Where do these two equalities for the expectation value come from precisely? :

$$\begin{align} \langle x\rangle &= \int_{-\infty} ^\infty \Psi^* x \Psi\,\mathrm{d}x \\ \langle x^2\rangle &= \int_{-\infty} ^\infty x^2 |\Psi|^2\,\mathrm{d}x \end{align}$$

Note: $\Psi$ is the wave function, and $\Psi^*$ is the complex conjugate of $\Psi$.

And are these two equalities also true? :

$$\begin{align} \langle x\rangle &= \int_{-\infty} ^\infty x |\Psi|^2\,\mathrm{d}x \\ \langle x^2\rangle &= \int_{-\infty} ^\infty \Psi^* x^2 \Psi\,\mathrm{d}x \end{align}$$

Don't we just have the equality $\Psi^* x \Psi = x |\Psi|^2$ ?

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    $\begingroup$ They come from the definition of the inner product on the $L^2$ space in the "position representation". $\endgroup$ – Phoenix87 Jun 21 '15 at 10:12
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    $\begingroup$ $\Psi=\Psi(x) \in \mathbb{C}$ so $\Psi^*(x)x\Psi(x)=x\Psi^*(x)\Psi(x)=x|\Psi(x)|^2$ $\endgroup$ – glS Jun 21 '15 at 10:36
  • $\begingroup$ Ya I have same problem dawgs $\endgroup$ – bodacydo Aug 14 '15 at 21:14

Yes,both the set of equalities are true but only in the position representation where operator(x)=x.In the momentum representation,where operator(x) takes a different form they are not true

  • $\begingroup$ Could you develop the part where you say "In the momentum representation they are not true"? What are $<p>$ and $<p^2>$ equal to? $\endgroup$ – Quantum Force Jun 22 '15 at 8:17

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