Fact-checked by Grok 2 weeks ago

Particle in a box

The particle in a box is a foundational model in quantum mechanics that describes the behavior of a single particle of mass m confined to a one-dimensional region between x = 0 and x = L, where the potential energy is zero inside the box and infinite outside, preventing the particle from escaping.^[1] The solutions to the time-independent Schrödinger equation for this system yield quantized energy levels E_n = n^2 \frac{\pi^2 \hbar^2}{2 m L^2} (where n = 1, 2, 3, \dots is the principal quantum number) and corresponding wavefunctions \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left( \frac{n \pi x}{L} \right), which represent standing waves that must fit exactly within the box boundaries.^[1] This quantization arises directly from the boundary conditions imposed by the infinite walls, ensuring the wavefunction vanishes at x = 0 and x = L, and contrasts sharply with classical mechanics, where the particle's energy could take any positive value.^[2] As one of the simplest exactly solvable problems in quantum mechanics, the particle in a box illustrates core principles such as wave-particle duality, the probabilistic interpretation of the wavefunction (via |\psi_n(x)|^2), and the exclusion of zero-point energy in the ground state E_1 = \frac{\pi^2 \hbar^2}{2 m L^2} > 0.^[2] It serves as an educational cornerstone for introducing the Schrödinger equation and normalization of wavefunctions, with higher n states featuring n-1 nodes and increasing energy quadratically with n.^[1] In practice, the model approximates real systems like electrons delocalized in linear conjugated polyenes (molecules with alternating single and double bonds), where the box length L corresponds to the chain length, enabling predictions of electronic transitions and UV-visible absorption spectra.^[3] Extensions of the model to finite potential wells, two-dimensional squares, or three-dimensional cubes address more realistic scenarios, such as tunneling through barriers or particles in quantum dots, with applications in optoelectronics, semiconductor physics, and nanotechnology.^[1] These variants retain the essence of quantization while incorporating effects like parity (even or odd wavefunctions) and degeneracy in higher dimensions, underscoring the model's versatility in bridging theoretical quantum mechanics to experimental observations in molecular and materials science.^[2]

Model Formulation

Infinite Potential Well

The infinite potential well, also known as the infinite square well, models the confinement of a particle within a finite region by imposing an idealized potential that is zero inside the well and infinite outside. In one dimension, the potential is defined as V(x) = 0 for $0 < x < L, where L is the width of the well, and V(x) = \infty for x \leq 0 or x \geq L.^[4] This setup ensures that the particle's wave function vanishes outside the well, as regions of infinite potential are classically and quantum mechanically forbidden. In higher dimensions, the model extends analogously, with the potential zero within a rectangular region and infinite elsewhere, confining the particle to that volume.^[4] The boundary conditions arising from this potential strictly enforce complete confinement: the wave function satisfies \psi(0) = 0 and \psi(L) = 0 at the walls in one dimension, with the wave function being zero outside the interval [0, L]. These conditions reflect the impenetrable nature of the infinite barriers, preventing any probability of finding the particle beyond the well's boundaries.^[4] Physically, the infinite potential well serves as an idealized representation of hard-wall confinement, approximating scenarios where particles are strongly localized by high barriers. For instance, it models electrons in semiconductor quantum dots, where three-dimensional confinement leads to discrete energy states.^[5] Similarly, in chemistry, it approximates the behavior of π electrons delocalized along linear conjugated molecules, such as polyenes, treating the molecular chain as a one-dimensional box.^[6] Within the well, the wave function must satisfy the normalization condition, ensuring the total probability of locating the particle inside the box integrates to unity: \int_0^L |\psi(x)|^2 \, dx = 1. This requirement maintains the probabilistic interpretation of the wave function as dictated by quantum mechanics.^[4]

Schrödinger Equation Setup

The time-independent Schrödinger equation governs the stationary states of a quantum system and, for a single particle in one dimension, takes the form

-\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + V(x) \psi(x) = E \psi(x),

where \hbar is the reduced Planck's constant, m is the particle's mass, V(x) is the potential, \psi(x) is the wave function, and E is the energy.^[7] This equation, derived from the foundational wave mechanics framework, describes how the wave function satisfies the energy eigenvalue problem under a given potential. In the infinite potential well model, the potential V(x) = 0 for $0 < x < L, reducing the equation inside the well to

-\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} = E \psi(x).

