Table of Contents
Overview#
In 1926, Erwin Schrödinger developed wave mechanics, providing a complete mathematical framework for quantum mechanics. His wave equation describes how quantum systems evolve in time.
The Schrödinger Equation#
Time-Dependent Form#
$$ i\hbar \frac{\partial \Psi}{\partial t} = \hat{H}\Psi $$For a particle in potential \(V(x)\):
$$ i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2} + V(x)\Psi $$Time-Independent Form#
For stationary states \(\Psi(x,t) = \psi(x)e^{-iEt/\hbar}\):
$$ \hat{H}\psi = E\psi $$$$ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi = E\psi $$Derivation Motivation#
From de Broglie Waves#
For a free particle wave:
$$ \Psi = Ae^{i(kx - \omega t)} $$Taking derivatives:
- \(\frac{\partial \Psi}{\partial t} = -i\omega\Psi\) → \(E = \hbar\omega\)
- \(\frac{\partial^2 \Psi}{\partial x^2} = -k^2\Psi\) → \(p = \hbar k\)
Energy Relation#
Kinetic energy:
$$ E = \frac{p^2}{2m} = \frac{\hbar^2 k^2}{2m} $$This leads naturally to the Schrödinger equation.
Key Concepts#
Wave Function \(\Psi\)#
- Complex-valued function
- Contains all information about the system
- \(|\Psi|^2\) gives probability density
Hamiltonian Operator#
$$ \hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) $$Total energy = Kinetic + Potential
Operators and Observables#
| Observable | Operator |
|---|---|
| Position | \(\hat{x} = x\) |
| Momentum | \(\hat{p} = -i\hbar\frac{\partial}{\partial x}\) |
| Energy | \(\hat{H}\) |
Important Solutions#
Free Particle#
\(V = 0\):
$$ \psi_k(x) = Ae^{ikx} $$Continuous energy spectrum.
Infinite Square Well#
$$ E_n = \frac{n^2\pi^2\hbar^2}{2mL^2} $$$$ \psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right) $$Harmonic Oscillator#
\(V = \frac{1}{2}m\omega^2x^2\):
$$ E_n = \hbar\omega\left(n + \frac{1}{2}\right) $$Ground state has zero-point energy!
Hydrogen Atom#
$$ E_n = -\frac{13.6 \text{ eV}}{n^2} $$Reproduces Bohr model results, plus angular momentum states.
Properties of Solutions#
Normalization#
$$ \int_{-\infty}^{\infty} |\Psi|^2 dx = 1 $$Orthogonality#
$$ \int \psi_m^* \psi_n dx = \delta_{mn} $$Completeness#
Any wave function can be expanded:
$$ \Psi = \sum_n c_n \psi_n $$Matrix Mechanics Equivalence#
Schrödinger proved his wave mechanics is equivalent to Heisenberg’s matrix mechanics (1925):
| Wave Mechanics | Matrix Mechanics |
|---|---|
| Wave functions | State vectors |
| Operators | Matrices |
| Differential equations | Matrix equations |
Both give identical predictions.
Interpretations#
Born Interpretation#
\(|\Psi(x)|^2\) is probability density.
Max Born received Nobel Prize (1954) for this interpretation.
Copenhagen Interpretation#
- Wave function is complete description
- Measurement causes collapse
- No underlying deterministic reality
Schrödinger’s Cat#
Famous thought experiment highlighting measurement paradox:
- Cat in superposition until observed
- Illustrates interpretation difficulties
Three-Dimensional Form#
$$ i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\Psi + V(\mathbf{r})\Psi $$Where:
$$ \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} $$Applications#
- Atomic structure - Electron orbitals
- Molecular chemistry - Chemical bonds
- Solid state physics - Band theory
- Quantum computing - Qubit evolution
- Quantum field theory - Foundation
Nobel Prize#
Erwin Schrödinger shared the Nobel Prize in Physics (1933) with Paul Dirac:
“For the discovery of new productive forms of atomic theory”
Legacy#
The Schrödinger equation is the fundamental equation of non-relativistic quantum mechanics, as central to quantum physics as Newton’s laws to classical mechanics.