Schrödinger equation

Schrödinger equation
Type	Fundamental equation
Key terms	time-dependent; time-independent; Hamiltonian
Concepts	normalization; boundary conditions; stationary states
Related	Hamiltonian operator; eigenvalue problem; unitary evolution
Domain	Quantum mechanics
Examples	infinite square well; harmonic oscillator
Methods	separation of variables
Wikidata	Q165498

The Schrödinger equation is the fundamental equation of non-relativistic quantum mechanics. It describes how the quantum wavefunction of a system evolves in time, analogous to how Newton’s laws govern motion in classical physics. The wavefunction Ψ(x,t) encodes the state of a particle or system; its absolute square |Ψ(x,t)|² gives the probability density for finding the particle at position x at time t. There are two main forms of the Schrödinger equation: the time-dependent equation for general evolution, and the time-independent equation for stationary states of definite energy. The operator in the equation, called the Hamiltonian, represents the total energy (kinetic plus potential) of the system. Solving the Schrödinger equation (often by separating variables) gives stationary states with quantized energy levels. Classic examples include a particle in an infinite potential well and the quantum harmonic oscillator. The Schrödinger equation is essential to understanding atoms, molecules, and modern technologies like semiconductors and lasers.

Historical Context and Evolution

By the early 20th century, experiments (such as the discrete spectral lines of atoms) showed that classical physics was incomplete. In 1926 the Austrian physicist Erwin Schrödinger proposed a wave equation for matter, influenced by de Broglie’s idea that particles have wave-like properties. Schrödinger’s equation unified earlier quantum ideas: electrons in atoms could be described by standing waves whose frequencies correspond to allowed energy levels. His work paralleled Heisenberg’s matrix mechanics (an equivalent formulation of quantum theory), and together they established the new quantum mechanics.

Schrödinger showed that solutions of his equation for the hydrogen atom reproduced the empirical energy levels found in experiments. The new theory replaced Bohr’s the quantized-orbit model with wavefunctions and probabilities. Schrödinger won the 1933 Nobel Prize in Physics for this work. Since then, the Schrödinger equation has been extended (for example, to include electron spin and relativistic effects via the Dirac equation), but the original non-relativistic form remains the starting point of most quantum theory. It underlies much of modern physics and chemistry, explaining everything from chemical bond structure to the behavior of electrons in a transistor.

The Schrödinger Equation: Time-Dependent and Time-Independent Forms

The time-dependent Schrödinger equation (TDSE) for a single particle of mass m in a potential V(x) (in one dimension) is:

$i\hbar {\frac {\partial }{\partial t}}\Psi (x,t)\;=\;-{\frac {\hbar ^{2}}{2m}}\,{\frac {\partial ^{2}}{\partial x^{2}}}\Psi (x,t)\;+\;V(x)\,\Psi (x,t).$

Here $\hbar$ is the reduced Planck constant ( $\hbar =h/2\pi$ ), $i$ is the imaginary unit, and $\Psi (x,t)$ is the complex wavefunction. This partial differential equation governs how $\Psi$ changes in time. In three dimensions, the spatial derivative term generalizes to the Laplacian $-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi$ . Physically, the right-hand side is the Hamiltonian operator ${\hat {H}}$ acting on $\Psi$ .

When the potential $V(x)$ does not depend on time, one often looks for solutions by separating variables: assume $\Psi (x,t)=\psi (x)\,\phi (t)$ . Substituting this into the TDSE and dividing by $\Psi$ allows one to set each side equal to a constant (called the energy $E$ ). The result is two simpler equations: one for time and one for space. The spatial part $\psi (x)$ must satisfy the time-independent Schrödinger equation (TISE):

${\hat {H}}\,\psi (x)=E\,\psi (x),$ or explicitly. $-{\frac {\hbar ^{2}}{2m}}\,{\frac {d^{2}\psi }{dx^{2}}}+V(x)\,\psi (x)=E\,\psi (x).$

This is an eigenvalue equation for the Hamiltonian: it says that certain special functions $\psi (x)$ (the energy eigenfunctions) yield definite values $E$ (the energy eigenvalues) when acted on by ${\hat {H}}$ . In three dimensions, ${\hat {H}}\psi ={\bigl (}-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} ){\bigr )}\psi (\mathbf {r} )=E\,\psi (\mathbf {r} )$ .