This simplified form represents a free particle confined by the boundaries, where the wave function must vanish at x = 0 and x = L due to the infinite barriers.^[8] Rearranging yields a standard differential equation whose general solution inside the well is an oscillatory function:

\psi(x) = A \sin(kx) + B \cos(kx),

with the wave number k = \sqrt{2mE}/\hbar.^[9] The constants A and B are determined by applying the boundary conditions. The condition \psi(0) = 0 immediately sets B = 0, leaving \psi(x) = A \sin(kx). The second condition, \psi(L) = 0, then requires A \sin(kL) = 0; for a non-trivial solution (A \neq 0), \sin(kL) = 0, so kL = n\pi where n = 1, 2, \dots.^[8] This quantization condition on k arises directly from the boundary constraints, implying that only discrete values of the wave number—and thus the energy E—are permissible, distinguishing the quantum solution from classical continuous motion.^[10]

One-Dimensional Solution

Wave Functions

The position-space wave functions for a particle confined in a one-dimensional infinite potential well of width L satisfy the boundary conditions \psi(0) = \psi(L) = 0, leading to sinusoidal solutions inside the well.^[11] The normalized spatial wave functions are

\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right)

for $0 < x < L, where n = 1, 2, 3, \dots, and \psi_n(x) = 0 elsewhere.^[11] Normalization requires that the probability density integrates to unity over the well:

\int_0^L |\psi_n(x)|^2 \, dx = 1.

Substituting the unnormalized form \psi_n(x) = A \sin\left( \frac{n\pi x}{L} \right) yields

\int_0^L A^2 \sin^2\left( \frac{n\pi x}{L} \right) \, dx = A^2 \frac{L}{2} = 1,

so A = \sqrt{2/L}.^[11] These wave functions are mutually orthogonal, satisfying

\int_0^L \psi_m^*(x) \psi_n(x) \, dx = \delta_{mn},

where \delta_{mn} is the Kronecker delta (1 if m = n, 0 otherwise). This property arises from the orthogonality of sine functions over the interval, using the identity \sin(\alpha)\sin(\beta) = \frac{1}{2} [\cos(\alpha - \beta) - \cos(\alpha + \beta)], which makes the integral vanish for m \neq n.^[12] The functions \psi_n(x) describe standing waves, with the quantum number n determining the number of half-wavelengths fitting within the box; specifically, there are n-1 nodes (points of zero amplitude) inside the well at positions x = kL/n for k = 1, \dots, n-1. Relative to the box center at x = L/2, the wave functions exhibit alternating parity: even parity for odd n and odd parity for even n.^[11] For time evolution in stationary states, the full wave function is

\psi_n(x,t) = \psi_n(x) \, e^{-i E_n t / \hbar},

where the phase factor ensures time-independent probability densities, characterizing these as stationary states.^[11]

Energy Eigenvalues

The energy eigenvalues for a particle of mass m confined to a one-dimensional infinite potential well of width L are quantized and given by

E_n = \frac{n^2 \pi^2 \hbar^2}{2 m L^2},

where n = 1, 2, 3, \dots is a positive integer quantum number representing the energy level, and \hbar = h / 2\pi is the reduced Planck's constant.^[13] This formula arises from substituting the allowed wave numbers k_n = n \pi / L into the free-particle energy-momentum relation E = \hbar^2 k^2 / 2m, enforcing boundary conditions that the wave function vanishes at x = 0 and x = L.^[13] The early detailed derivation of these eigenvalues and associated eigenfunctions appeared in the 1929 textbook by Condon and Morse, building on Schrödinger's wave mechanics framework.^[13] A key feature of these eigenvalues is the presence of zero-point energy, where the ground state energy E_1 = \pi^2 \hbar^2 / (2 m L^2) > 0, even at absolute zero temperature.^[13] This contrasts sharply with the classical counterpart, in which a particle can possess arbitrarily low energy, including zero, by coming to rest within the box.^[13] The non-zero ground state reflects the inherent uncertainty in the particle's position and momentum due to confinement, preventing the system from achieving classical rest.^[14] The energy levels exhibit specific dependencies on the system's parameters: E_n scales quadratically with the quantum number n, leading to increasingly spaced levels as n grows (e.g., E_2 = 4 E_1, E_3 = 9 E_1); it is inversely proportional to L^2, such that confining the particle to a smaller box raises all energies; and it is inversely proportional to m, meaning lighter particles occupy higher energy states for the same confinement.^[13] These scalings highlight how quantization enforces discrete allowed energies, with the spacing \Delta E_{n+1,n} = E_{n+1} - E_n = (2n+1) \pi^2 \hbar^2 / (2 m L^2) widening for higher n.^[13] In the semiclassical limit of large n, the discrete eigenvalues approach a continuous spectrum, corresponding to classical mechanics via the relation E_n \approx p^2 / 2m with momentum p \approx n \pi \hbar / L.^[13] This behavior aligns with Bohr's correspondence principle, where quantum predictions recover classical results for high quantum numbers or macroscopic systems.^[14]

