits below the barrier height, the result is governed by an evanescent decay across the barrier; when the energy reaches or exceeds the barrier, the calculator switches to the above-barrier transmission expression.

For the rectangular-barrier model, the decay constant is written in terms of the mass and the energy gap:

For a sub-barrier case, the transmission probability falls off through the hyperbolic sine term, so width and mass both work against transmission while higher energy works in its favor:

For energies at or above the barrier top, the calculator uses the corresponding oscillatory branch with sine instead of sinh. That is why the output can change shape abruptly once the particle energy crosses V₀.

In this context, the barrier strength parameter is a compact way to track how the mass, barrier height, and barrier width combine. If the probability is not changing the way you expect, revisit the units first and then the barrier assumptions. The barrier strength is not itself a probability; it is a useful intermediate quantity that grows when the barrier gets thicker or when the particle decays more strongly inside the barrier.

Worked example: a sub-barrier rectangular barrier

This rectangular-barrier worked example is best understood as a direction check rather than a number check. If the particle energy sits well below the barrier height, the result will usually be tiny and the barrier width will matter more than almost anything else. If the energy creeps toward V₀, the probability rises rapidly, and once the energy reaches the barrier top the calculator returns the near-match branch instead of an exponentially small tunneling value.

That makes the example useful for reasoning about model behavior: a narrower barrier should always be easier to cross, a heavier particle should always be harder to transmit, and a taller barrier should always suppress transmission. If any of those directions seem backward, the most likely cause is a units mismatch or a value entered into the wrong field. In a real sub-barrier case, the exponential sensitivity to width is usually the clearest signal, so even a modest change in thickness can outweigh a smaller change in energy.

So instead of checking a fake sum, use the example to verify whether the output changes in the direction quantum mechanics predicts for the barrier you are studying. A good sanity check is to ask whether the barrier is tall enough and thick enough to matter for the particle you chose; if it is not, the result should not look like a strong tunneling suppression in the first place.

Quantum tunneling sensitivity: which rectangular-barrier input changes transmission fastest?

This rectangular-barrier sensitivity section shows which input moves the transmission estimate the most when you hold the other values fixed. In practice, barrier width usually dominates because it sits inside the decay length, while particle mass reshapes the decay constant and particle energy decides whether you are far below the barrier or nearly at the top.

• Thinner barrier: transmission rises, sometimes sharply.
• Heavier particle: transmission falls because the wave decays more quickly inside the barrier.
• Higher particle energy: transmission rises and may switch the formula branch once the energy reaches the barrier height.

Those directional checks are more useful than a placeholder conservative-or-aggressive table because they map directly onto the calculator's real branches. If a tiny change in width makes the probability leap or collapse, that is a feature of the tunneling model, not a bug in the calculator. In many realistic scenarios, width is the fastest lever to test first, followed by energy, with mass acting as a slower but still important control on the decay constant.

How to read the rectangular-barrier probability

In this rectangular-barrier calculator, the reported number is the estimated probability that the particle makes it through the barrier under the model’s assumptions. Values close to 1 mean a transparent or nearly transparent barrier for the chosen inputs; values close to 0 mean strong suppression and a mostly reflecting barrier.

When the particle energy matches the barrier height, the calculator uses its near-match case and transmission approaches unity. That is a useful reminder that the output is tied to the energy regime as well as the barrier geometry.

If you need a record of several barrier cases, copy the inputs and the reported probability into your notes or report so you can compare them later. Writing the mass, energy, height, width, and transmission side by side is usually more useful than trying to reconstruct the scenario from memory after the fact. It also makes it easier to notice whether two cases were truly different or whether only one entry changed by accident.

It can also help to read the number qualitatively before worrying about the exact digits. A probability that is many orders of magnitude below one usually means the barrier is effectively opaque for that particle, while a value near one means the barrier is comparatively easy to cross under the selected conditions.

Assumptions and limits of the rectangular-barrier model

No rectangular-barrier tunneling calculator can capture every real-world correction. The model here assumes a one-dimensional barrier with a clean rectangular profile, so effects such as barrier shape variation, interference from neighboring structures, surface states, or multi-particle behavior are outside the scope of the estimate.

• Input interpretation: read each input label literally; changing the meaning of mass, energy, or barrier dimensions changes the transmission estimate.
• Unit conversions: convert source data carefully before entering values, especially when moving between eV, joules, nanometers, and meters.
• Linearity: the output is not a weighted sum of the inputs; width and mass can suppress tunneling much faster than a linear rule would suggest.
• Rounding: displayed probabilities may be rounded; small differences are normal.
• Missing factors: surface effects, barrier shape variations, and other quantum corrections may not be represented.

If you use the output for research, safety, medical, legal, or financial decisions, treat it as a starting estimate and verify it against authoritative sources. The best use of this calculator is to make the tunneling assumptions explicit so you can see which inputs drive the probability and explain the result clearly. For a more complete physical picture, compare this result with whatever experimental conditions or higher-fidelity model you are already using.