Question

if A^ and B^ are hermitian and anti-hermitian operators respectively.check the hermiticity of the commutator [A^,B^] of the two operators .

Answer 1

Answer on Question #68641, Physics / Quantum Mechanics |

Question

if A^ and B^ are hermitian and anti-hermitian operators respectively, check the hermiticity of the commutator [A^,B^] of the two operators.

Solution

We have

$\hat{A}^{+} = \hat{A}$ — hermitian operator,

$\hat{B}^{+} = -\hat{B}$ — anti-hermitian operator.

We remind the definition of the commutator

$[\hat{A},\hat{B}] = \hat{A}\hat{B} -\hat{B}\hat{A}$

and the property of hermitian conjugation

$(\hat{A}\hat{B})^{+} = \hat{B}^{+}\hat{A}^{+}$.

Now we can check the hermiticity of the commutator

$[\hat{A},\hat{B}]^{+} = (\hat{A}\hat{B} -\hat{B}\hat{A})^{+} = (\hat{A}\hat{B})^{+} - (\hat{B}\hat{A})^{+} = \hat{B}^{+}\hat{A}^{+} - \hat{A}^{+}\hat{B}^{+} = -\hat{B}\hat{A} +\hat{A}\hat{B} = [\hat{A},\hat{B}]$

Thus, commutator $[\hat{A},\hat{B}]$ is hermitian.

Answer: commutator $[\hat{A},\hat{B}]$ is hermitian.

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