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I am currently reading the paper "Gravitation and quantum mechanics for macroscopic objects" by F. Karolyhazy (1966). In his paper, he uses certain notation that I haven't come across before (he also eats up some mathematics here and there but that's another story). He is speaking of the development of initial states of a quantum mechanical system to one of the states that he denotes as follows:$$ \frac{1}{(\Psi_k, \Psi_k)^{1/2}} \Psi_k $$ where $k=1, 2$. What does this notation imply?

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$(\cdot, \cdot)$ is an alternative, more mathematical way of denoting the scalar prodict (here in Hilbert space). The vector is thus normalized, i.e. divided by its norm – yuggib Aug 31 '14 at 11:58
Ah ok :) thank you :) – Artemisia Aug 31 '14 at 12:00
up vote 5 down vote accepted

This is the same notation that you'll find in Weinberg's books.

$$(\psi, \chi)$$

is the inner product of the two states $\psi$ and $\chi$, and corresponds to

$$\langle \psi \mid \chi \rangle$$.

So, the above corresponds literally to

$$ \frac{1}{\sqrt{\langle \psi_k \mid \psi_k \rangle}} \left| \psi_k \right>$$

This new object is just the normalized version of the state $\psi_k$, whose inner product with itself is unity.

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In math, $(\cdot, \cdot)$ is linear in the first argument and conjugate linear in the second, whereas in modern physics $\langle \cdot | \cdot \rangle$ is linear in the second argument. I wonder which one Weinberg (and Karolyhazy) use. – Chris White Aug 31 '14 at 14:30

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