Question

The parity operator $hat{P}$ is a linear operator defined by the relation $langlemathbf{r}|hat{P}|psi angle = langle-mathbf{r}|psi angle$. (a) Show that $hat{P}$ is both Hermitian and unitary, i.e., $hat{P} = hat{P}^{dagger}$, $hat{P}^2 = hat{mathbb{1}}$. (b) Find eigenvalues $p$ of $hat{P}$. (c) Show that $hat{P}$ anti-commutes with the position operator $hat{mathbf{r}}$, i.e., $hat{P}hat{mathbf{r}} + hat{mathbf{r}}hat{P} = hat{mathbb{0}}$, or, equivalently, $hat{P}hat{mathbf{r}}hat{P} = -hat{mathbf{r}}$. (d) Show that $langlemathbf{p}|hat{P}|psi angle = langle-mathbf{p}|psi angle$ and that $hat{P}$ anti-commutes with the momentum operator $hat{mathbf{p}}$.

          The parity operator $hat{P}$ is a linear operator defined by the relation $langlemathbf{r}|hat{P}|psi
angle = langle-mathbf{r}|psi
angle$.
(a) Show that $hat{P}$ is both Hermitian and unitary, i.e., $hat{P} = hat{P}^{dagger}$, $hat{P}^2 = hat{mathbb{1}}$.
(b) Find eigenvalues $p$ of $hat{P}$.
(c) Show that $hat{P}$ anti-commutes with the position operator $hat{mathbf{r}}$, i.e., $hat{P}hat{mathbf{r}} + hat{mathbf{r}}hat{P} = hat{mathbb{0}}$, or, equivalently, $hat{P}hat{mathbf{r}}hat{P} = -hat{mathbf{r}}$.
(d) Show that $langlemathbf{p}|hat{P}|psi
angle = langle-mathbf{p}|psi
angle$ and that $hat{P}$ anti-commutes with the momentum operator $hat{mathbf{p}}$.

The parity operator hatP is a linear operator defined by the relation langlemathbfr|hatP|psi
angle = langle-mathbfr|psi
angle.
(a) Show that hatP is both Hermitian and unitary, i.e., hatP = hatP^dagger, hatP^2 = hatmathbb1.
(b) Find eigenvalues p of hatP.
(c) Show that hatP anti-commutes with the position operator hatmathbfr, i.e., hatPhatmathbfr + hatmathbfrhatP = hatmathbb0, or, equivalently, hatPhatmathbfrhatP = -hatmathbfr.
(d) Show that langlemathbfp|hatP|psi
angle = langle-mathbfp|psi
angle and that hatP anti-commutes with the momentum operator hatmathbfp.

Added by Manuela F.

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