Hermitian Operator | Quantum Mechanics

An operator  is said to be hermitian if Â$^{\dagger}=\hat{A}$ and it should satisfy the following relation.

Dirac representation:

$\left(\prec\phi|\hat{A}|\psi\succ\right)^{\dagger}=\prec\psi|\hat{A}^{\dagger}|\phi\succ=\prec\psi|\hat{A}|\phi\succ$

Schrodinger representation:

$\left(\phi,\hat{A}\psi\right)=\left(\hat{A}\phi,\psi\right)$

$\Longrightarrow\intop\phi^{*}\hat{A}\psi d\text{${\tau}$}$=$\intop\left(\hat{A}\phi\right)^{*}\psi d \text{${\tau}$}$

Properties:

(1) Eigenvalues of hermitian operators are real.

(2) Eigenfunctions corresponding to different eigenvalues of a hermitian operator are orthogonal.

(3) All physical observable in quantum mechanics are represented by hermitian operators.

Example:

$\left(\phi,\hat{p}_{x}\psi\right)$

= $\intop_{-\infty}^{\infty}\phi^{*}\left(-i\hbar\frac{\partial}{\partial x}\right)\psi dx$

= $\intop_{-\infty}^{\infty}\phi^{*}\left(-i\hbar\frac{\partial\psi}{\partial x}\right)dx$

= $-i\hbar\left[\phi^{*}\psi\right]_{-\infty}^{\infty}+i\hbar\intop_{-\infty}^{\infty}\left(\frac{d\phi^{*}}{dx}\right)\psi dx$

= $-i\hbar\intop_{-\infty}^{\infty}\left(\frac{d\phi^{*}}{dx}\right)\psi dx$

= $\intop_{-\infty}^{\infty}\hat{p}_{x}^{*}\phi^{*}\psi dx$

Hence, the momentum operator $\hat{p}_{x}$ is hermitian in nature.

Similarly, we can see that $\frac{\partial}{\partial x}$ is anti-hermitian

i.e. $\left[\frac{\partial}{\partial x}\right]^{\dagger}=-\frac{\partial}{\partial x}$