Schrödinger Equation (Time-Independent)

📊 advanced

$$\hat{H}\psi = E\psi$$

Eigenvalue equation for quantum states, where $\hat{H}$ is the Hamiltonian operator.

Variables & Units

$\hat{H}$

Hamiltonian operator

Unit: $\text{J}$

$\psi$

Wavefunction

Unit: $\text{m}^{-3/2}$ (1D: $\text{m}^{-1/2}$)

$E$

Energy eigenvalue

Unit: $\text{J}$

Key Information

📐 Derived From

From classical Hamiltonian $H = T + V$ with $p \to -i\hbar\nabla$.

💡 Example

For particle in a 1D box of length $L$, $\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$, $E_n = \frac{n^2h^2}{8mL^2}$.

🔧 Applications

Atomic orbitals, semiconductor physics, quantum chemistry.

⚠️ Constraints

Non-relativistic, spinless particle.

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