Probability Distributions

The position probability density for a particle in the nth energy eigenstate is given by

|\psi_n(x)|^2 = \frac{2}{L} \sin^2 \left( \frac{n \pi x}{L} \right),

for $0 < x < L, where L is the length of the box. This distribution features n-1 nodes, located at x = k L / n for integers k = 1, 2, \dots, n-1, where the probability of finding the particle is zero. For the ground state (n=1), the distribution has no nodes within the box and reaches its maximum value of $2/L at the center, x = L/2. The momentum-space wave function \phi_n(p) is the Fourier transform

\phi_n(p) = \frac{1}{\sqrt{2\pi \hbar}} \int_0^L \psi_n(x) e^{-i p x / \hbar} \, dx.

This yields a continuous distribution peaked at p = \pm \frac{n \pi \hbar}{L}, corresponding to the standing wave's superposition of counter-propagating plane waves, but broadened by the finite well size. Thus, |\phi_n(p)|^2 is nonzero over a range of p, with bimodal peaks of equal integrated probability.^[15] These distributions satisfy the Heisenberg uncertainty principle \Delta x \Delta p \geq \hbar/2. For the ground state, the position uncertainty is \Delta x \approx L / (2 \sqrt{3}) and the momentum uncertainty is \Delta p \approx \pi \hbar / L, yielding a product \Delta x \Delta p \approx 0.906 \hbar > \hbar/2. In general, \Delta p \approx n \pi \hbar / L for higher states, ensuring the principle holds with increasing spread as n grows.

Higher-Dimensional Solutions

Rectangular Boxes

The rectangular box model extends the infinite potential well to higher dimensions, confining a particle within a three-dimensional region defined by $0 < x < a, $0 < y < b, and $0 < z < c, where the potential energy V(x,y,z) = 0 inside this volume and V = \infty elsewhere.^[16] This setup assumes impenetrable walls aligned with the coordinate axes, allowing the Schrödinger equation to be solved exactly through separation of variables. The time-independent Schrödinger equation in three dimensions,

-\frac{\hbar^2}{2m} \nabla^2 \psi(x,y,z) = E \psi(x,y,z),

is separable under these boundary conditions, yielding \psi(x,y,z) = X(x) Y(y) Z(z), where each function X(x), Y(y), and Z(z) satisfies the one-dimensional infinite well equation along its respective dimension with lengths a, b, and c.^[17]^[5] The normalized wave function for the state labeled by quantum numbers n_x, n_y, n_z = 1, 2, 3, \dots is

\psi_{n_x n_y n_z}(x,y,z) = \sqrt{\frac{8}{a b c}} \sin\left(\frac{n_x \pi x}{a}\right) \sin\left(\frac{n_y \pi y}{b}\right) \sin\left(\frac{n_z \pi z}{c}\right),

valid only inside the box and zero outside.^[16] The corresponding energy eigenvalues are additive, E_{n_x n_y n_z} = E_{n_x}^{(x)} + E_{n_y}^{(y)} + E_{n_z}^{(z)}, where each term follows the one-dimensional form, giving

E_{n_x n_y n_z} = \frac{\pi^2 \hbar^2}{2m} \left( \frac{n_x^2}{a^2} + \frac{n_y^2}{b^2} + \frac{n_z^2}{c^2} \right).

