Schrödinger equation

Codex  0 Created 2026-09-24 Updated 2026-10-20

The Schrödinger equation governs the evolution of a quantum state between instances of measurement in quantum mechanics. For a particle of mass

m > 0

moving in a prescribed real potential energy

V (r, t)

, its position wavefunction satisfies

i ℏ \frac{\partial ψ}{\partial t} = H (t) ψ, H (t) = - \frac{ℏ ^{2}}{2 m} \nabla^{2} + V (r, t), ℏ = \frac{h}{2 π} .

(1)

Here

h

is the Planck constant,

\nabla^{2}

is the Laplace operator, and the Hamiltonian operator combines kinetic energy with potential energy. Specify an initial wavefunction, an operator domain and any boundary conditions: the differential expression alone does not completely specify a physical model. The Born rule then turns the evolving probability amplitude into position probabilities through

ρ = ∣ ψ ∣^{2}

and integration over a region.

This scalar Schrödinger operator describes nonrelativistic quantum mechanics, with relevant momenta much smaller than

m c

, where

c

is the speed of light. It omits the constant rest energy

m c^{2}

, assumes fixed particle number, and suppresses active spin dynamics and coupling to a magnetic vector potential. A free electron with a fixed, uncoupled spin state can still use this scalar model; the Pauli equation supplies the additional spin components and magnetic coupling when needed. A prescribed real

V

describes a closed evolution model or an approximation in which the source of that potential energy is treated as external. Dissipation, particle creation and a general measurement in quantum mechanics need further modelling.

For a time-independent self-adjoint Hamiltonian operator, the spectral theorem defines unitary time evolution by

ψ (t) = e^{- i H t /ℏ} ψ (0), ∥ ψ (t) ∥^{2} = ∥ ψ (0) ∥^{2} .

(2)

Initial states in the operator domain satisfy the differential equation in the Hilbert space sense; the unitary operator also evolves all other states in that Hilbert space. Time-dependent Hamiltonian operators need suitable regularity and domain hypotheses; their evolution is not generally the exponential of

- i t H (t) /ℏ

The local reason for conservation is especially tangible. Multiply the Schrödinger equation by the complex conjugate

ψ^{*}

, subtract its conjugate multiplied by

ψ

, and use the fact that the real potential energy terms cancel. For smooth wavefunctions this gives the probability continuity equation

\partial_{t} ρ + \nabla \cdot j = 0, j = \frac{ℏ}{m} Im (ψ^{*} \nabla ψ) .

(3)

The divergence theorem identifies the change in probability inside a region with minus the outward probability current. Decay at infinity, or appropriate probability-conserving boundary conditions, preserves wavefunction normalization. Conservation of total probability does not require a stationary probability density.

Its usefulness comes from being a linear partial differential equation: quantum superpositions of solutions are solutions, while different quantum phases can produce interference. For a fixed Hamiltonian operator, the time-independent Schrödinger equation finds the energy eigenstates; each then acquires just its own time-dependent quantum phase. One eigenvalue problem therefore organizes both allowed energies and the evolution of arbitrary quantum superpositions, with a continuous spectrum included when required. This is the attractive bridge between a spatial wave problem and a clock: energy differences determine the rate at which relative quantum phases change. It makes atomic spectral lines, quantum tunnelling and quantum scattering calculable from a specified model.

For an explicit example without confinement, take a free particle on the whole line,

V = 0

, initially in a Gaussian wave packet with position standard deviation

σ_{0}

and mean momentum

ℏ k_{0}

ψ (x, 0) = \frac{e ^{- x^{2} / (4 σ_{0}^{2}) + i k_{0} x}}{( 2 π σ _{0}^{2} ) ^{1/4}}, t_{s} = \frac{2 m σ _{0}^{2}}{ℏ}, τ = \frac{t}{t _{s}}, v_{0} = \frac{ℏ k _{0}}{m} .

