What Is the Schrodinger Equation?

The Schrodinger equation is the fundamental partial differential equation of non-relativistic quantum mechanics, governing how the quantum state of a physical system evolves over time. Postulated by Austrian physicist Erwin Schrodinger in 1925 and published in 1926, it introduced the concept of the wave function, a mathematical object whose squared magnitude gives the probability of finding a particle at a given position when a measurement is performed. The equation unified earlier quantum hypotheses by treating the electron not as a point particle following a classical trajectory but as a wave whose amplitude and phase evolve according to a deterministic rule, reconciling the wave-particle duality that had been observed in diffraction experiments.

The equation draws its formal structure from classical Hamiltonian mechanics, replacing classical energy expressions with quantum mechanical operators. It forms the theoretical foundation for atomic physics, molecular chemistry, solid-state electronics, and quantum computing, and its solutions, the energy eigenstates and their superpositions, describe everything from the discrete spectral lines of hydrogen to the band structure of crystalline materials.

Time-Dependent and Time-Independent Forms

The time-dependent Schrodinger equation describes how the wave function of a system changes at every instant and applies to any quantum system evolving under a Hamiltonian that may itself vary with time. For the common case of a conservative system, where the potential energy does not change with time, the equation separates into a spatial part and a temporal part. The spatial component, the time-independent Schrodinger equation, reduces to an eigenvalue problem in which the energy eigenvalues correspond to the allowed, or stationary, energy levels of the system and the eigenfunctions describe the corresponding probability amplitudes in space. For the hydrogen atom, Schrodinger's original solution, detailed in an arXiv analysis of his 1926 method, yields discrete energy levels that match the observed emission spectrum exactly, providing the first rigorous derivation of the Balmer series from first principles.

Wave Function and Quantum States

The wave function is a complex-valued function of position and time whose physical content lies in its squared modulus: the Born interpretation states that the probability of detecting a particle in an infinitesimal volume element is proportional to the squared magnitude of the wave function integrated over that element. The wave function must satisfy normalization, boundary conditions appropriate to the potential, and continuity of both the function and its first derivative at interfaces between potential regions. Superpositions of stationary states produce interference effects with observable consequences, including quantum tunneling through potential barriers that would be classically forbidden. The relationship between the Schrodinger equation and phase-space formulations of quantum mechanics is examined in a quant-ph preprint on Schrodinger equation and phase space, which connects the wave equation approach to alternative operator formalisms such as the Wigner distribution. A mathematical exposition of the Schrodinger equation by Terry Tao, available through UCLA's mathematics preprints, treats dispersive properties of solutions in the context of rigorous analysis.

Application to Electrons in Materials

The Schrodinger equation's most consequential engineering application is in describing electrons in semiconductor structures. Solving the equation for an electron in a periodic crystal potential yields the Bloch theorem and the band structure theory that underlies transistor operation, p-n junction behavior, and the optical properties of semiconductors. In heterostructure devices such as quantum wells and quantum dots, the time-independent Schrodinger equation with a position-dependent effective mass gives the quantized energy levels that determine laser emission wavelengths and electron confinement in high-electron-mobility transistors.

Applications

The Schrodinger equation has applications in a range of fields, including:

• Semiconductor device physics, including band structure computation and quantum well energy-level design
• Quantum chemistry, where molecular orbital methods solve the many-electron Schrodinger equation numerically
• Quantum computing, where qubit Hamiltonians are designed by engineering the Schrodinger equation's eigenstates
• Atomic and nuclear physics modeling of bound-state energy levels
• Scanning tunneling microscopy, which exploits quantum tunneling predicted by barrier solutions to the equation