The Schrodinger equation provides a mathematical language to describe how the quantum state of a physical system changes over time. It transformed physics by replacing classical trajectories with evolving probability waves, unifying quantum mechanics through a powerful differential equation that remains the bedrock of modern quantum chemistry, condensed matter physics, and solid-state electronics.
where, \[ \hbar = \frac{h}{2\pi} = 1.054573 \times 10^{-34} J s \]
Erwin Schrödinger was an Austrian physicist that formulated The Schrodinger equation in 1925 and published in early 1926. It marks a pivotal turning point in physics by establishing wave mechanics as a core framework of quantum theory.
Theoretical foundations
The development of the equation relied on bridging classical mechanics, optics, and emerging quantum hypotheses:
‣ The crisis of the Bohr model:
By the early 1920s, the old quantum theory of Niels Bohr and Arnold Sommerfeld could not adequately explain the spectra of
multi-electron atoms or the intensities of spectral lines.
In 1924, Louis de Broglie proposed that all matter exhibits wave-particle duality. He postulated that a
particle with momentum \(p\) possesses an associated wavelength given by \(\lambda = \frac{h}{p}\), where \(h\) is Planck's constant.
Schrödinger built heavily on William Rowan Hamilton’s 19th-century work, which demonstrated a mathematical
analogy between geometric optics (light acting as rays) and classical mechanics (particles acting as trajectories). Schrödinger reasoned that
just as geometric optics is a limit of physical wave optics, classical mechanics must be a limit of a more fundamental wave mechanics.
During a vacation in Arosa, Switzerland, in late 1925, Schrödinger sought a clean wave equation to describe de Broglie's matter waves for a bound electron:
‣ The Relativistic Failure:
Schrödinger initially attempted to construct a relativistic wave equation. However,
when applied to the hydrogen atom, it yielded predictions that disagreed with experimental data because it omitted electron spin (this equation
was later recognized as the Klein-Gordon equation).
Stripping away relativity, Schrödinger focused on a non-relativistic approach. By combining the classical energy conservation equation with a wave ansatz,
he successfully derived his famous time-independent equation. When applied to the hydrogen atom, it flawlessly reproduced the correct Bohr energy levels.
In 1926, Schrödinger published a series of four papers titled "Quantisierung als Eigenwertproblem" (Quantization as an Eigenvalue Problem):
‣ The Eigenvalue Breakthrough:
He showed that quantization was not an arbitrary rule forced onto physics, but a natural consequence of solving differential equations under specific
boundary conditions (eigenvalues).
Months earlier, Werner Heisenberg, Max Born, and Pascual Jordan had developed matrix mechanics.
Schrödinger initially disliked the abstract, non-visual nature of Heisenberg's matrices.
In May 1926, Schrödinger formally proved that his wave mechanics and Heisenberg’s matrix mechanics were mathematically equivalent, despite their vastly different
conceptual frameworks.