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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.

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    $\begingroup$ I’m not sure what you expect. An explanation of bras and kets?… $\endgroup$
    – ZeroTheHero
    Commented Oct 31 at 19:09
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    $\begingroup$ $\langle x|\Psi\rangle$ is the $\psi(x)$ you’re familiar with. $\endgroup$
    – Ghoster
    Commented Oct 31 at 19:45
  • $\begingroup$ @Ghoster. Thanks that's made everything clear now. $\endgroup$
    – user441992
    Commented Oct 31 at 19:56
  • $\begingroup$ A ket is just notation for a vector. You can write $v$ or $|v\rangle$ or $\textbf v$ or $\vec v$ -- different fields (and different authors within) use different notations. There is nothing special about the bra-ket notation. Understanding this can save a lot of time thinking about issues which actually are none. See also the comments here. $\endgroup$
    – Tobias Fünke
    Commented Oct 31 at 20:46

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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 $$\psi(x) := \langle x \lvert \psi \rangle.$$

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  • $\begingroup$ I very often read statements like this. Could you explain what "abstract vectors" are (it is a honest question, I really don't get it). $\endgroup$
    – Tobias Fünke
    Commented Oct 31 at 20:51
  • $\begingroup$ @TobiasFünke By abstract I mean to emphasize that a vector is a mathematical object that exists independent of any basis. So by “abstract vector” I mean an element of a vector space $V$ where the vector space is defined as satisfying the usual vector space axioms. $\endgroup$
    – Silly Goose
    Commented Oct 31 at 21:13
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    $\begingroup$ ...ok, so just "vector". thanks for clarifying, appreciated. $\endgroup$
    – Tobias Fünke
    Commented Oct 31 at 21:50

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