So, initially when I first learned Quantum Mechanics, the expectation value of the operator corresponding to a physical observable was given by : $$\langle \hat{Q} \rangle = \frac{\int_{-\infty}^{\infty}\psi^{*}(x)\hat{Q}\psi(x)dx}{\int_{-\infty}^{\infty}\psi^{*}(x)\psi(x)dx}$$ where $\psi(x)$ represents the state of a Quantum System in position space. Or equivalently, $$\langle \hat{Q} \rangle = \langle\psi|\hat{Q}|\psi\rangle$$ And for a normalized wavefunction : $$\int_{-\infty}^{\infty} |\psi(x)|^2 dx = \int_{-\infty}^{\infty}\psi^{*}(x)\psi(x)dx=1$$ But in some cases I Found : $$\langle \hat{Q} \rangle = \langle\phi|\hat{Q}|\psi\rangle$$ where $|\psi\rangle$ and $|\phi\rangle$ describes two different states. Cannot get this thing when do we use these two different notations
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4$\begingroup$ An expectation value in a given state is always calculated "sandwiching" the operator between a bra and a ket corresponding to the same state. The last expression you mentioned is called matrix element and it's not an expectation value $\endgroup$Lagrangiano– Lagrangiano2025-04-27 07:57:47 +00:00Commented Apr 27, 2025 at 7:57
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3$\begingroup$ Found in some cases? Which reference? Which page? $\endgroup$Qmechanic– Qmechanic ♦2025-04-27 08:41:21 +00:00Commented Apr 27, 2025 at 8:41
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$\begingroup$ If you are following Griffiths, then you might be confusing expectation values with the hermiticity of operators given in section 3.2.1 of 3rd ed $\endgroup$GedankenExperimentalist– GedankenExperimentalist2025-04-27 09:01:51 +00:00Commented Apr 27, 2025 at 9:01
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$\begingroup$ @Qmechanic 3.2.1 Giffiths, "most books require an ostensibly stronger condition" for Hermitian Operator. Can refer to this. $\endgroup$19thdimensioncosmicstellar– 19thdimensioncosmicstellar2025-04-27 10:37:07 +00:00Commented Apr 27, 2025 at 10:37
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$\begingroup$ @GedankenExperimentalist exactly from where my confusion rose. Perfectly said. $\endgroup$19thdimensioncosmicstellar– 19thdimensioncosmicstellar2025-04-27 10:37:47 +00:00Commented Apr 27, 2025 at 10:37
2 Answers
When we write a state $\psi$ overlapped with another state $\phi$ or a state $\hat{Q}\phi$, we simply mean what is the or how much is the likelihood (not exactly this, actually it's modulus squared)* for the state $\psi$ to be the same as $\phi$ (or $\hat{Q}\phi$). So $\psi$ overlapped with $\psi$ will be $1$, it's overlap with $\hat{Q}\psi$ will be the expectation value of $\hat{Q}$ in state $\psi$ and it's overlap with $\hat{Q}\phi$ will be the expectation value of $\psi$ being equal to $\hat{Q}\phi$.
*The likelihood or the probability is actually the modulus squared of the amplitude.
The value $$ A_w = \frac{\langle \phi|\hat{A}|\psi\rangle}{\langle \phi|\psi\rangle} $$
is called a weak value and it is used to describe measurements in which states are pre and post selected and the measurement device couples weakly to the measured system. For more detail, see this paper and references therein:
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$\begingroup$ It needs to be normalized $\endgroup$flippiefanus– flippiefanus2025-04-27 14:07:02 +00:00Commented Apr 27, 2025 at 14:07
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$\begingroup$ I have normalized the weak value $\endgroup$alanf– alanf2025-04-27 19:51:20 +00:00Commented Apr 27, 2025 at 19:51