The general time-dependent solution can then be built from these stationary states. For a given energy eigenstate, the time-dependent part is $\phi (t)\propto e^{-iEt/\hbar }$ . Thus a stationary state has the form. $\Psi (x,t)=\psi (x)\,e^{-iEt/\hbar }.$ The probability density $|\Psi |^{2}=|\psi (x)|^{2}$ is independent of time for such a state. A general solution of the TDSE is a linear combination (superposition) of these stationary states, each evolving with its own phase factor $e^{-iE_{n}t/\hbar }$ .

The Hamiltonian and Energy

In quantum mechanics, the Hamiltonian operator ${\hat {H}}$ corresponds to the total energy of the system. For a particle of mass m moving in a potential V(x), the Hamiltonian is the sum of kinetic and potential energy operators. In coordinate representation, ${\hat {H}}={\hat {T}}+{\hat {V}}=-{\frac {\hbar ^{2}}{2m}}\,{\frac {d^{2}}{dx^{2}}}\;+\;V(x).$ Here $-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}$ is the kinetic energy operator (analogous to $p^{2}/2m$ , with momentum $p\to -i\hbar {\frac {d}{dx}}$ ), and $V(x)$ acts as a multiplication operator. The Hamiltonian is a Hermitian (self-adjoint) operator, which ensures that its eigenvalues $E$ are real. These eigenvalues represent the allowed energy levels of the system.

Operating the Hamiltonian on a wavefunction $\psi$ appears in both forms of the Schrödinger equation. In the TISE, ${\hat {H}}\psi =E\psi$ explicitly shows the energy eigenvalue problem. In the TDSE, ${\hat {H}}$ governs time evolution via $i\hbar \partial \Psi /\partial t={\hat {H}}\Psi$ . Thus, ${\hat {H}}$ plays a dual role: it generates the system’s energy spectrum and it drives the time evolution of states.

Wavefunctions: Normalization and Boundary Conditions

The wavefunction $\Psi (x,t)$ is generally a complex-valued function. Physically, it is a probability amplitude: its absolute square $|\Psi (x,t)|^{2}$ is interpreted (per Max Born’s rule) as the probability density of finding the particle near position $x$ at time $t$ . Thus the wavefunction must be normalizable, meaning the total probability over all space is finite (and usually set to 1). Mathematically, this is expressed by the normalization condition: $\int _{-\infty }^{\infty }|\Psi (x,t)|^{2}\,dx\;=\;1.$ For a stationary (time-independent) state $\psi (x)$ , one often writes $\int |\psi (x)|^{2}dx=1$ . If a wavefunction is not normalized, it can always be scaled (multiplied by a constant) so that this condition holds.

In addition to normalization, physically acceptable wavefunctions must satisfy certain boundary conditions and regularity conditions:

Single-valued and continuous: $\Psi (x,t)$ must return a single value at each $x$ , and be continuous. (Wavefunctions that double-back on themselves or are discontinuous are unphysical.)
Square-integrable: $\Psi$ must be normalizable, which usually means $\Psi \to 0$ as $|x|\to \infty$ . For bound states in a finite or infinite potential well, $\Psi$ tends to zero outside the classically allowed region.
Finite and well-behaved derivatives: In regions where the potential $V(x)$ is finite, $\Psi (x)$ and its first derivative $\Psi '(x)$ must be continuous. (At a point of infinite potential, the wavefunction itself must go to zero at that boundary.)

These conditions ensure the wavefunction is physically reasonable. For example, the infinite potential well (particle in a box) requires $\Psi =0$ at the walls. More generally, any bound state wavefunction vanishes at infinity so that the probability of finding the particle far away is zero.

Importantly, the time evolution given by the Schrödinger equation preserves normalization and continuity if ${\hat {H}}$ is Hermitian. Thus if $\Psi$ is normalized at one time, it remains so as it evolves.