(4)

Unlike a single plane wave, this wavefunction satisfies wavefunction normalization. A Fourier transform writes it as a quantum superposition of plane waves. Their dispersion relation is

ω (k) = ℏ k^{2} / (2 m)

, so each Fourier component gains the factor

e^{- i ℏ k^{2} t / (2 m)}

. Completing the square in the resulting Gaussian integral gives

ψ (x, t) = \frac{1}{( 2 π σ _{0}^{2} ) ^{1/4} 1 + i τ} exp [- \frac{( x - v _{0} t ) ^{2}}{4 σ _{0}^{2} ( 1 + i τ )} + i k_{0} x - \frac{i ℏ k _{0}^{2} t}{2 m}],

(5)

with the square root chosen continuously from

t = 0

. Its position probability density and standard deviation are

ρ (x, t) = \frac{e x p [ - ( x - v _{0} t ) ^{2} / ( 2 σ _{t}^{2} )]}{2 π σ _{t}}, σ_{t} = σ_{0} 1 + τ^{2} .

(6)

The centre moves with the group velocity

ω^{'} (k_{0}) = v_{0}

and the wave packet spreads because different wavenumbers have different group velocities. This is unitary wave dispersion, without a random force or loss of probability. Independently, the initial momentum standard deviation is

Δ p = ℏ/ (2 σ_{0})

and the initial position-momentum covariance vanishes. Free evolution gives

x (t) = x (0) + p (0) t / m

in the Heisenberg picture, hence

σ_{t}^{2} = σ_{0}^{2} + \frac{t ^{2}}{m ^{2}} (Δ p)^{2} = σ_{0}^{2} + \frac{ℏ ^{2} t ^{2}}{4 m ^{2} σ _{0}^{2}} .

(7)

This independently checks the spreading law. The Gaussian integral of

ρ

is one at every time; the decreasing peak is exactly offset by increasing width.

As hypothetical preparation parameters, choose a free electron with

σ_{0} = 1 nm

and

k_{0} = 2 nm^{- 1}

. Using

ℏ ≃ 1.054571817 \times 1 0^{- 34} J s

and

m ≃ 9.1093837 \times 1 0^{- 31} kg

gives

t_{s} ≃ 17.276 fs, v_{0} ≃ 2.31535 \times 1 0^{5} m s^{- 1}, Δ p ≃ 5.27286 \times 1 0^{- 26} kg m s^{- 1} .

(8)

The rounded electron mass uses the 2022 adjustment.^[ref] At

t = t_{s}

, the centre is

4 nm

and

σ_{t} = 2 nm

; at

t = 2 t_{s}

, they are

8 nm

and

5 nm

. The mean kinetic energy stays

⟨ H ⟩ = \frac{ℏ ^{2}}{2 m} (k_{0}^{2} + \frac{1}{4 σ _{0}^{2}}) ≃ 0.1619 eV .

(9)

It includes both mean motion and the momentum spread. The mean speed is about

7.72 \times 1 0^{- 4} c

and the momentum spread corresponds to

1.93 \times 1 0^{- 4} c

, making the nonrelativistic model accurate for the appreciably occupied part of this ideal Gaussian distribution. Its mathematically unbounded tails are an idealization.

Figure 1.
Translation and spreading of a free Gaussian wave packet
. Calculated electron position probability densities for the hypothetical $1 nm$ preparation above, together with the position standard deviation. The area under every curve is one. The dashed width curve illustrates pure translation without spreading for comparison; it is not the free solution. No curve represents measured data. **Source**. **Details**.

The complex width also explains where the probability goes. Write

ψ = ρ e^{i θ}

, where

θ

is the dimensionless quantum phase. Differentiating the exact solution gives

\partial_{x} θ = k_{0} + \frac{τ ( x - v _{0} t )}{2 σ _{0}^{2} ( 1 + τ ^{2} )}, \frac{j}{ρ} = \frac{ℏ}{m} \partial_{x} θ = v_{0} + \frac{t}{t _{s}^{2} + t ^{2}} (x - v_{0} t) .

(10)

Relative to the moving centre, the probability current

j_{rel} = j - v_{0} ρ

points outward on both sides for

t > 0

. This is a flux statement, not an additional assumption about individual particle trajectories. At

t = t_{s}

and

x - v_{0} t = \pm σ_{0}

, the relative flow speeds

j_{rel} / ρ

are

\pm σ_{0} / (2 t_{s})

, about

\pm 2.8942 \times 1 0^{4} m s^{- 1}

. The probability within a moving interval of half-width

σ_{0}

falls from about

0.6827

initially to

0.5205

t_{s}

, exactly because of this outward probability current.

