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The Schrödinger equation is the fundamental governing equation of non-relativistic quantum mechanics, describing how the quantum state of a physical system evolves over time.

1. Core Formulations

Time-Dependent Schrödinger Equation

The TDSE describes the dynamic evolution of a wave function $\psi(\mathbf{r}, t)$ in a state space:

$$ i\hbar \frac{\partial \psi(\mathbf{r}, t)}{\partial t} = \hat{H} \psi(\mathbf{r}, t) $$

Where $\hat{H}$ is the Hamiltonian operator, representing the total energy of the system. For a single particle of mass $m$ moving in a potential $V(\mathbf{r}, t)$, the explicit coordinate-space Hamiltonian is:

$$ \hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r}, t) $$

Time-Independent Schrödinger Equation

When the potential $V(\mathbf{r})$ is stationary (independent of time), we can look for separable solutions of the form $\Psi(\mathbf{r}, t) = \psi(\mathbf{r})\phi(t)$. This separation of variables reduces the problem to an eigenvalue problem of the form:

$$ \hat{H} \psi(\mathbf{r}) = E \psi(\mathbf{r}) $$
$$ -\frac{\hbar^2}{2m}\nabla^2 \psi(\mathbf{r}) + V(\mathbf{r})\psi(\mathbf{r}) = E \psi(\mathbf{r}) $$

Where $E$ represents the definite energy eigenvalues of the stationary states. The temporal component simply evolves as a phase factor: $\phi(t) = e^{-iEt/\hbar}$.

2. Properties of Solutions

  • Linearity & Superposition: If $\psi_1$ and $\psi_2$ are solutions to the TDSE, then any linear combination $c_1\psi_1 + c_2\psi_2$ is also a solution.

  • Normalization: The total probability of finding the particle somewhere in space must equal $1$: $$\int_{-\infty}^{\infty} |\psi(x,t)|^2 dx = 1$$

  • Orthogonality: Stationary states belonging to distinct energy eigenvalues are orthogonal: $$\int_{-\infty}^{\infty} \psi_m^*(x) \psi_n(x) dx = \delta_{mn}$$

3. Particular Solutions (1D Systems)

The Free Particle

For a particle experiencing zero potential everywhere, the hamiltonian simplifies to:

$$ \hat{H} = \frac{\hat{p}^2}{2m} $$

Since $[\hat{H}, \hat{p}] = 0$, we can choose the eigenstate of the hamiltonian to be those of the impulse operator, which acts as:

$$ \hat{p} \ket{k} = \hbar k \ket{k} $$
$$ \hat{H} \ket{k} = \frac{\hbar^2 k^2}{2m} \ket{k} $$
$$ E_k = \frac{\hbar^2 k^2}{2m} $$

$$-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} = E \psi$$

  • General Solution: $$\psi(x) = A e^{ikx} + B e^{-ikx}$$

Where the wavenumber $k$ is defined by:

$$k^2 = \frac{2mE}{\hbar^2}$$

  • Physical Meaning: Represents a superposition of forward- and backward-propagating plane waves with continuous energy spectra ($E \ge 0$). Note that these idealized states cannot be normalized across infinite space without forming wave packets.

Infinite Potential Well

Finite Potential Step

Finite Potential Barrier

A particle confined to a 1D box of length $L$, where: $$V(x) = \begin{cases} 0 & 0 < x < L \\ \infty & \text{otherwise} \end{cases}$$

Applying boundary conditions ($\psi(0) = \psi(L) = 0$) yields quantized energy levels and strictly bounded wave functions:

  • Normalized Eigenfunctions: $$\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right), \quad n = 1, 2, 3, \dots$$

  • Energy Quantization: $$E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$$

Case C: Quantum Harmonic Oscillator

A particle subject to a parabolic restoring potential, modeling smooth localized fluctuations: $$V(x) = \frac{1}{2}m\omega^2 x^2$$

Solving this requires transforming the TISE into Hermite’s differential equation.

  • Energy Quantization: $$E_n = \hbar\omega\left(n + \frac{1}{2}\right), \quad n = 0, 1, 2, \dots$$

Note: The ground state ($n=0$) possesses a non-zero minimum energy $E_0 = \frac{1}{2}\hbar\omega$, known as the zero-point energy, a direct consequence of the uncertainty principle.

  • Eigenfunctions: $$\psi_n(x) = \left( \frac{m\omega}{\pi \hbar} \right)^{1/4} \frac{1}{\sqrt{2^n n!}} H_n(\xi) e^{-\xi^2 / 2}$$

Where $\xi = \sqrt{\frac{m\omega}{\hbar}} x$ and $H_n(\xi)$ are the physicist’s Hermite polynomials.

4. Particular Solutions (3D Systems)