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I am reading book called "A Unified Grand Tour of Theoretical Physics" by Ian D Lawrie. I have started bra ket notation and state vectors. Could somebody explain to me how $$P(a,b,c..| \Psi) = |\langle a,b,c...|\Psi\rangle| ^2.$$

$a,b,c$ are called observable quantities. $\Psi$ has been normalised. I come from a background of using wavefunctions and not state vectors and Dirac notation. For me a probability comes from $\Psi \Psi^*$ or $\Psi \Psi$. Perhaps I haven't understood what a state vector is.

Tobias Fünke
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1 Answers1

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In the framework of textbook quantum mechanics, states are abstract vectors $\lvert \psi \rangle$ of a Hilbert space $\mathcal{H}$. More technically, a state is an equivalence class of vectors defined by the equivalence relation $\lvert \psi \rangle \sim \lvert \psi' \rangle \iff \lvert \psi \rangle = c\lvert \psi'\rangle$ for some $c \in \mathbb{C}$. We usually agree to use a normalized state as a representative of an equivalence class of such vectors. This abstract vector is a ket.

A wave function is just the components of one of these abstract vectors in the position basis $$\psi(x) := \langle x \lvert \psi \rangle.$$

Silly Goose
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