Finite Potential Well Calculator
Calculate finite potential well with our free science calculator. Uses standard scientific formulas with unit conversions and explanations.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Finite Potential Well Calculator
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Formula: E_n (infinite) = n^2 * pi^2 * hbar^2 / (2 * m * L^2)
Worked example — Ground State Energy ~ 0.056 eV | 1 bound state | Penetration depth ~ 1.2 nm
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 z0 = (L/2) * sqrt(2*m*V0) / hbar = 5e-9 * sqrt(2 * 0.067 * 9.109e-31 * 0.3 * 1.602e-19) / 1.0546e-34 = 5e-9 * sqrt(5.854e-51) / 1.0546e-34 = 5e-9 * 2.419e-26 / 1.0546e-34 = 1.147 Max bound states = floor(2*1.147/pi) + 1 = floor(0.73) + 1 = 1 E1 (infinite) = pi^2 * hbar^2 / (2*m*L^2) = 0.084 eV E1 (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 z0 = 1e-15 * sqrt(2 * 1838.7 * 9.109e-31 * 4e7 * 1.602e-19) / 1.0546e-34 = 1e-15 * sqrt(2.153e-11) / 1.0546e-34 = 1e-15 * 4.640e-6 / 1.0546e-34 = 44.0 Max bound states ~ floor(2*44/pi) + 1 = 28 + 1 = 29 E1 (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.
What is wavefunction penetration and why does it matter?
Wavefunction penetration refers to the exponential decay of the particle wavefunction into the classically forbidden region outside the finite well, where the potential energy exceeds the particle total energy. The penetration depth equals hbar divided by the square root of 2m(V0 minus E), where V0 is the barrier height and E is the particle energy. Higher energy states penetrate deeper because the effective barrier is smaller. This penetration is crucial in semiconductor physics for understanding quantum tunneling through thin barriers, coupling between adjacent quantum wells, and the operation of tunnel diodes and scanning tunneling microscopes. It also explains alpha particle decay in nuclear physics.
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.
How do you solve the finite potential well equation numerically?
The finite well requires solving transcendental equations that arise from matching boundary conditions at the well edges. For even parity states, the equation is z times tan(z) equals the square root of (z0 squared minus z squared), and for odd parity, negative z times cot(z) equals the square root of (z0 squared minus z squared). These can be solved graphically by plotting both sides and finding intersections, or numerically using root-finding algorithms like Newton-Raphson or bisection methods. The parameter z is related to energy by z equals (L/2) times the square root of (2mE) divided by hbar. Each intersection gives one bound state energy. Modern computational tools like Python or MATLAB can solve these efficiently.
What is the role of symmetry in finite potential well solutions?
Symmetry plays a crucial role in the finite potential well because a symmetric well (V of x equals V of negative x) has solutions that are either even (symmetric) or odd (antisymmetric) functions. The ground state is always even, the first excited state is odd, the second is even, and so on alternating. This parity classification simplifies the problem because even and odd solutions satisfy different transcendental equations. The even solutions involve tangent functions while odd solutions involve cotangent functions. This symmetry property also determines selection rules for transitions between energy levels when the well is perturbed by electromagnetic radiation, which is important for spectroscopic applications.
How does the finite well relate to more realistic quantum potentials?
The finite square well is an idealized model with sharp potential boundaries, but it captures the essential physics of more realistic smooth potentials. The harmonic oscillator potential (parabolic shape) better describes molecular vibrations and atomic traps. The Morse potential describes diatomic molecular bonds more accurately, including anharmonic effects and dissociation. The Woods-Saxon potential provides a smooth approximation to the nuclear potential. The Coulomb potential describes hydrogen-like atoms. Despite its simplicity, the finite well teaches key concepts including quantized energy levels, wavefunction penetration, bound state counting, and the transition from discrete to continuous spectra that appear in all these more complex systems.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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