The mathematical structure of quantum mechanics is based in large part on linear algebra

 

• In quantum mechanics the expression ⟨φ|ψ⟩ is typically interpreted as the probability amplitude for the state ψ to collapse into the state φ. Mathematically, this means the coefficient for the projection of ψ onto φ. It is also described as the projection of state ψ onto state φ.
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•  in the Schrödinger picture, there is a linear time evolution operator U with the property that if an electron is in state |ψ⟩ right now, at a later time it will be in the state U|ψ⟩, the same U for every possible |ψ⟩.
• Wave function normalization is scaling a wave function so that its norm is 1.