I know that an inner product between two vectors is defined like:

$$\langle a | b\rangle = {a_1}^\dagger b_1+{a_2}^\dagger b_2+\dots$$

but because a transpose of a component for example $a_1$ is again only $a_1$ the above simplifies to:

$$\langle a | b\rangle = \overline{a_1} b_1+\overline{a_2} b_2+\dots$$

Where $\overline{a_1}$ is a complex conjugate of $a_1$. Furthermore we can similarly define an inner product for two complex functions like this:

$$\langle f | g \rangle = \int\limits_{-\infty}^\infty \overline{f} g\, dx$$

In the Griffith's book (page 96) there is an equation which describes expectation value and we can write this as an inner product of a function $\Psi$ with a $\widehat{x} \Psi$:

\begin{align*} \langle x \rangle = \int\limits_{-\infty}^{\infty}\Psi\,\,\widehat{x}\Psi\,\,dx = \int\limits_{-\infty}^{\infty} \Psi\,\,(\widehat{x}\Psi)\,\, dx \equiv \underbrace{\langle\Psi |\widehat{x} \Psi \rangle}_{\rlap{\text{expressed as an inner product}}} \end{align*}

In Zettili's book (page 173) the expectation value is defined like a fraction:

\begin{align*} \langle \widehat{x} \rangle = \frac{\langle\Psi | \widehat{x} | \Psi \rangle}{\langle \Psi | \Psi \rangle} \end{align*}

Main question: I know the meanning of the definition in Griffith's book but i simply have no clue what Zetilli is talking about. What does this fraction mean and how is it connected to the definition in the Griffith's book.

Sub question: I noticed that in Zetilli's book they write expectation value like $\langle \widehat{x}\rangle$ while Griffith does it like this $\langle x \rangle$. Who is right and who is wrong? Does it matter? I think Griffith is right, but please express your oppinion.


2 Answers 2


If the wave function $\Psi$ is normalized, then $\langle\Psi|\Psi\rangle$ should equal 1. Griffiths' definition assumes the wave function is already normalized, while Zetilli accounts for all possibilities by dividing out the normalization constant. So if the wave function $\Psi$ is normalized, Zetilli's definition will reduce to Griffiths' definition.

As for the sub question, that's just a matter of notation, which is just a matter of convention, which doesn't really have an objective "right" and "wrong." They're both right as long as they're consistent with the rest of the notation in their respective books.

  • $\begingroup$ Your answer went up while I was typing mine. $\endgroup$ May 24, 2013 at 18:29
  • $\begingroup$ Thank you both. I will accept the one who anwsered first. $\endgroup$
    – 71GA
    May 24, 2013 at 18:30

Main question: the difference between the two is that Griffiths is assuming that the wave function is normalized: $\langle\Psi^|\Psi\rangle = 1$. Zetilli appears not to be making that assumption. If the wave function is not normalized, then you have to divide by $\langle\Psi^|\Psi\rangle$ to normalize your answer, as Zetilli does. If it is normalized, then Zetilli's formula reduces to Griffiths's.

Sub question: that's a matter of notation. Usually, the $\hat x$ notation is meant to emphasize that $x$ is an operator, rather than a scalar. The way Zetilli writes it is probably more technically correct, because he is saying that the expectation value of the operator is that expression involving the operator. But no one would be confused by the way Griffiths writes it.


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