gen-define-the-schrodinger-equation-and-give-one-short-worked-ex-aa71

Define the Schrodinger equation and give one short worked example.

# The Schrödinger Equation The **time-dependent Schrödinger equation** is the fundamental equation of quantum mechanics: $$i\hbar\frac{\partial\Psi}{\partial t} = \hat{H}\Psi$$ where: - $\Psi$ is the wave function - $\hbar$ is the reduced Planck constant - $\hat{H}$ is the Hamiltonian operator (total energy) - $i$ is the imaginary unit For time-independent problems, this reduces to: $$\hat{H}\Psi = E\Psi$$ where $E$ is the energy eigenvalue. ## Worked Example: Particle in a 1D Box **Problem:** Find the allowed energies for a particle of mass $m$ confined to a box of length $L$ (with infinite potential walls). **Solution:** Inside the box (0 < x < L), the Hamiltonian is just kinetic energy: $$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi$$ This gives: $\frac{d^2\psi}{dx^2} = -k^2\psi$ where $k^2 = \frac{2mE}{\hbar^2}$ General solution: $\psi(x) = A\sin(kx) + B\cos(kx)$ Boundary conditions: $\psi(0) = 0$ and $\psi(L) = 0$ From $\psi(0) = 0$: $B = 0$ From $\psi(L) = 0$: $\sin(kL) = 0$, so $kL = n\pi$ where $n = 1, 2, 3...$ **Result:** The allowed energies are: $$E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}, \quad n = 1, 2, 3...$$ This shows **energy quantization** - only discrete energy levels are allowed!