Finite Potential Well Calculator
Calculate finite potential well with our free science calculator. Uses standard scientific formulas with unit conversions and explanations.
Calculator
Adjust values & calculateAll Bound State Energies
Formula
The infinite well energy provides an upper bound for finite well energies. Actual finite well energies are found by solving transcendental equations from matching boundary conditions. The penetration depth is delta = hbar / sqrt(2m(V0-E)). The parameter z0 = (L/2)*sqrt(2mV0)/hbar determines the number of bound states.
Last reviewed: December 2025
Worked Examples
Example 1: GaAs/AlGaAs Quantum Well
Example 2: Nuclear Potential Well for a Neutron
Background & Theory
The Finite Potential Well Calculator applies the following established principles and formulas. Physics is the fundamental natural science concerned with matter, energy, and the interactions between them. Classical mechanics, founded on Newton's three laws of motion, provides the framework for analyzing the motion of objects. The first law states that an object remains at rest or in uniform motion unless acted upon by a net external force. The second law quantifies this relationship: F = ma, where force equals mass times acceleration in SI units of newtons (N = kgยทm/sยฒ). The third law establishes that every action produces an equal and opposite reaction. Kinematics describes motion without reference to its causes. The four fundamental equations relate displacement s, initial velocity u, final velocity v, acceleration a, and time t: v = u + at, s = ut + ยฝatยฒ, vยฒ = uยฒ + 2as, and s = ยฝ(u + v)t. These assume constant acceleration and are foundational for solving projectile motion, free fall, and linear dynamics problems. Energy conservation underpins much of physics. Kinetic energy is KE = ยฝmvยฒ, where m is mass in kilograms and v is speed in meters per second. Gravitational potential energy is PE = mgh, where g โ 9.81 m/sยฒ near Earth's surface and h is height in meters. The work-energy theorem states that the net work done on an object equals its change in kinetic energy: W = ฮKE. Electricity and circuits rely on Ohm's law: V = IR, where voltage V is in volts, current I in amperes, and resistance R in ohms. Electrical power is P = IV = IยฒR = Vยฒ/R, measured in watts. Wave mechanics connects frequency f, wave speed v, and wavelength ฮป through f = v/ฮป, with frequency in hertz (Hz). Pressure is defined as force per unit area, P = F/A, in pascals (Pa = N/mยฒ). The ideal gas law PV = nRT links pressure, volume, moles n, the gas constant R = 8.314 J/(molยทK), and absolute temperature in kelvin. Gravitational force between two masses follows Newton's law of universal gravitation: F = Gmโmโ/rยฒ, where G = 6.674ร10โปยนยน Nยทmยฒ/kgยฒ is the gravitational constant.
History
The history behind the Finite Potential Well Calculator traces back through the following developments. The history of physics spans over two millennia, beginning with the natural philosophy of ancient Greece. Aristotle (384โ322 BCE) proposed that all matter consisted of four elements and that objects moved toward their natural place, with heavier objects falling faster than lighter ones. While largely incorrect, his systematic approach to explaining nature dominated Western thought for nearly 2,000 years. The Scientific Revolution overturned Aristotelian physics. Galileo Galilei (1564โ1642) performed groundbreaking experiments on inclined planes and falling bodies, demonstrating that all objects fall with the same acceleration regardless of mass, and established the principle of inertia. His use of mathematics to describe motion was revolutionary. Isaac Newton synthesized these developments in his landmark Principia Mathematica (1687), laying out the three laws of motion and the law of universal gravitation. Newton's framework unified terrestrial and celestial mechanics, explaining planetary orbits with the same equations governing a falling apple. His calculus provided the mathematical language for expressing rates of change. The 19th century brought two major theoretical achievements. James Clerk Maxwell formulated his equations of electromagnetism between 1861 and 1862, unifying electricity, magnetism, and optics, and predicting the existence of electromagnetic waves traveling at the speed of light. Thermodynamics was developed by Carnot, Clausius, and Kelvin, establishing the laws governing heat, work, and entropy. The 20th century produced two revolutions that fundamentally altered the classical picture. Albert Einstein published the special theory of relativity in 1905, showing that space and time are not absolute but relative to the observer, and that mass and energy are equivalent via E = mcยฒ. His general theory of relativity in 1915 reinterpreted gravity as the curvature of spacetime. Simultaneously, quantum mechanics emerged from the work of Planck, Bohr, Heisenberg, and Schrรถdinger, revealing that at atomic scales energy is quantized and particles exhibit wave-particle duality. These developments culminated in the Standard Model of particle physics, which describes all known fundamental particles and three of the four fundamental forces.
Frequently Asked Questions
Formula
E_n (infinite) = n^2 * pi^2 * hbar^2 / (2 * m * L^2)
The infinite well energy provides an upper bound for finite well energies. Actual finite well energies are found by solving transcendental equations from matching boundary conditions. The penetration depth is delta = hbar / sqrt(2m(V0-E)). The parameter z0 = (L/2)*sqrt(2mV0)/hbar determines the number of bound states.
Worked Examples
Example 1: GaAs/AlGaAs Quantum Well
Problem: An electron (mass ratio 0.067 of free electron mass) is in a GaAs quantum well of width 10 nm and depth 0.3 eV. Find the ground state energy and number of bound states.
