Proof that the expectation eigenvalue of a Hermitian operator is always real
정리
증명
$A$를 에르미트 연산자라고 하자. $A$의 기댓값은
$$ \braket{A \rangle = \int \psi^{\ast}A\psi dx = \langle \psi | A\psi} $$
실수임을 보이려먼 $\braket{\psi | A\psi}-\braket{\psi | A\psi}^{\ast}=0$임을 보이면 된다.
$$\begin{align*} \braket{\psi | A\psi}^{\ast} &= \braket{A\psi | \psi} \\ &= \int (A\psi)^{\ast}\psi dx \\ &= \int \psi^{\ast}A^{\ast}\psi dx \\ &= \int \psi^{\ast} A \psi dx \\ &= \braket{\psi | A \psi} \end{align*}$$
따라서
$$ \braket{\psi | A\psi}-\braket{\psi | A\psi}^{\ast}=\braket{\psi | A\psi}-\braket{\psi | A\psi}=0 $$
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