I'm trying to find the expectation value of the operator $\hat W(x_1,x_2)=\hat x_1 \hat x_2$ with respect to the eigenstates of a system composed of two one dimensional quantum harmonic oscillators. The eigenstate of the total system will be $|n_1n_2\rangle=|n_1\rangle \otimes|n_2\rangle $, with $|n_1\rangle$, $|n_2\rangle$ the eigenstates of each individual oscillator, so the expectation value will be
$$\left((|n_1\rangle \otimes|n_2\rangle )^\dagger,\hat W(|n_1\rangle \otimes|n_2\rangle) \right)$$ Two questions have arisen to me with this:
- Is the bra corresponding to a ket formed by a tensor product just the tensor product of the bras, $\left(|n_1 n_2\rangle \right)^\dagger=\left(|n_1\rangle \otimes|n_2\rangle \right)^\dagger=\langle n_1| \otimes \langle n_2| =\langle n_1 n_2| $?
- Are operators corresponding to different Hilbert spaces associative with respect to the tensor product of different states? That is, $ \hat x_1 \hat x_2 (|n_1\rangle \otimes|n_2\rangle)=\hat x_1 |n_1\rangle \otimes\hat x_2 |n_2\rangle?$
- How do the inner products behave with respect to the tensor products? Would it be just $\big(\langle n_1| \otimes \langle n_2|\big) \big(\hat x_1 |n_1\rangle \otimes\hat x_2 |n_2\rangle \big)= \langle n_1|\hat x_1 |n_1\rangle\otimes\ \langle n_2|\hat x_2 |n_2\rangle$?