# Schrödinger equation

The Schrödinger equation is the fundamental differential equation of non-relativistic quantum mechanics that determines how a system's wave function changes in time and, for time-independent potentials, what its allowed energies are. It was formulated by the Austrian physicist [Erwin Schrödinger](https://www.edgepedia.org/erwin-schrodinger) in early 1926 in the paper "Quantisierung als Eigenwertproblem" (Quantization as an eigenvalue problem), submitted to *Annalen der Physik* on 26 January 1926<sup>[1](https://www.uzh.ch/en/researchinnovation/excellence/nobelprize/schroedinger.html)</sup>. The equation plays a role in quantum mechanics analogous to Newton's second law in classical mechanics: once the potential energy of a particle is specified, solving the equation yields the wave function from which all predictions about the particle follow<sup>[2](https://openstax.org/books/university-physics-volume-3/pages/7-3-the-schrodinger-equation)</sup>.

| Key fact | Detail |
|---|---|
| Time-dependent form (1D) | \( -\frac{\hbar^{2}}{2m}\frac{\partial^{2}\Psi}{\partial x^{2}} + U(x,t)\Psi = i\hbar\frac{\partial\Psi}{\partial t} \)<sup>[2](https://openstax.org/books/university-physics-volume-3/pages/7-3-the-schrodinger-equation)</sup> |
| Time-independent form | \( -\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}} + U(x)\psi = E\psi \), an eigenvalue equation for the energy<sup>[2](https://openstax.org/books/university-physics-volume-3/pages/7-3-the-schrodinger-equation)</sup> |
| Origin | First paper submitted to *Annalen der Physik* on 26 January 1926; formulated the time-independent wave equation<sup>[1](https://www.uzh.ch/en/researchinnovation/excellence/nobelprize/schroedinger.html)</sup> |
| Recognition | Nobel Prize in Physics 1933, shared with Paul A.M. Dirac, "for the discovery of new productive forms of atomic theory"<sup>[3](https://www.nobelprize.org/prizes/themes/the-nobel-prize-in-physics-1901-2000/)</sup> |
| Interpretation | \( |\Psi|^{2} \) is the probability density of finding the particle at a location (Born's interpretation)<sup>[4](https://openstax.org/books/university-physics-volume-3/pages/7-1-wave-functions)</sup> |
| Particle in a box | \( E_{n} = n^{2}\pi^{2}\hbar^{2}/(2mL^{2}) \), \( \psi_{n}(x) = \sqrt{2/L}\,\sin(n\pi x/L) \), n = 1, 2, 3, ...<sup>[5](https://openstax.org/books/university-physics-volume-3/pages/7-4-the-quantum-particle-in-a-box)</sup> |
| Harmonic oscillator | \( E_{n} = (n + \tfrac{1}{2})\hbar\omega \) with Hermite-polynomial wave functions<sup>[6](https://openstax.org/books/university-physics-volume-3/pages/7-5-the-quantum-harmonic-oscillator)</sup> |

## The time-dependent equation

For a particle of mass m moving in one dimension in a potential energy function U(x, t), the time-dependent Schrödinger equation reads

\[ -\frac{\hbar^{2}}{2m}\frac{\partial^{2}\Psi(x,t)}{\partial x^{2}} + U(x,t)\Psi(x,t) = i\hbar\frac{\partial\Psi(x,t)}{\partial t}, \]

where \( \hbar \) is the reduced [Planck constant](https://www.edgepedia.org/planck-constant), \( \hbar = h/2\pi \)<sup>[2](https://openstax.org/books/university-physics-volume-3/pages/7-3-the-schrodinger-equation)</sup><sup> • </sup><sup>[7](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/b5106a499ae03e36b5a2e002355668f9_MIT22_02S12_lec_ch6.pdf)</sup>. The equation can be obtained by starting from the classical energy relation \( E = p^{2}/2m + U \) and using the de Broglie relations \( p = \hbar k \) and \( E = \hbar\omega \)<sup>[2](https://openstax.org/books/university-physics-volume-3/pages/7-3-the-schrodinger-equation)</sup>. In the general operator form used across quantum mechanics, \( i\hbar\,\partial_{t}|\psi(t)\rangle = \hat{H}(t)|\psi(t)\rangle \), where the Hamiltonian operator \( \hat{H} \) describes all interactions and is the sum of kinetic and potential energy<sup>[8](https://ocw.mit.edu/courses/5-74-introductory-quantum-mechanics-ii-spring-2009/f61b11cc95a86973d9815f85093088db_MIT5_74s09_lec01.pdf)</sup>.