This energy spectrum reflects the independent quantization in each direction, with the one-dimensional solutions providing the foundational components.^[17]^[5] Degeneracy arises when distinct sets of (n_x, n_y, n_z) produce the same energy value, which depends on the box dimensions; for example, if a = b = c, multiple combinations can yield identical sums of squares. In the special case of a cubic box where a = b = c = L, the energy simplifies to

E_{n_x n_y n_z} = \frac{\pi^2 \hbar^2}{2m L^2} (n_x^2 + n_y^2 + n_z^2),

and the wave function becomes \psi_{n_x n_y n_z}(x,y,z) = \sqrt{8/L^3} \sin(n_x \pi x / L) \sin(n_y \pi y / L) \sin(n_z \pi z / L).^[16]^[17] This configuration exhibits higher degeneracy, such as the six-fold degeneracy for states where \{n_x, n_y, n_z\} are permutations of three distinct integers.^[18]

Non-Rectangular Boundaries

For non-rectangular boundaries, the Schrödinger equation lacks separability in Cartesian coordinates, necessitating alternative coordinate systems, special functions, or numerical methods to obtain solutions for the particle's wave functions and energy levels.^[19] Unlike rectangular geometries where the wave function factors into independent one-dimensional components, arbitrary shapes introduce coupled variables that require perturbative expansions or computational discretization for accurate treatment.^[20] In two dimensions, a circular box of radius R confines the particle within polar coordinates, where the radial wave function involves Bessel functions of the first kind. The boundary condition J_m(k \rho) = 0 at \rho = R determines the wave numbers k_{m,n} as the n-th roots of the m-th order Bessel equation, leading to quantized energies E_{m,n} = \frac{\hbar^2 k_{m,n}^2}{2m}, with m as the angular momentum quantum number and n indexing the radial nodes.^[19] This exact solution highlights angular degeneracy for |m| values and provides a model for systems with rotational symmetry, such as certain quantum dots. Extending to three dimensions, a spherical box of radius R employs spherical coordinates, where the radial part of the wave function is described by spherical Bessel functions j_l(kr), satisfying j_l(k r) = 0 at r = R. The energy levels are proportional to the squares of the zeros of these functions, E_{n,l} \propto \left( \frac{z_{l,n}}{R} \right)^2, with quantum numbers n for radial nodes, l for orbital angular momentum, and m for its projection.^[21] This formulation captures spherical symmetry and is foundational for modeling particles in isotropic potentials. For more complex geometries like triangular or irregular domains, analytical solutions are generally unavailable, prompting the use of numerical techniques such as the finite element method (FEM) to discretize the domain and solve the eigenvalue problem approximately.^[20] Variational methods, which minimize the energy functional with trial wave functions adapted to the boundary, offer another approach for estimating ground and excited states in non-separable shapes.^[22] Graphene quantum dots with hexagonal or triangular edges exemplify such applications, where these methods reveal confinement effects on electronic states beyond simple rectangular approximations.^[23] These non-rectangular treatments incur higher computational demands due to the need for mesh generation in FEM or extensive basis set expansions in variational approaches, contrasting with the algebraic simplicity of rectangular cases.^[20]

Applications

Conjugated Systems in Chemistry

The particle-in-a-box model finds significant application in chemistry for describing the behavior of π electrons in linear conjugated polyenes, such as those found in molecules like ethylene and β-carotene. In this approximation, the delocalized π electrons are treated as non-interacting particles confined to a one-dimensional box along the chain of carbon atoms, with infinite potential barriers at the ends. This simple quantum mechanical framework, introduced in the free-electron theory by Hückel and further developed for organic dyes and polyenes, provides qualitative insights into the electronic transitions responsible for UV-visible absorption spectra.^[24] The length of the box L is approximated as L \approx (N+1) l, where N is the number of double bonds in the polyene chain and l is the average C-C bond length in the conjugated system, approximately 140 pm. The energy levels for the π electrons are given by

E_n \approx \frac{n^2 h^2}{8 m L^2},

where n is the quantum number, h is Planck's constant, and m is the electron mass; this form uses h rather than \hbar as in the historical Hückel-inspired approximations for simplicity in early calculations. For a polyene with N double bonds (and thus $2N π electrons), the lowest-energy electronic transition is from the highest occupied molecular orbital (HOMO, n=N) to the lowest unoccupied molecular orbital (LUMO, n=N+1), with an energy difference

\Delta E_{N \to N+1} = (2N + 1) \frac{h^2}{8 m L^2}.

