How to understand wavefunction in quantum mechanics in math I am reading some introduction on quantum mechanics. I don't understand all but I get the point that the wavefunction tells some probability aspects. In one book, they show one example of the wavefunction $f(x)$ in position space as a complex function, so they said the probability of finding the particle is $f^*(x) f(x) = |f(x)|^2$. In other book, the same example shown but in so-called bra and ket vector form, I know if I calculate the absolute square, I should get the same answer. But I am still learning the bra, ket notation, so I wonder if $\langle f(x)|f(x)\rangle$ or $|\langle f(x)|f(x)\rangle|^2$ gives the probability? If the last one gives the probability, what is $\langle f(x)|f(x)\rangle$? Is $\langle f(x)|f(x)\rangle = f^*(x)\cdot f(x)$ ?
 A: "$| f(x) \rangle$" does not mean anything and is not proper bra-ket notation. For translating back and forth beteween wavefunction and bra-ket notation, here is the #1 thing to keep in mind:
$$
f(x) = \langle x \mid f \rangle
$$
So, the probability density to find the particle at $x$ is
$$
\left|f(x)\right|^2 = \left| \langle x \mid f \rangle \right|^2
$$
Since $\langle a \mid b \rangle = \langle b \mid a \rangle^*$, this can also be written
$$
|f(x)|^2 = \langle f \mid x \rangle \langle x \mid f \rangle
$$
Remember, this represents a probability density in $x$. What this means is that
$$
\int dx\, A(x) \left| f(x) \right|^2 = \left< f \right| \left\{ \int dx \, A(x) \left| x \rangle \langle x \right| \right\} \left| f \right>
$$
should be the expected value of the function A(x). The quantity in the brackets is an operator:
$$
\hat A = \int dx \, A(x) \left| x \rangle \langle x \right|
$$
(Edit: As pointed out by Trimok, the above is not true for most operators. It is only true for any operator that is diagonal in the x basis, or equivalently that can be written as a function of the operator $\hat x$. These are the only kind of operator for which expectation value and higher moments can be computed using $|f(x)|^2$ as a probability density function.)
The expectation value of this operator is 
$$
\langle A \rangle = \left< f \right| \hat A \left| f \right>
$$
A: In Bra-Ket language:
Wave function is defined to be the coefficient of expansion of arbitary state in space base ket. $|\alpha \rangle=\int dx | x\rangle \langle x | \alpha \rangle=\int dx | x\rangle f_{\alpha}(x)$ . For a general state  $|\alpha \rangle$, $f_{\alpha}$ is the wave function. The probability is $f^*f=|\langle x | \alpha\rangle|^2$.