**The wave function is complex.** For a free particle, U is constant and the equation admits the plane-wave solution \( \Psi(x,t) = A e^{i(kx - \omega t)} \). The solution necessarily contains the imaginary unit because the differential equation itself contains \( i \); the real plane waves \( A\sin(kx - \omega t) \) and \( A\cos(kx - \omega t) \) do not satisfy the equation. Position probability density depends only on \( |\Psi(x,t)|^{2} \), which is real<sup>[2](https://openstax.org/books/university-physics-volume-3/pages/7-3-the-schrodinger-equation)</sup>, but physical predictions for other observables require the full wave function, including its phase<sup>[18](https://www.damtp.cam.ac.uk/user/tong/qm/qmhtml/S1.html)</sup>.

## Interpretation of the wave function

The solution of the equation is a wave function, a mathematical object that cannot itself be observed. In Born's interpretation, the square of the wave function represents the probability density of finding the particle around a specific location in space; the probability of finding the particle somewhere is 1 (the normalization condition)<sup>[4](https://openstax.org/books/university-physics-volume-3/pages/7-1-wave-functions)</sup>. Schrödinger himself struggled to interpret his new quantity, first calling it a "shadow wave" and then a charge density, until [Max Born](https://www.edgepedia.org/max-born) proposed the probability reading; Born's role in deciphering the meaning of the theory won him half of the [Nobel Prize in Physics](https://www.edgepedia.org/nobel-prize-in-physics) in 1954<sup>[9](https://educationalgames.nobelprize.org/educational/physics/quantised_world/final-5.html)</sup>.

Schrödinger also wrote down the complex conjugate equation in 1926 and showed that together they imply a continuity equation \( \partial\rho/\partial t + \mathrm{div}\,\vec{j} = 0 \) for the probability density \( \rho = |\psi|^{2} \) and the probability current \( \vec{j} \)<sup>[10](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/01%253A_Reviewing_Elementary_Stuff/1.03%253A_Wave_Equations_Wavepackets_and_Superposition)</sup>.

## The time-independent equation

When the potential energy does not depend on time, the wave function separates into a space-only part and a time-only part, \( \Psi(x,t) = \psi(x)e^{-iEt/\hbar} \), where the exponential is sometimes called the time-modulation factor<sup>[2](https://openstax.org/books/university-physics-volume-3/pages/7-3-the-schrodinger-equation)</sup>. Substituting this form reduces the time-dependent equation to the time-independent Schrödinger equation,

\[ -\frac{\hbar^{2}}{2m}\frac{d^{2}\psi(x)}{dx^{2}} + U(x)\psi(x) = E\psi(x), \]

an eigenvalue equation in which \( \psi(x) \) are the eigenstates of the Hamiltonian and E the energy eigenvalues<sup>[2](https://openstax.org/books/university-physics-volume-3/pages/7-3-the-schrodinger-equation)</sup><sup> • </sup><sup>[8](https://ocw.mit.edu/courses/5-74-introductory-quantum-mechanics-ii-spring-2009/f61b11cc95a86973d9815f85093088db_MIT5_74s09_lec01.pdf)</sup>. Its solutions are the stationary states, states of definite energy whose probability density does not vary in time<sup>[11](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/01%253A_Reviewing_Elementary_Stuff/1.05%253A_Electron_in_a_Box)</sup>. Acceptable solutions must be continuous, have a continuous first derivative (except where the potential is infinite), and must not diverge at infinity so that \( |\psi|^{2} \) can be normalized<sup>[2](https://openstax.org/books/university-physics-volume-3/pages/7-3-the-schrodinger-equation)</sup>.

For a time-independent Hamiltonian, any initial state evolves as \( |\psi(t)\rangle = e^{-i\hat{H}(t-t_{0})/\hbar}|\psi(t_{0})\rangle \); expanding in energy eigenstates gives \( |\psi(t)\rangle = \sum_{n} c_{n}(t_{0})e^{-iE_{n}(t-t_{0})/\hbar}|n\rangle \)<sup>[12](https://site.physics.georgetown.edu/~jkf/grad_quant2/html/lecture25.html)</sup>. A linear combination of stationary states is also a valid solution, and such superpositions can describe a localized particle whose probability density oscillates in time<sup>[2](https://openstax.org/books/university-physics-volume-3/pages/7-3-the-schrodinger-equation)</sup><sup> • </sup><sup>[8](https://ocw.mit.edu/courses/5-74-introductory-quantum-mechanics-ii-spring-2009/f61b11cc95a86973d9815f85093088db_MIT5_74s09_lec01.pdf)</sup>.

