Expectation value of a vector?

[phrygian]

Homework Statement

Prove that for a particle in a potential V(r) the rate of change of the expectation value of the orbital angular momentum L is equal to the expectation value of the torque:

d/dt <L> = <N>

Where N = r x(-del V)

N, r, and L are vectors.

Homework Equations

The Attempt at a Solution

I know how to solve this, but do not understand the equation. How can a vector have an expectation value? A vector is not an operator, I thought only an operator can have an expectation value?

Thanks for the help

 

[kuruman]

That's three equations in one. You write out the components and the expectation value of the x component of the torque will be equal to the time derivative of the expectation value of L_x and so on.

 

[arkajad]

In other words:

$$d/dt \begin{pmatrix}\langle L_x\rangle\\ \langle L_y\rangle\\ \langle L_z\rangle \end{pmatrix}= \begin{pmatrix}\langle N_x\rangle \\\langle N_y\rangle \\ \langle N_z\rangle \end{pmatrix}$$

where $L_x,L_y,L_z,N_x,N_y,N_z$ are operators