The corresponding absorption wavelength is then

\lambda \approx \frac{8 m c L^2}{h (2N + 1)},

where c is the speed of light; substituting L \approx (N+1) l yields \lambda \propto N approximately, predicting a red-shift (longer wavelengths) for longer polyene chains. This qualitatively matches experimental observations, such as the π → π* absorption at approximately 170 nm for ethylene (N=1) and progressive red-shifts to around 217 nm for 1,3-butadiene (N=2) and 258 nm for 1,3,5-hexatriene (N=3).^[24] Despite its success in capturing the trend of spectral shifts with chain length, the model has notable limitations. It overestimates transition energies because it neglects electron-electron interactions and assumes uniform bond lengths, ignoring the bond-length alternation typical in polyenes that affects the effective potential. Refinements to the free-electron model, such as incorporating periodic potentials or varying box lengths, improve accuracy but still simplify the complex molecular orbital interactions.^[25]

Quantum Wells in Semiconductors

Quantum wells in semiconductors provide a practical realization of one-dimensional electron confinement through heterostructures, where a narrow layer of a smaller-bandgap material, such as GaAs, is sandwiched between wider-bandgap barriers like AlGaAs, creating a Type I band alignment with approximately 67% conduction band offset and 33% valence band offset.^[26] These structures confine electrons primarily along the growth direction (z-axis), with typical well widths ranging from 5 to 20 nm to ensure significant quantization effects while maintaining manufacturability via techniques like molecular beam epitaxy.^[27] Unlike the free electron mass m_e, charge carriers in these wells experience an effective mass m^*, which for electrons in GaAs is approximately $0.067 m_e (where m_e is the vacuum electron mass), reflecting the curvature of the band structure near the conduction band minimum.^[26] The quantized energy levels in these quantum wells can be approximated using the one-dimensional infinite potential well model, yielding subband energies given by

E_n = \frac{n^2 \pi^2 \hbar^2}{2 m^* w^2} + E_g,

where n = 1, 2, 3, \dots is the quantum number, w is the well width, \hbar is the reduced Planck's constant, and E_g is the bandgap energy of the well material (about 1.42 eV for GaAs at room temperature); this results in discrete subbands separated by tens of meV, such as ~40 meV for the ground state in a 10 nm GaAs well.^[26] For thin wells, the finite height of the barriers (typically 200-300 meV for Al_{0.3}Ga_{0.7}As/GaAs) allows a reasonable approximation to the infinite well, as wavefunction penetration into the barriers is minimal, though exact solutions require matching boundary conditions and account for slight energy lowering and tunneling.^[26] A key application of semiconductor quantum wells is in laser diodes, where the discrete subband structure facilitates population inversion by enabling selective carrier injection into higher subbands (e.g., n=2) while the lower subband (n=1) empties via fast phonon scattering, promoting stimulated emission at photon energies matching the subband separation \Delta E_{1 \to 2} and wavelengths \lambda \approx hc / \Delta E_{1 \to 2}.^[26] Compared to bulk semiconductor lasers, quantum well lasers exhibit advantages such as lower threshold current density (due to enhanced density of states at the band edge), higher differential gain, improved temperature stability, and greater modulation bandwidth, enabling efficient operation at room temperature.^[28] The first quantum well laser was demonstrated in 1977 by Nick Holonyak Jr. and colleagues at the University of Illinois, using a GaAs/AlGaAs double heterostructure that operated continuously at room temperature.^[29] These devices have since become integral to fiber-optic telecommunications, particularly for 1.55 μm lasers based on InGaAsP/InP quantum wells, which align with the low-loss window of silica fibers and support high-speed data transmission over long distances.^[30]

Quantum Dots in Nanotechnology

Quantum dots (QDs) are semiconductor nanocrystals typically 2–10 nm in size, where electrons and holes experience three-dimensional (3D) confinement analogous to a particle in a 3D box model, leading to quantized energy levels and size-dependent optical properties.^[31] This confinement arises from the nanoscale dimensions, which are comparable to the exciton Bohr radius in the material, resulting in discrete electronic states rather than continuous bands as in bulk semiconductors. The simple particle-in-a-box approximation captures the leading quantum size effects, with confinement energy scaling as E \propto 1/L^2, where L is the characteristic length (e.g., edge length for cubic or radius for spherical QDs). For spherical QDs, a common approximation, the energy eigenvalues are given by