## What the equation predicts

**Quantization of energy.** For a particle confined to a one-dimensional box of length L with impenetrable walls, the boundary conditions \( \psi(0) = \psi(L) = 0 \) force the wave number to satisfy \( kL = n\pi \), giving the discrete energies \( E_{n} = n^{2}\pi^{2}\hbar^{2}/(2mL^{2}) \) and standing-wave functions \( \psi_{n}(x) = \sqrt{2/L}\,\sin(n\pi x/L) \). A bound particle cannot have zero kinetic energy<sup>[5](https://openstax.org/books/university-physics-volume-3/pages/7-4-the-quantum-particle-in-a-box)</sup>. For the harmonic oscillator potential \( U(x) = \tfrac{1}{2}m\omega^{2}x^{2} \), normalizability and symmetry give \( E_{n} = (n + \tfrac{1}{2})\hbar\omega \) with wave functions built from [Hermite polynomials](https://www.edgepedia.org/hermite-polynomials)<sup>[6](https://openstax.org/books/university-physics-volume-3/pages/7-5-the-quantum-harmonic-oscillator)</sup>. Confinement is the general mechanism: for a free particle a continuum of energies solves the time-independent equation, while a bound potential leaves only a discrete set<sup>[13](https://phys.libretexts.org/Bookshelves/Nuclear_and_Particle_Physics/Introduction_to_Applied_Nuclear_Physics_%28Cappellaro%29/04%3A_Energy_Levels/4.01%3A_Bound_Problems)</sup>.

**Atoms.** Applied to the hydrogen atom with a Coulomb potential, the equation yields the discrete bound-electron levels \( U_{n} \approx -13.6/n^{2} \) eV, reproducing the observed spectrum<sup>[14](https://ocw.mit.edu/courses/3-014-materials-laboratory-fall-2006/6514132b7d3be6c117fede9b0d8acc5a_g1_2_fcenter.pdf)</sup>. Schrödinger succeeded in solving the three-dimensional equation with a Coulomb potential and found the Bohr energy levels of hydrogen, one of the great problems of quantum physics at the time<sup>[10](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/01%253A_Reviewing_Elementary_Stuff/1.03%253A_Wave_Equations_Wavepackets_and_Superposition)</sup><sup> • </sup><sup>[1](https://www.uzh.ch/en/researchinnovation/excellence/nobelprize/schroedinger.html)</sup>.

**Classical limit.** Ehrenfest's theorem, derived from the Schrödinger equation, states that the expectation values of position and momentum obey \( m\,d\langle x\rangle/dt = \langle p\rangle \) and \( d\langle p\rangle/dt = -\langle dV/dx\rangle \), the classical equations of motion. [Quantum mechanics](https://www.edgepedia.org/quantum-mechanics) corresponds to classical mechanics when the spatial extent of the wave function is negligible compared with the variation length-scale of the potential<sup>[15](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_%28Fitzpatrick%29/03%3A_Fundamentals_of_Quantum_Mechanics/3.04%3A_Ehrenfest%27s_Theorem)</sup>.

## History and status

Schrödinger built on [Louis de Broglie](https://www.edgepedia.org/louis-de-broglie)'s 1925 doctoral thesis proposing that matter, such as electrons, has wave properties<sup>[1](https://www.uzh.ch/en/researchinnovation/excellence/nobelprize/schroedinger.html)</sup>. His first paper presented the time-independent equation for a particle in an external field; Schrödinger and others quickly showed the complete equivalence of his wave mechanics with Heisenberg's matrix mechanics for non-relativistic quantum mechanics<sup>[16](https://www.theorie.physik.uni-goettingen.de/~schoenh/MBGQM.pdf)</sup>. He shared the 1933 Nobel Prize in Physics with Paul A.M. Dirac "for the discovery of new productive forms of atomic theory"<sup>[3](https://www.nobelprize.org/prizes/themes/the-nobel-prize-in-physics-1901-2000/)</sup>.

