Schrödinger Equation · aiwiki.page

The Schrödinger equation is a fundamental equation of quantum mechanics that describes how a quantum state changes with time. In its familiar position-space form, it is a linear differential equation for a wave function, whose amplitudes determine probabilities for measurement outcomes. Its time-independent form identifies states of definite energy and their allowed energies. The equation provides a mathematical framework for describing particles, atomic structure, and other quantum phenomena. (ocw.mit.edu)

Historical development

Erwin Schrödinger introduced his wave equation in 1926 while working at the University of Zurich. He was influenced by Louis de Broglie’s proposal that matter possesses wave properties. Schrödinger submitted the first paper in his series, “Quantization as an Eigenvalue Problem,” to Annalen der Physik on January 26, 1926. The approach made it possible to calculate the energy levels of an electron in an atom from a wave equation rather than prescribed classical orbits. (uzh.ch)

Schrödinger shared the 1933 Nobel Prize in Physics with Paul Adrien Maurice Dirac for their contributions to atomic theory. The subsequent probabilistic interpretation distinguished the wave function from an ordinary material wave: it supplies probability amplitudes rather than a directly observable distribution of matter. (nobelprize.org)

Time-dependent form

The general state-vector equation is

i\hbar\frac{d}{dt}|\Psi(t)\rangle =\hat H(t)|\Psi(t)\rangle.

Here $i$ is the imaginary unit, $\hbar=h/(2\pi)$ is the reduced Planck constant, and $\hat H$ is the Hamiltonian operator, representing the system’s total energy. The state vector belongs to a Hilbert space. Specifying the Hamiltonian and an initial state determines the subsequent state through this equation. The equation is a postulate of quantum mechanics, not a consequence of classical mechanics alone. (damtp.cam.ac.uk)

For a single nonrelativistic particle of mass $m$ , moving in a scalar potential $V(\mathbf r,t)$ without magnetic coupling, the position-space expression is

i\hbar\frac{\partial\Psi(\mathbf r,t)}{\partial t} = \left[-\frac{\hbar^2}{2m}\nabla^2 +V(\mathbf r,t)\right]\Psi(\mathbf r,t).

The Laplacian $\nabla^2$ sums second spatial derivatives. Its coefficient supplies the kinetic-energy operator; multiplication by $V$ supplies potential energy. This equation is first order in time and second order in space. Spatial boundary conditions and the initial wave function are essential parts of a particular physical problem. (damtp.cam.ac.uk)

Probability and linear evolution

According to the Born rule, a normalized single-particle wave function gives the position probability density

\rho(\mathbf r,t)=|\Psi(\mathbf r,t)|^2, \qquad \int_{\mathbb R^3}|\Psi(\mathbf r,t)|^2\,d^3r=1.

Integrating this density over a region gives the probability of finding the particle there. The wave function is generally complex-valued; its magnitude alone does not specify the complete state, because relative phases affect interference. (ocw.mit.edu)

Linearity means that any linear combination of solutions for the same Hamiltonian is also a solution, subject to the required physical conditions. This supports quantum superposition. For a self-adjoint Hamiltonian with an appropriate domain, evolution preserves normalization. When the Hamiltonian is time-independent, the evolution operator is

U(t,t_0)= \exp\!\left[-\frac{i}{\hbar}\hat H(t-t_0)\right].

Thus, the state evolves deterministically even though the probabilities calculated from it need not select a unique measurement outcome. (live.ocw.mit.edu)

Time-independent equation and stationary states

When the Hamiltonian does not depend explicitly on time, solutions of definite energy can be separated into spatial and temporal factors:

\Psi(\mathbf r,t)=\psi(\mathbf r)e^{-iEt/\hbar}.

Substitution yields the time-independent equation,

\hat H\psi=E\psi.

This is an eigenvalue problem: $E$ is an energy eigenvalue and $\psi$ is its eigenfunction. Such a state is stationary because its position probability density is independent of time, although the wave function’s overall phase continues to change. A superposition of different energy eigenstates can instead have a time-dependent density. (ocw.mit.edu)

Boundary conditions and normalizability determine which solutions are physically admissible. Bound states commonly have discrete energies, providing a mechanism for energy quantization. Unconfined particles can have continuous energy spectra, so discreteness is not a universal consequence of the equation. (live.ocw.mit.edu)

Representative solutions and applications

The particle in a box illustrates how confinement produces discrete energies. For an infinite one-dimensional well of width $L$ ,

E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}, \qquad n=1,2,3,\ldots.

Its wave functions vanish at the walls and form standing-wave patterns inside the well. The quantum harmonic oscillator provides another exactly solvable model, with equally spaced energies $E_n=\hbar\omega(n+\tfrac12)$ . (live.ocw.mit.edu)

For hydrogen, the attractive Coulomb potential produces discrete bound-state energies and spatial wave functions called atomic orbitals. These are not classical trajectories. Potential-step and barrier problems describe scattering and quantum tunneling, including transmission through regions inaccessible to a classical particle of the same energy. (damtp.cam.ac.uk)

Many-particle equations involve wave functions depending on all particle coordinates. Interactions make most realistic problems difficult to solve exactly. Perturbation theory, variational methods, and numerical approximations extend the framework to many-electron atoms, molecules, and solids. (ocw.mit.edu)

Scope and limitations

The familiar scalar, single-particle equation is nonrelativistic and does not by itself include intrinsic spin. Spin-dependent descriptions require additional components and suitable Hamiltonians; relativistic electron dynamics are described by the Dirac equation. Processes involving particle creation and annihilation require quantum field theory. Nevertheless, the abstract Schrödinger equation remains a statement of quantum-state evolution when the state space and Hamiltonian are appropriately generalized. (cambridge.org)