E_{n,l} = \frac{\hbar^2 \pi^2}{2 m^* R^2} \left( \frac{\chi_{n,l}}{\pi} \right)^2,

where R is the radius, m^* is the effective mass of the charge carrier, and \chi_{n,l} are the zeros of the spherical Bessel functions j_l(\chi) = 0, with quantum numbers n (radial) and l (angular momentum). This confinement energy adds to the bulk bandgap, E_g, shifting the effective bandgap to higher values for smaller R. In real multi-electron QDs, Coulomb interactions between electrons and holes introduce corrections, but the box model reliably predicts the dominant $1/R^2 confinement term. The size-tunable properties enable precise control over emission wavelengths; for example, in CdSe QDs, reducing the radius from approximately 6 nm to 2 nm blue-shifts the fluorescence from around 650 nm (red) to 450 nm (blue), spanning the visible spectrum due to increased confinement energy. Quantum yield is enhanced in these confined systems, often reaching 50–90% with core-shell structures that passivate surface traps.^[32] QDs are synthesized via colloidal methods (e.g., hot-injection for monodisperse particles) or self-assembly in epitaxial growth, enabling scalable production. In nanotechnology applications, QDs serve as tunable emitters in light-emitting diodes (LEDs) for displays and lighting, where their narrow emission lines and high stability improve color purity and efficiency.^[31] They enhance solar cells by broadening absorption spectra and reducing thermalization losses, with power conversion efficiencies exceeding 15% in QD-sensitized devices.^[31] For biomarkers, water-soluble QDs like Qdot nanocrystals, developed in the 1990s and commercialized since the late 1990s, enable multiplexed imaging in biology due to their bright, photostable fluorescence across multiple colors from a single excitation wavelength.