Stationary States and Separation of Variables

Stationary states are solutions of the Schrödinger equation with a definite energy. When the potential $V(x)$ is time-independent, one seeks solutions of the form. $\Psi (x,t)=\psi (x)\,T(t),$ where $\psi (x)$ depends only on position and $T(t)$ depends only on time. Substituting into the TDSE yields. $i\hbar \psi (x)\,{\frac {dT}{dt}}=-{\frac {\hbar ^{2}}{2m}}T(t)\,{\frac {d^{2}\psi }{dx^{2}}}+V(x)\,\psi (x)T(t).$ Dividing both sides by $\psi (x)T(t)$ separates the variables: $i\hbar \,{\frac {1}{T}}{\frac {dT}{dt}}\;=\;-{\frac {\hbar ^{2}}{2m}}\,{\frac {1}{\psi }}{\frac {d^{2}\psi }{dx^{2}}}+V(x).$ The left side depends only on time and the right side only on space $x$ . For this equality to hold at all times and positions, both sides must equal a constant, which we call $E$ . This leads to two ordinary differential equations:

Time equation: $i\hbar \,{\frac {dT}{dt}}=E\,T(t).$

Its solution is  $T(t)=e^{-iEt/\hbar }$  (up to a constant factor).

Space equation (time-independent SE): $-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}+V(x)\,\psi (x)=E\,\psi (x).$

This is exactly the TISE, with  $E$  now identified as the energy eigenvalue.

Thus separation of variables shows that each stationary state has a spatial part $\psi (x)$ satisfying ${\hat {H}}\psi =E\psi$ and a time part $e^{-iEt/\hbar }$ . The full solution is $\Psi (x,t)=\psi (x)e^{-iEt/\hbar }$ . Because the complex phase $e^{-iEt/\hbar }$ has unit magnitude, the probability density $|\Psi (x,t)|^{2}=|\psi (x)|^{2}$ is time-independent. Hence the term stationary: all physical observables (probabilities, expectation values) are constant in time for such states.

Stationary states have definite energy $E$ . In a typical bound system, only certain discrete energies $E_{n}$ give acceptable (normalizable) solutions $\psi _{n}(x)$ . These energies are the eigenvalues of ${\hat {H}}$ , and the corresponding wavefunctions $\psi _{n}$ are orthonormal: $\int \psi _{m}^{*}(x)\,\psi _{n}(x)\,dx=\delta _{mn}.$ Any general (time-dependent) solution of the Schrödinger equation can be expressed as a linear combination of these eigenstates: $\Psi (x,t)=\sum _{n}c_{n}\,\psi _{n}(x)\,e^{-iE_{n}t/\hbar }.$ The coefficients $c_{n}$ are fixed by the initial condition $\Psi (x,0)$ . Each $|c_{n}|^{2}$ then gives the probability of finding the system in the eigenstate $\psi _{n}$ with energy $E_{n}$ . The energy expectation value is $\sum _{n}|c_{n}|^{2}E_{n}$ . In quantum mechanics, stationary states form the basis for most analysis: they are like “modes” of the system.

Examples

Infinite Potential Well (Particle in a Box)

A classic example illustrating quantization is the particle in an infinite potential well. Here a particle of mass m is confined to the region $0<x<L$ by infinitely high walls. The potential is defined as: $V(x)=<<<M0>>>$ Because $V=\infty$ outside, the wavefunction must vanish at $x=0$ and $x=L$ : $\psi (0)=\psi (L)=0$ . Inside the well (where $V=0$ ), the time-independent Schrödinger equation becomes. $-{\frac {\hbar ^{2}}{2m}}\,{\frac {d^{2}\psi }{dx^{2}}}=E\,\psi .$ This is a simple second-order ODE with general solution $\psi (x)=A\sin(kx)+B\cos(kx)$ , where $k^{2}=2mE/\hbar ^{2}$ . Imposing $\psi (0)=0$ forces $B=0$ . The other boundary $\psi (L)=0$ requires $\sin(kL)=0$ , giving $kL=n\pi$ for integer $n=1,2,3,\dots$ . Thus the allowed wavefunctions are. $\psi _{n}(x)=A_{n}\,\sin {\Bigl (}{\tfrac {n\pi x}{L}}{\Bigr )},$ with corresponding energies. $E_{n}={\frac {\hbar ^{2}k^{2}}{2m}}={\frac {\hbar ^{2}}{2m}}{\Bigl (}{\frac {n\pi }{L}}{\Bigr )}^{2}={\frac {n^{2}\pi ^{2}\hbar ^{2}}{2mL^{2}}}.$ Each $n$ gives a stationary state called the $n$ th mode. The normalization constant $A_{n}$ is found by requiring $\int _{0}^{L}|\psi _{n}|^{2}dx=1$ , which yields $A_{n}={\sqrt {2/L}}$ . For example, the ground state ( $n=1$ ) is $\psi _{1}(x)={\sqrt {2/L}}\sin(\pi x/L)$ , with one half-period of a sine wave and no node inside the box; the first excited state ( $n=2$ ) is ${\sqrt {2/L}}\sin(2\pi x/L)$ , which has one node in the middle, etc.

