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iℏ∂∂tΨ(r,t)=H^Ψ(r,t)i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t)=\hat{H}\Psi(\mathbf{r},t)iℏ∂t∂​Ψ(r,t)=H^Ψ(r,t)

iℏ∂∂ti\hbar\frac{\partial}{\partial t}iℏ∂t∂​

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Ψ(r,t)\Psi(\mathbf{r},t)Ψ(r,t)

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H^\hat{H}H^

Schrödinger Equation

Click on formula components below to explore their properties

Full Formula Properties

Category: Quantum Mechanics

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Baby Fast Definition

It's the rule that tells us how the quantum 'cloud' that describes a particle changes over time, keeping the total chance of finding the particle equal to 100 %.

The Schrödinger equation is quantum mechanics' central postulate: it shows how the wavefunction Ψ changes in time while preserving probability, with the Hamiltonian operator converting energy into temporal phase rotation.

Role:

Governs the time evolution of a quantum system's wavefunction

Domain:

Complex Hilbert space L²(ℝ³) × ℝ → ℂ

Binding:

Wavefunction Ψ encodes all measurable information; Hamiltonian Ĥ links energy to dynamics

Variance:

Wavefunction's phase rotates in time while |Ψ|² stays constant

Geometric:

Unitary flow on an infinite-dimensional complex sphere

Invariant:

Total probability ∫|Ψ|² d³r = 1

Limits:

As ℏ→0 solutions approach classical trajectories

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