The equation is non-relativistic: Schrödinger's and Heisenberg's quantum mechanics was valid for the relatively low velocities and energies of orbital electrons in atoms, and does not satisfy the requirements of Einstein's rules for fast-moving particles, which later required relativistic extensions<sup>[3](https://www.nobelprize.org/prizes/themes/the-nobel-prize-in-physics-1901-2000/)</sup>. Within its domain it remains the working equation of non-relativistic quantum mechanics, and a century after its publication the [University of Vienna](https://www.edgepedia.org/university-of-vienna) marks 1926 as the moment wave mechanics gave a new language to nature at the smallest scales<sup>[17](https://schroedinger100.univie.ac.at/)</sup>.

## References

1. [Erwin Schrödinger – Nobel Prize in Physics 1933, University of Zurich](https://www.uzh.ch/en/researchinnovation/excellence/nobelprize/schroedinger.html)
2. [7.3 The Schrödinger Equation, University Physics Volume 3, OpenStax](https://openstax.org/books/university-physics-volume-3/pages/7-3-the-schrodinger-equation)
3. [The Nobel Prizes in Physics 1901–2000, NobelPrize.org](https://www.nobelprize.org/prizes/themes/the-nobel-prize-in-physics-1901-2000/)
4. [7.1 Wave Functions, University Physics Volume 3, OpenStax](https://openstax.org/books/university-physics-volume-3/pages/7-1-wave-functions)
5. [7.4 The Quantum Particle in a Box, University Physics Volume 3, OpenStax](https://openstax.org/books/university-physics-volume-3/pages/7-4-the-quantum-particle-in-a-box)
6. [7.5 The Quantum Harmonic Oscillator, University Physics Volume 3, OpenStax](https://openstax.org/books/university-physics-volume-3/pages/7-5-the-quantum-harmonic-oscillator)
7. [Lecture notes, Chapter 6: Time Evolution in Quantum Mechanics, MIT 22.02 (Spring 2012)](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/b5106a499ae03e36b5a2e002355668f9_MIT22_02S12_lec_ch6.pdf)
8. [Lecture 1: Time-Dependent Quantum Mechanics, MIT 5.74 (Spring 2009), Andrei Tokmakoff](https://ocw.mit.edu/courses/5-74-introductory-quantum-mechanics-ii-spring-2009/f61b11cc95a86973d9815f85093088db_MIT5_74s09_lec01.pdf)
9. [Schrödinger and Wave Mechanics, Nobel Prize educational physics](https://educationalgames.nobelprize.org/educational/physics/quantised_world/final-5.html)
10. [1.3: Wave Equations, Wavepackets and Superposition, Physics LibreTexts (Fowler)](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/01%253A_Reviewing_Elementary_Stuff/1.03%253A_Wave_Equations_Wavepackets_and_Superposition)
11. [1.5: Electron in a Box, Physics LibreTexts (Fowler)](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/01%253A_Reviewing_Elementary_Stuff/1.05%253A_Electron_in_a_Box)
12. [Phys 506 Lecture 25: Time-dependent Schrödinger equation, Georgetown University](https://site.physics.georgetown.edu/~jkf/grad_quant2/html/lecture25.html)
13. [4.1: Bound Problems, Physics LibreTexts (Cappellaro)](https://phys.libretexts.org/Bookshelves/Nuclear_and_Particle_Physics/Introduction_to_Applied_Nuclear_Physics_%28Cappellaro%29/04%3A_Energy_Levels/4.01%3A_Bound_Problems)
14. [MIT 3.014 Materials Laboratory: F-center notes](https://ocw.mit.edu/courses/3-014-materials-laboratory-fall-2006/6514132b7d3be6c117fede9b0d8acc5a_g1_2_fcenter.pdf)
15. [3.4: Ehrenfest's Theorem, Physics LibreTexts (Fitzpatrick)](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_%28Fitzpatrick%29/03%3A_Fundamentals_of_Quantum_Mechanics/3.04%3A_Ehrenfest%27s_Theorem)
16. [3.3: 1926: Wave Mechanics and beyond, University of Göttingen lecture notes](https://www.theorie.physik.uni-goettingen.de/~schoenh/MBGQM.pdf)
17. [100 years of Schrödinger's equation, University of Vienna](https://schroedinger100.univie.ac.at/)
18. [1 Introduction‣ Quantum Mechanics by David Tong](https://www.damtp.cam.ac.uk/user/tong/qm/qmhtml/S1.html)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum formalism and states › Quantum states and wave functions*

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