Figure 2.
Curved quantum phase drives spreading probability current
. The same free Gaussian wave packet at $t = t_{s}$ . With $y = (x - v_{0} t) / σ_{0}$ , removing the carrier $k_{0} x$ and a spatially constant quantum phase leaves $y^{2} /8$ . The right panel plots $t_{s} j_{rel} = (σ_{0} ρ) y /2$ : negative to the left, positive to the right. Flux through the marked moving interval endpoints explains its decreasing probability. **Source**. **Details**.

Historically, the 1926 Quantisierung als Eigenwertproblem (Erste Mitteilung) by Erwin Schrödinger began with the stationary hydrogen atom problem. It distinguishes discrete negative energies from a positive continuous spectrum; it is not yet the general evolution equation displayed above.^[ref] The opening of quantum mechanics discusses its original vibrating-string analogy.

In the 12 December 1933 lecture The fundamental idea of wave mechanics by Erwin Schrödinger, Erwin Schrödinger explains the move from independent rays to waves through diffraction, then applies that analogy to atomic physics. His closing pages explicitly acknowledge the difficulty of reconciling wave language with observed particle tracks. This retrospective physical argument motivates the model; it does not derive the modern Born rule from classical mechanics.^[ref]

The dates separate proposal from measurement: the de Broglie wavelength belongs to the 1924 thesis, the Schrödinger equation to 1926, and the Davisson–Germer experiment's electron diffraction beams to 1927.^[ref] That linked experiment describes the prepared nickel crystal, evacuated electron gun, angular collector scans, retarding-potential selection and heating controls. Its measured collector electric current tests wave-dependent scattering probabilities; it neither measures a wavefunction directly nor establishes this equation uniquely.

Barack Obama  11 Created 2022-11-02 Updated 2025-04-18

This is a section about Schrödinger equation!

Schrödinger equation is a very important subject about which there is a lot to say.

For example, this sentence. And then another one.

Wikipedia Bot  1

The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is a key principle in understanding wave functions and the behavior of particles at the quantum level. There are two forms of the Schrödinger equation: 1. **Time-dependent Schrödinger equation**: This form is used to describe how the quantum state evolves over time.

Ciro Santilli  40 Updated 2025-07-16

Existence and uniqueness: mathoverflow.net/questions/212913/existence-and-uniqueness-for-two-dimensional-time-dependent-schr%C3%B6dinger-equation

The partial differential equation of non-relativistic quantum mechanics.

Experiments explained:

via the Schrödinger equation solution for the hydrogen atom it predicts:
- spectral line basic lines, plus Zeeman effect
Schrödinger equation solution for the helium atom: perturbative solutions give good approximations to the energy levels
double-slit experiment: I think we have a closed solution for the max and min probabilities on the measurement wall, and they match experiments

Experiments not explained: those that the Dirac equation explains like:

To get some intuition on the equation on the consequences of the equation, have a look at:

The easiest to understand case of the equation which you must have in mind initially that of the Schrödinger equation for a free one dimensional particle.

Then, with that in mind, the general form of the Schrödinger equation is:

i ℏ \frac{\partial ψ ( x , t )}{\partial t} = \hat{H} [ψ (x, t)]

(1)

Equation 1.

Schrodinger equation

where:

$ℏ$ is the reduced Planck constant
$ψ$ is the wave function
$t$ is the time
$\hat{H}$ is a linear operator called the Hamiltonian. It takes as input a function $ψ$ , and returns another function. This plays a role analogous to the Hamiltonian in classical mechanics: determining it determines what the physical system looks like, and how the system evolves in time, because we can just plug it into the equation and solve it. It basically encodes the total energy and forces of the system.

The

x

argument of

ψ

could be anything, e.g.:

we could have preferred polar coordinates instead of linear ones if the potential were symmetric around a point
we could have more than one particle, e.g. solutions of the Schrodinger equation for two electrons, which would have e.g. $x_{1}$ and $x_{2}$ for different particles. No matter how many particles there are, we have just a single $ψ$ , we just add more arguments to it.
we could have even more generalized coordinates. This is much in the spirit of Hamiltonian mechanics or generalized coordinates

Note however that there is always a single magical

t

time variable. This is needed in particular because there is a time partial derivative in the equation, so there must be a corresponding time variable in the function. This makes the equation explicitly non-relativistic.

The general Schrödinger equation can be broken up into a trivial time-dependent and a time-independent Schrödinger equation by separation of variables. So in practice, all we need to solve is the slightly simpler time-independent Schrödinger equation, and the full equation comes out as a result.