Quantum mechanics MOC

Schrödinger equation

The Schrödinger equation governs the time evolution quantum mechanical wavefunction in the Schrödinger picture. If |Ψ(𝑡)⟩ is the state of a system at time 𝑡, then

𝑖ℏ𝑑𝑑𝑡|Ψ(𝑡)⟩=ˆ𝐻(𝑡)|Ψ(𝑡)⟩

where ˆ𝐻(𝑡) is the Hamiltonian operator for the system. This is a linear differential system and thus solutions are given by a Quantum time evolution operator.

Time-independent Schrödinger equation

A stationary state is one for which every Observable is independent of time. They are precisely the solutions to the time-independent Schrödinger equation as obtained from Separation of variables

ˆ𝐻|Ψ(𝑡)⟩=𝐸|Ψ(𝑡)⟩=𝑖ℏ𝑑𝑑𝑡|Ψ(𝑡)⟩

and are thus are energy eigenstates.¹

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develop | en | SemBr

Footnotes

1. 2018. Introduction to Quantum Mechanics, §2.1, pp. 27–28 ↩