References

[1]
The Quantum Particle in a Box – University Physics Volume 3
Energy states of a particle in a box are quantized and indexed by principal quantum number. The quantum picture differs significantly from the classical picture ...
[2]
[PDF] Begin Quantum Mechanics: Free Particle in a 1D Box
Sep 15, 2017 · The particle in a box is the problem that we can most easily understand completely. This is where we begin to become comfortable with some of ...
[3]
[PDF] Quantum theory: techniques and applications
Example 1. Using the particle in a box solutions. The wavefunctions of an electron in a conjugated polyene (a molecule with alternating single and double bonds) ...
[4]
[PDF] Quantum Physics I, Lecture Note 11 - MIT OpenCourseWare
Mar 17, 2016 · Here we introduce another instructive toy model, the infinite square well potential. This forces a particle to live on an interval of the real ...Missing: arxiv. | Show results with:arxiv.
[5]
[PDF] Part 2. The Quantum Particle in a Box - MIT OpenCourseWare
All details, such as atoms, are ignored. In quantum dots, electrons are confined in all three dimensions, in quantum wires, electrons are confined in only two ...
[6]
[PDF] ( )ψ ( ) = Eψ ( ) ( ) and φ2 ( ) Aφ1 ( ) + Bφ2 ( ) x ( ) = ( ) and ( ). ( ) ( ) = 0,
We've solved some simple quantum mechanics problems! The P-I-B model is a good approximation for some important cases, e.g. pi-bonding electrons on aromatics.
[7]
[PDF] 1 The Schrödinger equation - MIT OpenCourseWare
Sep 13, 2013 · Any of the three boxed equations above is referred to as the time-independent Schrödinger equation. h Page 6 is nowhere infinite, ψ = ψ ′ = 0 ...<|control11|><|separator|>
[8]
[PDF] The Schrödinger Equation in One Dimension
Let us now apply the TISE to a simple system - a particle in an infinitely deep potential well. Particle in a One-Dimensional Rigid Box (Infinite Square Well).
[9]
[PDF] 5. The Schrödinger Equations - Physics
In general, “solving the time-independent Schrödinger equation” means finding both the eigenfunctions ψ(x) and the corresponding eigenvalues E.
[10]
[PDF] Chapter 7 The Schroedinger Equation in One Dimension In classical ...
In quantum mechanics the equation of motion is the time-dependent Schroedinger equation. If we know a particles wave function at t = 0, the time-dependent ...
[11]
A Particle in a Box
A Particle in a Box. ... So the wave function will be normalized if we choose \bgroup\color{black}$C ...
[12]
[PDF] Problem Set 2 CHM 305, Fall 2023 1 1D Particle-in-a-Box
(b) Show that the particle-in-a-box wavefunctions are orthogonal, e.g. that. ∫ a. 0 ψ. ∗ n(x)ψm(x)dx = 0 for m ̸= n. You may want to use the trigonometric ...Missing: orthogonality | Show results with:orthogonality
[13]
Quantum mechanics : Condon, Edward Uhler, 1902-1974
Apr 14, 2010 · Quantum mechanics. by: Condon, Edward Uhler, 1902-1974; Morse, Philip M. (Philip McCord), 1903-1985. Publication date: 1929 ... PDF. Uplevel ...
[14]
https://doi.org/10.1007/BF01400234
[15]
[PDF] Quantum Mechanics in Multidimensions
In the case of the particle in a box, nx +1 is the number of wave function nodes along x and ny +1 is the number of nodes along y. Each set of quantum ...
[16]
Three-Dimensional Wave Mechanics - Richard Fitzpatrick
Three-Dimensional Wave Mechanics. Up to now, we have only discussed wave mechanics for a particle moving in one dimension. However, the generalization to a ...Missing: function | Show results with:function
[17]
Particles in Two-Dimensional Boxes - Galileo
The solution to the first equation just gives the phase time dependence,. φ(t)=Ae−iEtℏ. and the second is the time independent Schrödinger equation as before.
[18]
The Particle in a Box (and in a Circular Box) - ResearchGate
The particle in a box problem in 1 and 2 dimensions is treated both for the Cartesian problem (square, rectangle) but for circular boundary conditions.
[19]
[PDF] Quantum particle in the wrong box (or: the perils of finite ... - arXiv
In this paper we demonstrate that, under mild assumptions, they converge to the solution of the Schrödinger equation generated by a specific Hamiltonian which ...
[20]
[PDF] Quantum particle in a spherical well confined by a cone - arXiv
Jul 4, 2022 · Abstract. We consider the quantum problem of a particle in either a spherical box or a finite spherical well confined by a circular cone ...<|control11|><|separator|>
[21]
(PDF) Quantum mechanics of particles trapped in a Lamé circle or ...
Mar 4, 2021 · The model is potentially useful for describing quantum dots that deviate from simple geometric shapes, or for demonstrating methods of ...Missing: irregular | Show results with:irregular
[22]
Electronic and Optical Properties of Graphene Quantum Dots
Aug 6, 2025 · The electronic structure and optical properties of hexagonal armchair and zigzag-edged graphene quantum dots (GQDs) are investigated within ...Missing: rectangular | Show results with:rectangular
[23]
https://www.researchgate.net/publication/273309746_Electronic_and_Optical_Properties_of_Graphene_Quantum_Dots_The_Role_of_Many-Body_Effects
[24]
[PDF] Optical Physics of Quantum Wells - Stanford Electrical Engineering
The actual energy of the first allowed electron energy level in a typical 100 Å GaAs quantum well is about 40 meV, which is close to the value that would be ...
[25]
Quantum Wells - RP Photonics
The thickness of such a quantum well is typically ≈ 5–20 nm. Such thin layers can be fabricated with molecular beam epitaxy (MBE) or metal–organic chemical ...
[26]
Quantum Well Lasers - an overview | ScienceDirect Topics
They exhibit advantages such as lower threshold current, tunable emission wavelengths, and high modulation frequencies. AI generated definition based on: ...
[27]
Milestone-Proposal:THE FIRST DEMONSTRATION OF THE ...
Sep 22, 2022 · The first quantum-well laser diode was demonstrated in 1977 by Professor Nick Holonyak, Jr. and his graduate students in the Electrical Engineering Research ...
[28]
[PDF] TWO REVOLUTIONARY OPTICAL TECHNOLOGIES - Nobel Prize
Oct 6, 2009 · The optical communication window has evolved from 870 nm to 1.3 μm and, finally, to 1.55 μm where fiber losses are lowest (see Fig. 3). Gradient ...<|separator|>
[29]
[PDF] QUANTUM DOTS – SEEDS OF NANOSCIENCE - Nobel Prize
Oct 4, 2023 · Brus presented a model describing the effect of particle size on electron and hole redox potentials for surface chemical reactions. Using an ...
[30]
https://www.nobelprize.org/uploads/2018/06/advanced-physicsprize2009.pdf