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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 beolgerm said:I 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##.olgerm said:Is it expected value of quantity A?
NO! It'sDrClaude said:Sorry 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.vanhees71 said: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!
I will put more context to make my point more clear.DrClaude said:It wouldn't lead necessarily to a "physical meaning," but I was hoping for more context. Looks more like a set-up for the matrix representation of ##\hat{A}##.
The physical meaning of ##\psi_1(x)\hat{A}\psi_2(x)## is the probability amplitude for a particle described by wavefunctions ##\psi_1(x)## and ##\psi_2(x)## to be in a state described by the operator ##\hat{A}##. This quantity is used in quantum mechanics to calculate the probabilities of different outcomes of measurements on a system.
The uncertainty principle states that certain pairs of physical quantities, such as position and momentum, cannot be precisely measured at the same time. In the case of ##\psi_1(x)\hat{A}\psi_2(x)##, the product of the wavefunctions and the operator represents the simultaneous measurement of two quantities. Therefore, the uncertainty principle applies to this quantity as well.
Yes, ##\psi_1(x)\hat{A}\psi_2(x)## can have a negative value. This indicates that the particle described by the wavefunctions ##\psi_1(x)## and ##\psi_2(x)## has a negative probability of being in the state described by the operator ##\hat{A}##. However, the overall probability of the particle being in any state must still be positive.
The physical meaning of ##\psi_1(x)\hat{A}\psi_2(x)## can change depending on the operator used. For example, if ##\hat{A}## represents the position of a particle, then ##\psi_1(x)\hat{A}\psi_2(x)## represents the probability amplitude for the particle to be at a specific position. However, if ##\hat{A}## represents the momentum of a particle, then ##\psi_1(x)\hat{A}\psi_2(x)## represents the probability amplitude for the particle to have a specific momentum.
The product of two wavefunctions and an operator, ##\psi_1(x)\hat{A}\psi_2(x)##, is a fundamental concept in quantum mechanics. It allows us to calculate the probabilities of different outcomes of measurements on a system described by the wavefunctions ##\psi_1(x)## and ##\psi_2(x)##. This quantity also takes into account the inherent uncertainty of quantum systems, as described by the uncertainty principle.