Physical meaning of ##\psi_1(x)\hat{A}\psi_2(x)##

[amjad-sh]

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?

 

[vanhees71]

I don't see any physical meaning in this expression.

 

[DrClaude]

Inside an integral?

 

[vanhees71]

What should it mean there?

 

[DrClaude]

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 $\hat{A}$.

 

[haushofer]

I think bra-ket notation is missing.

 

[olgerm]

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.
Is it expected value of quantity A?

 

[DrClaude]

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.

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)
$$
\int \psi_1^*(x) \hat{A} \psi_2(x) \, dx
$$

Is it expected value of quantity A?

No. That would only be the case for $\psi_1 = \psi_2$.

 

[DaTario]

If $\hat A$ is a perturbing Hamiltonian, a search in "Fermi's Golden Rule" may present useful information related to the expressions mentioned by DrClaude in #8. They may be related to transition probabilities.

 

[vanhees71]

Well, if you want to calculate
$$\langle \psi_1|\hat{A}|\psi_2 \rangle=\int_{\mathbb{R}} \mathrm{d} x_1 \int_{\mathbb{R}} \mathrm{d} x_1 \psi_1^*(x_1) A(x_1,x_2) \psi_2(x_2).$$
Now defining the operator in position representation as
$$\hat{A} \psi(x)=\int_{\mathbb{R}} \mathrm{d} x' \langle x|\hat{A}|x' \rangle \psi(x')$$

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 $\psi_1 = \psi_2$.

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!

 

[DrClaude]

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!

Of course. I'll fix my typo.

 

[amjad-sh]

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 $\hat{A}$.

I will put more context to make my point more clear.
$$\hat{H}=\dfrac{p^2}{2m}-\dfrac{\partial_z^2}{2m}+V(z) +\gamma V'(z)(\hat{z} \times \vec{p})$$
The eigenfunctions of the Hamiltonian $\hat{H}$ have the following form:
$\vec{\varphi}_{k\sigma}(z)=\begin{cases} (e^{ikz}+(r_0+r_x\sigma_x+r_y\sigma_y)e^{-ikz})\xi_{\sigma} \quad\text{if} \quad z<0\\ (t_0+t_x\sigma_x+t_y\sigma_y)e^{ik'z}\xi_{\sigma} \quad \text{if} \quad z>0 \end{cases}$

Where $V(z)=V\theta(z)$,$\xi_{\pm}$ are the eigenfunctions of $\sigma_x$ and the terms $(r_x\sigma_x+r_y\sigma_y)e^{-ikz}\xi_{\sigma}$ and $(t_x\sigma_x+t_y\sigma_y)e^{ik'z}\xi_{\sigma}$ are the terms added due to the presence of Rashba spin-orbit coupling term at z=0.
The presence of the Rashba spin-orbit coupling term at the interface induces lateral spin and charge currents flowing along the x-y plane.
The interference between the spin-orbit coupling part of the wavefunction such as $(r_x\sigma_x +r_y\sigma_y)\xi_{\sigma}$ with the part not related to the spin-orbit coupling term is the fundamental reason behind the induction of the lateral charge and spin currents.
I speculated that the quantum overlap between $e^{ikz}\xi_{\sigma}$ and $r_x\sigma_x\xi_{\sigma}$ or $r_y\sigma_y\xi_y$ induces a charge current flowing along the x or y directions, respectively, as $r_x \propto p_x$ and $r_y \propto p_y$.
In the case of spin currents, I speculated that instead of quantum overlapping the non-zero value, for example, of the integral below:
$$\int (e^{-ikz}\xi_{\sigma}^{\dagger})\sigma_{x/y/z}(r_{x/y}\sigma_{x/y}\xi_{\sigma})dz$$
induces a spin current flowing along the x/y direction and spin-polarized along the x/y/z-axis.
I want to know what physical meaning may this integral give.