Key features:

Quantized energies: $E_{n}\propto n^{2}$ . The lowest energy is $E_{1}={\frac {\pi ^{2}\hbar ^{2}}{2mL^{2}}}$ . The particle cannot have zero energy because of the boundary constraint.
Standing-wave wavefunctions: They form a complete set for functions on $[0,L]$ . Any initial wavefunction can be expanded in these sine states.
Nodes and antinodes: Higher states have more oscillations. At the nodes $\psi =0$ and the probability density is zero.

This simple model is often used to illustrate quantum confinement and the idea that bound particles have discrete energy levels. It also shows boundary conditions in action: $\psi$ must vanish at the infinite barriers.

Quantum Harmonic Oscillator

Another fundamental example is the quantum harmonic oscillator, where a particle experiences a quadratic (parabolic) potential. In one dimension, $V(x)={\tfrac {1}{2}}m\omega ^{2}x^{2}$ for a mass $m$ and angular frequency $\omega$ . This models, for instance, small vibrations of atoms in a molecule (a diatomic molecule’s bond) or many modes in solid-state physics.

The time-independent Schrödinger equation becomes. $-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}+{\frac {1}{2}}m\omega ^{2}x^{2}\,\psi =E\,\psi .$ Solving this exactly involves special functions (Hermite polynomials) and is more complex than the infinite well. The key results are:

Eigenfunctions: The $n$ th state has the form

 $\psi _{n}(x)=N_{n}\,H_{n}{\bigl (}{\sqrt {\tfrac {m\omega }{\hbar }}}\,x{\bigr )}\,\exp \!{\Bigl (}-{\frac {m\omega x^{2}}{2\hbar }}{\Bigr )},$ 
where  $H_{n}$  is the  $n$ th Hermite polynomial and  $N_{n}$  is a normalization constant. The ground state ( $n=0$ ) is a simple Gaussian  $\psi _{0}(x)\propto e^{-m\omega x^{2}/(2\hbar )}$ . Excited states are this Gaussian times a polynomial factor, leading to wavefunctions with increasing numbers of nodes.

Energy levels: A striking result is that the energies are equally spaced:

 $E_{n}=\hbar \omega {\Bigl (}n+{\tfrac {1}{2}}{\Bigr )},\quad n=0,1,2,\dots$ 
The lowest energy (ground state) is  $E_{0}={\tfrac {1}{2}}\hbar \omega$ , known as the zero-point energy. Even in its lowest state, the oscillator has nonzero energy, reflecting the quantum uncertainty that prevents the particle from sitting exactly at rest at the bottom of the well.

The equally spaced spectrum $\hbar \omega ,\,2\hbar \omega ,\,3\hbar \omega ,\dots$ is unique to the harmonic potential. Physically, it means that the oscillator can absorb or emit energy only in quanta of $\hbar \omega$ , which is foundational for understanding molecular vibrations and phonons in solids.

The harmonic oscillator is also important because it is mathematically tractable (solved by special functions) and because many more complicated potentials can be approximated as harmonic near a minimum (using a Taylor expansion).

Solving the Schrödinger Equation

Besides these textbook cases, most real-world Schrödinger equations cannot be solved with simple formulas. The standard analytical method is separation of variables when the potential is time-independent and problems have enough symmetry (like the examples above). One factors the wavefunction into time and space parts as described earlier. For multidimensional problems, one often also separates variables if the potential is separable in different coordinates (e.g., radial vs. angular in hydrogen atom).