Solution: L = 10 nm, V0 = 0.3 eV, m = 0.067 * me\nz0 = (L/2) * sqrt(2*m*V0) / hbar\n= 5e-9 * sqrt(2 * 0.067 * 9.109e-31 * 0.3 * 1.602e-19) / 1.0546e-34\n= 5e-9 * sqrt(5.854e-51) / 1.0546e-34\n= 5e-9 * 2.419e-26 / 1.0546e-34 = 1.147\nMax bound states = floor(2*1.147/pi) + 1 = floor(0.73) + 1 = 1\nE1 (infinite) = pi^2 * hbar^2 / (2*m*L^2) = 0.084 eV\nE1 (finite) ~ 0.056 eV (reduced due to finite barriers)
Result: Ground State Energy ~ 0.056 eV | 1 bound state | Penetration depth ~ 1.2 nm
Example 2: Nuclear Potential Well for a Neutron
Problem: A neutron (mass ratio 1838.7) is in a nuclear potential well of width 2 fm (0.002 nm) and depth 40 MeV. Estimate the ground state energy and bound states.
Solution: L = 0.002 nm = 2e-15 m, V0 = 40 MeV = 4e7 eV, m = 1838.7 * me\nz0 = 1e-15 * sqrt(2 * 1838.7 * 9.109e-31 * 4e7 * 1.602e-19) / 1.0546e-34\n= 1e-15 * sqrt(2.153e-11) / 1.0546e-34\n= 1e-15 * 4.640e-6 / 1.0546e-34 = 44.0\nMax bound states ~ floor(2*44/pi) + 1 = 28 + 1 = 29\nE1 (infinite) = pi^2 * hbar^2 / (2*m*L^2*eV) ~ 5.1 MeV
Result: E1 ~ 5.1 MeV | ~29 bound states | Deep well with many levels
Frequently Asked Questions
What is a finite potential well in quantum mechanics?
A finite potential well is a quantum mechanical model where a particle is confined in a region of space by potential energy barriers of finite height. Unlike the infinite potential well where the walls are infinitely high and the particle is completely trapped, a finite well allows the particle wavefunction to penetrate into the classically forbidden barrier regions. This penetration is called quantum tunneling and has no classical analog. The finite well is more physically realistic than the infinite well because no real potential barrier is truly infinite. This model is used to describe electrons in semiconductor quantum wells, nucleons in nuclear potentials, and atoms in optical traps.
How do energy levels in a finite well differ from an infinite well?
Energy levels in a finite potential well are always lower than the corresponding levels in an infinite well of the same width. This is because the wavefunction extends beyond the well boundaries into the barrier region, effectively making the particle wavelength longer and its energy lower. The deeper and wider the well, the closer the finite well energies approach the infinite well values. Additionally, a finite well has a limited number of bound states determined by the well depth and width, while an infinite well has infinitely many bound states. For very shallow or narrow wells, there may be only one bound state. The number of bound states can be estimated from the dimensionless parameter z0.
How do you determine the number of bound states in a finite well?
The number of bound states depends on the dimensionless parameter z0 equals (L/2) times the square root of (2mV0) divided by hbar, where L is the well width, m is the particle mass, and V0 is the well depth. The approximate number of bound states is the integer part of (2z0/pi) plus 1. This means even the shallowest finite well always has at least one bound state in one dimension. A deeper or wider well supports more bound states. For a symmetric well, bound states alternate between even and odd parity solutions. The exact energies must be found by solving transcendental equations graphically or numerically because no closed-form analytical solution exists for the finite well.
What are the applications of finite potential wells in semiconductor physics?
Finite potential wells are the fundamental model for semiconductor quantum wells used in modern optoelectronic devices. Quantum well lasers confine electrons and holes in thin semiconductor layers (typically 2 to 20 nanometers) sandwiched between wider bandgap materials, creating discrete energy levels that enable efficient light emission at specific wavelengths. Quantum well infrared photodetectors use intersubband transitions for thermal imaging. High electron mobility transistors (HEMTs) use quantum wells to create two-dimensional electron gases with superior mobility. Multiple quantum well structures form superlattices with unique electronic and optical properties. The finite well model predicts the quantized energy levels that determine device operating wavelengths and performance.
How does the finite well model apply to nuclear physics?
In nuclear physics, the finite potential well models the nuclear potential that binds nucleons (protons and neutrons) within the nucleus. The nuclear potential is approximately a finite square well with a depth of about 40 to 50 MeV and a radius of a few femtometers. Bound states of this well correspond to the energy levels of nucleons in the nucleus. The model explains why nuclei have discrete energy levels and why certain numbers of nucleons (magic numbers: 2, 8, 20, 28, 50, 82, 126) are particularly stable. The finite well model also predicts alpha decay rates through barrier penetration and explains nuclear reactions where particles tunnel through the Coulomb barrier.
What is the difference between bound and unbound states in a finite well?
Bound states have energies below the well depth (E less than V0) and their wavefunctions are localized near the well, decaying exponentially in the barrier regions. These states are normalizable and represent particles that are trapped in the well. Unbound or scattering states have energies above the well depth (E greater than V0) and their wavefunctions extend to infinity as traveling waves. These represent particles that are not trapped but may still be affected by the well. At the well boundary, unbound states can be reflected or transmitted with probabilities determined by the energy and well parameters. The transition between bound and unbound states occurs at E equals V0.
References
Reviewed by Manoj Kumar, Mathematics Educator ยท Editorial policy