Braket Notation: Is <φ|x+y+z|φ> = <φ|x|φ> + <φ|y|φ> + <φ|z|φ>?

[brydustin]

If x,y,z are the position operators.

Is it true that:

<φ|x|φ> + <φ|y|φ> + <φ|z|φ> = <φ | x+y+z| φ> ?

So that if, for example, one wanted to compute <φ|r|φ> (where r =x+y+z), then they would just have to sum the parts.

I know that for scalars, a and b, we have the following:

(a+b)|φ> = a|φ> + b|φ>
But I don't know for sure if this is related at all to the case for operators (especially when they are sandwiched between the bra and the ket.

 

[Chopin]

Yes, it's true that \langle \psi | A + B | \psi \rangle = \langle \psi | A | \psi \rangle + \langle \psi | B | \psi \rangle. But I'm not sure where you're going with the substitution r = x + y + z. If you're trying to convert to spherical coordinates, that's not right--you need to do r = \sqrt{x^2 + y^2 + z^2} instead.

 

[brydustin]

Yes, it's true that \langle \psi | A + B | \psi \rangle = \langle \psi | A | \psi \rangle + \langle \psi | B | \psi \rangle. But I'm not sure where you're going with the substitution r = x + y + z. If you're trying to convert to spherical coordinates, that's not right--you need to do r = \sqrt{x^2 + y^2 + z^2} instead.

I am given the set of overlaps <φ_i|φ_j> and <φ_i|x|φ_j> for all i and j (as well as y and z sets). So that's four sets (the overlaps, and the set with x y and z).

So if I'm given those sets, how do I combine the sets {<φ_i|x|φ_j>, <φ_i|y|φ_j>, <φ_i|z|φ_j>} to get the single set {<φ_i|r|φ_j>}

 

[Chopin]

And the definition of r is x + y + z? Or something else?

 

[brydustin]

Could the following be a solution?


<φ|r|φ> = <φ|x|φ>i-hat + <φ|y|φ>j-hat + <φ|z|φ>k-hat

where i-hat is the unit vector of the x-axis, j-hat is the unit vector of the y-axis, and k-hat is the unit vector of the z-axis.

 

[brydustin]

And the definition of r is x + y + z? Or something else?

I double checked my notes:

r = x*i-hat +y*j-hat + z*k-hat *(by definition)

 

[Chopin]

Ah, I see, they're defining r as a vector-valued operator. In that case, there's actually no new math here, it's just a notation trick. Specifically, it's a way of unifiying three separate operators together into one object that's easy to keep track of. So we have three operators \hat{x}, \hat{y}, \hat{z}, and we're defining the new object \hat{\textbf{r}} = (\hat{x}, \hat{y}, \hat{z}).

In this case, applying the operator to a state is really applying three separate operators to the state, and using them to make an ordered triple. \langle \psi | \hat{\textbf{r}} | \psi \rangle = \langle \psi | (\hat{x}, \hat{y}, \hat{z}) | \psi \rangle = (\langle \psi | \hat{x} | \psi \rangle, \langle \psi | \hat{y} | \psi \rangle, \langle \psi | \hat{z} | \psi \rangle).

 

[brydustin]

Final question:

What does it mean to do the following:

(<φ|r|φ> - <ψ|r|ψ>)^2

Obviously if <ψ|r|ψ> is a vector, then the difference of two vectors is straightforward. What does it mean to take their square?

 

[Chopin]

Well, think about what it means to take the square of a normal vector. Since x^2 = x\cdot x (where x is a normal number), then the square of a vector must be \textbf{r}^2 = \textbf{r} \cdot \textbf{r}. By the definition of the dot product, \textbf{r} \cdot \textbf{r} = r_x^2 + r_y^2 + r_z^2. The generalization to the quantum case should be pretty straightforward.