When separation of variables is not possible or leads to intractable equations, physicists use approximate and numerical methods. Common approaches include:

Perturbation theory: If the potential $V$ can be written as a simple solvable part plus a small extra part, one expands the solution in powers of the small perturbation. This works for many slightly anharmonic systems.
Variational method: One guesses a trial wavefunction with adjustable parameters and minimizes the energy expectation value to approximate the ground state.
WKB approximation: A semiclassical method that approximates the wavefunction when the quantum number is large.
Matrix methods / Basis expansion: Expanding $\Psi$ in some known basis (like the harmonic oscillator states or plane waves) and reducing Schröinger’s equation to a matrix eigenvalue problem.
Computational methods: Finite-difference or finite-element schemes discretize space and convert the differential equation to large matrix problems that can be solved by computers.

These methods are essential in atomic, molecular, and condensed-matter physics, where potentials can be complicated. In advanced studies (quantum chemistry, solid-state physics), one seldom solves the Schrödinger equation by hand; instead, sophisticated codes compute energy levels and wavefunctions for electrons in molecules and materials.

Significance and Applications

The Schrödinger equation lies at the heart of quantum mechanics. It explains the structure and spectra of atoms and molecules, dictating chemical bonds and reactions. Every electron in an atom, crystal, or nanoscale device is described by a Schrödinger wavefunction.

Practical applications abound. Semiconductor devices (transistors, diodes) rely on quantum effects that are modeled by Schrödinger’s equation in potential wells and barriers. Lasers and LEDs come from electronic transitions in atoms and solids, whose energies are predicted by Schrödinger theory. Quantum wells, wires, and dots (engineered nanostructures) are designed using solutions of Schrödinger’s equation in confined geometries. Modern fields like quantum computing build on quantum states and their evolution (unitary time development given by Schrödinger’s equation) to manipulate information.

Even outside engineering, Schrödinger’s equation explains fundamental phenomena: the opacity of matter (electron orbitals), magnetism (electron spin and orbitals), electrical conductivity (band theory), to name a few. It also underlies spectroscopy: the absorption and emission lines in astronomy and chemistry correspond to transitions between energy eigenvalues.

On a deeper level, Schrödinger’s equation embodies several core principles: wave-particle duality (particles behave as waves), superposition (quantum states can add together), and quantization (only certain states are allowed). It made probabilistic interpretation necessary (Born’s rule for |Ψ|²) and introduced the notion that physical quantities correspond to operators with eigenvalues.

Debates and Open Questions

While the Schrödinger equation itself is well-established and extremely successful, it raises foundational questions. One major issue is the interpretation of the wavefunction $\Psi$ . What does $\Psi$ really represent? The most common Copenhagen interpretation says $|\Psi |^{2}$ gives probabilities and that physical properties (like position or momentum) become definite only upon measurement (the controversial “collapse” of $\Psi$ ). Other interpretations (many-worlds, de Broglie-Bohm pilot-wave, objective collapse models, etc.) offer different pictures of reality but all agree on Schrödinger’s equation’s predictions. Debates continue about whether the wavefunction is merely a tool for calculating probabilities or something physically real.

Another open question is reconciling the Schrödinger equation with relativity and field theory. Schrödinger’s equation is non-relativistic; it does not account for effects at speeds close to light or for the creation and annihilation of particles. Relativistic quantum mechanics (the Dirac and Klein-Gordon equations) and quantum field theory generalize the Schrödinger approach. In heavy atomic systems or high-energy contexts, Dirac’s equation is needed to correctly describe spin and relativistic energy. There are also ongoing inquiries in quantum gravity and quantum cosmology about how a Schrödinger-like equation might govern the universe at the largest scales or earliest times.

On the practical side, computing Schrödinger solutions for very many particles is famously hard. The many-body problem (quantum interactions of many particles, like electrons in a solid) cannot be solved exactly. Approximation schemes (density functional theory, quantum Monte Carlo, etc.) are used, but giving exact solutions is usually impossible beyond a few particles. Thus one frontier is finding better computational methods and approximations for strongly correlated systems.