Enter values to see results
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nm^-0.5
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nm^-1
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eV
Enter values to see results
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nm^-0.5
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nm^-1
—
—
eV
The Wave Function Probability Calculator computes the quantum mechanical wave function and probability density for a particle in an infinite square well (particle in a box) — the most fundamental exactly solvable problem in quantum mechanics. Understanding wave functions and probability densities is essential to all of quantum mechanics, and the particle in a box serves as the foundation for understanding more complex quantum systems.
In quantum mechanics, the state of a particle is described by a wave function psi(x), a complex-valued function whose squared magnitude |psi(x)|^2 gives the probability density of finding the particle at position x. For a particle confined to a one-dimensional box of length L with infinitely high walls, the Schrodinger equation yields exactly solvable standing wave solutions: psi_n(x) = sqrt(2/L) * sin(n*pi*x/L), where n = 1, 2, 3... is the quantum number.
The corresponding quantized energy levels are E_n = n^2 * pi^2 * hbar^2 / (2*m*L^2). The energy grows as n^2, so the second level has 4 times the ground state energy, the third has 9 times, and so on. Crucially, the minimum energy (n=1 ground state) is not zero but E_1 = pi^2 * hbar^2 / (2*m*L^2), the zero-point energy, a direct consequence of the uncertainty principle.
The particle in a box is not just a textbook exercise: it models pi electrons in conjugated organic molecules (explaining their UV-visible spectra), electrons in quantum dots (whose tunable size gives tunable fluorescence colors), and serves as the conceptual basis for understanding quantum confinement in nanomaterials, quantum wells in semiconductor lasers, and electron behavior in carbon nanotubes.
This calculator computes the wave function value, probability density, and the probability of finding the particle in a small interval at position x for the nth energy level of a box of length L.
For a 1D infinite square well: psi_n(x) = sqrt(2/L) * sin(n*pi*x/L). Probability density: |psi_n(x)|^2 = (2/L)*sin^2(n*pi*x/L). Probability in [x, x+dx]: P ~ |psi|^2 * dx. Energy: E_n = n^2 * pi^2 * hbar^2 / (2*m*L^2) in eV.
Nodes (zero probability) occur at x = k*L/n for k = 0, 1, ..., n. The ground state (n=1) has one maximum at x = L/2. Level n=2 has nodes at 0, L/2, L and maxima at L/4 and 3L/4. Higher n leads to more oscillations. Probability is higher near antinodes and zero at nodes. Energy level spacing decreases with increasing L^2.
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At the center (x = L/2) of a 1 nm box, the n=1 wave function is at its maximum. The ground state energy is 0.376 eV, comparable to chemical bond energies in nanoscale systems.
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For n=2, a node exists at x = L/2 = 0.5 nm. The probability density is exactly zero there — the electron has zero probability of being found at the center of the box in the n=2 state.
A wave function psi(x) is a mathematical description of the quantum state of a particle. It is complex-valued and its squared magnitude |psi(x)|^2 gives the probability density (probability per unit length in 1D) of finding the particle at position x.
A node is a point where the wave function is exactly zero, meaning the probability of finding the particle there is zero. The nth energy level has n-1 interior nodes (not counting the walls).
The minimum energy of a quantum system, which is nonzero even at absolute zero temperature. For the particle in a box, E_1 = pi^2*hbar^2/(2mL^2) > 0. It arises because perfectly localizing a particle requires infinite momentum uncertainty.
Energy scales as 1/L^2. A smaller box gives higher energy levels (larger confinement). This is why quantum dots emit higher energy (bluer) light when smaller: the confined electron energy levels are higher.
It is an idealization. Real systems have finite potential wells (electrons can tunnel out). But it captures the essential physics of quantum confinement and gives qualitatively correct energy spectra for quantum dots and pi-electron systems.
Quantum dots are semiconductor nanocrystals where electrons and holes are confined in all three dimensions. The 3D particle-in-a-box model predicts that smaller dots emit higher energy (shorter wavelength, bluer) light, which is observed experimentally.
The integral of |psi|^2 over all space equals 1, ensuring total probability = 100%. The sqrt(2/L) prefactor normalizes the particle-in-a-box wave functions so this condition is satisfied.
Probability density |psi(x)|^2 has units of 1/length. Actual probability requires multiplying by a length interval dx: P(x, x+dx) = |psi(x)|^2 * dx. The probability is dimensionless and between 0 and 1.
For bosons (integer spin), yes. For fermions like electrons (half-integer spin), the Pauli exclusion principle forbids two identical fermions from occupying the same quantum state, so each n level holds at most 2 electrons (spin up and down).
In a 3D box with sides Lx, Ly, Lz the energy is E_nx,ny,nz = (pi^2*hbar^2/2m)*(nx^2/Lx^2 + ny^2/Ly^2 + nz^2/Lz^2). Degenerate states (same energy, different quantum numbers) appear when dimensions are equal.
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