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What is the physical meaning of ##\psi_1(x)\hat{A}\psi_2(x)##, where ##\hat{A}## is an observable and, ##\psi_1(x)## and ##\psi_2(x)## are arbitrary wavefunctions?
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Sorry for being pedantic, but this mix of notation doesn't make sense. It should beI think he means:
physical meaning of ##<\psi_1(x)|\hat{A}|\psi_2(x)>##, where ##\hat{A}## is an observable and, ##\psi_1(x)## and ##\psi_2(x)## are arbitrary wavefunctions.
No. That would only be the case for ##\psi_1 = \psi_2##.Is it expected value of quantity A?
NO! It'sSorry for being pedantic, but this mix of notation doesn't make sense. It should be
$$
\langle \psi_1 | \hat{A} | \psi_2 \rangle
$$
or (which is why I asked about an integral)
$$
\int \psi_1(x) \hat{A} \psi_2(x) \, dx
$$
No. That would only be the case for ##\psi_1 = \psi_2##.
Of course. I'll fix my typo.NO! It's
$$
\int \psi_1^*(x) \hat{A} \psi_2(x) \, dx
$$
Then it's a matrix element of ##\hat{A}##. The complex conjugation is of utmost importance!