Finite Potential Well Calculator
Calculate finite potential well with our free science calculator. Uses standard scientific formulas with unit conversions and explanations.
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.