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The Schrödinger Equation Calculator evaluates the time-independent wavefunction and probability density for a particle in an infinite potential well — the prototypical solution to the Schrödinger equation. This equation, formulated by Erwin Schrödinger in 1926, is the fundamental equation of non-relativistic quantum mechanics: $$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi = E\psi$$
For the infinite square well (V = 0 inside the box, V = ∞ outside), the boundary conditions ψ(0) = ψ(L) = 0 yield the normalized solutions: $$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\!\left(\frac{n\pi x}{L}\right)$$ with corresponding energies Eₙ = n²π²ħ²/(2mL²). The wavefunction completely describes the quantum state of the particle.
The physical meaning of the wavefunction comes from Born's interpretation: the probability density |ψ(x)|² gives the probability per unit length of finding the particle at position x. The probability of finding the particle between x₁ and x₂ is the integral of |ψ|² over that interval. Normalization ensures the total probability over the entire box equals 1.
This calculator evaluates ψ(x) and |ψ(x)|² at any position x for any quantum state n. It reveals key quantum features: the wavefunction oscillates with n half-wavelengths fitting in the box, nodes appear at x = kL/n, and the probability density shows where the particle is most and least likely to be found. For even n, there is a node at the center; for odd n, the center is an antinode (local maximum of probability).
The time-independent Schrödinger equation is a second-order differential equation that acts as an eigenvalue problem: the allowed wavefunctions are eigenfunctions and the allowed energies are eigenvalues. This mathematical structure underlies all of quantum mechanics, from atomic orbitals to solid-state band theory to quantum computing.
Understanding the Schrödinger equation solutions is essential preparation for studying the hydrogen atom, the harmonic oscillator, tunneling through barriers, and perturbation theory. This calculator provides a hands-on tool for exploring wavefunctions and building quantum mechanical intuition.
The calculator solves the time-independent Schrödinger equation for the infinite square well:
Wavefunction:
$$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\!\left(\frac{n\pi x}{L}\right)$$
This satisfies the boundary conditions ψ(0) = ψ(L) = 0 and is normalized so that ∫₀ᴸ|ψ|²dx = 1.
Probability Density:
$$|\psi_n(x)|^2 = \frac{2}{L}\sin^2\!\left(\frac{n\pi x}{L}\right)$$
Normalization Constant:
$$A = \sqrt{\frac{2}{L}}$$
Energy Eigenvalue:
$$E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$$
Probability in Half-Box:
$$P(0 \leq x \leq L/2) = \frac{1}{2}$$
By symmetry, the probability of finding the particle in either half of the box is always 1/2, regardless of quantum number.
The wavefunction ψ(x) at your chosen position gives the quantum amplitude — it can be positive or negative. The probability density |ψ|² is always non-negative and tells you the likelihood of finding the particle near that position. At nodes, both ψ and |ψ|² are zero. The maximum value of |ψ| equals √(2/L) and occurs at antinodes. For n = 1, the particle is most likely found at the center (x = L/2). For n = 2, the center is a node and the particle is most likely near x = L/4 and x = 3L/4.
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At the center of a 1 nm box, the ground-state wavefunction reaches its maximum value ψ = √(2/L) ≈ 1.414×10⁹ m⁻½. The probability density is 2/L = 2×10¹⁸ m⁻¹.
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For n = 2, x = L/2 is a node where ψ = 0 and the probability density vanishes. The energy is 4 times the ground-state energy (1.50 eV).
The time-independent Schrödinger equation –(ħ²/2m)d²ψ/dx² + Vψ = Eψ determines the allowed wavefunctions and energy levels of a quantum system. It is an eigenvalue equation: the Hamiltonian operator acting on ψ returns E times ψ. For the infinite square well, the solutions are sinusoidal standing waves with quantized energies proportional to n².
The wavefunction ψ(x) itself is a complex amplitude with no direct physical meaning. Its squared modulus |ψ(x)|² gives the probability density — the probability per unit length of finding the particle at position x. This is Born's rule, one of the foundational postulates of quantum mechanics. The total probability must integrate to 1 (normalization).
The infinite potential at the walls means the particle cannot exist there (it would require infinite energy). Mathematically, continuity of ψ requires ψ(0) = ψ(L) = 0, since ψ = 0 outside the box. These boundary conditions are what quantize the allowed wavelengths and energies, just as a vibrating string fixed at both ends has quantized modes.
Yes, for the infinite square well, the probability of finding the particle in the left half (0 to L/2) equals exactly 1/2 for every energy eigenstate. This follows from the symmetry of sin²(nπx/L) about x = L/2. However, this is not true for superposition states or for asymmetric potentials.
Atomic orbitals are the 3D analogue of the particle-in-a-box wavefunctions. Instead of a box potential, the electron is bound by the Coulomb potential of the nucleus. The mathematics is more complex (spherical harmonics, radial wavefunctions), but the core concepts — quantization, nodes, normalization, probability interpretation — are identical to what you learn from the 1D box.
The normalization constant A = √(2/L) ensures that the total probability of finding the particle somewhere in the box equals 1: ∫₀ᴸ|ψ|²dx = A²∫₀ᴸsin²(nπx/L)dx = A²(L/2) = 1, giving A = √(2/L). Without normalization, the wavefunction would not have a proper probabilistic interpretation.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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