dipole moment

KFC

1. The problem statement, all variables and given/known data
I we know the eigenstates of the system be [tex]|\psi_1\rangle[/tex] and [tex]|\psi_2\rangle[/tex]. Current state of the system is

[tex]|\Psi\rangle = c_1 |\psi_1\rangle + c_2 |\psi_2\rangle[/tex]

Try to find the expectation value of electric dipole moment [tex]\mu[/tex] (assume it is real) and write it in matrix form

2. The attempt at a solution
The expectation value of something is just the integral of that operator in given state, so

[tex]\langle \mu \rangle = \int \Psi^* \mu \Psi d^3x = \int (c_1^* \psi_1^* + c_2^* \psi_2^*)\mu(c_1 \psi_1 + c_2 \psi_2) = |c_1|^2 + |c_2|^2 + \int c_1^*c_2\psi_1^*\psi_2d^3x + \int c_1c_2^*\psi_1\psi_2^*d^3x[/tex]


The last two terms are zero because the eigenstates are orthogonal to each other, right?

so

[tex]\langle \mu \rangle = \mu|c_1|^2 + \mu|c_2|^2 [/tex]


Is this correct? But what does it mean by writing it as matrix form?

weejee

Isn't mu an operator defined by mu = ex, where e is the electron charge and x is the position operator. I think you treated mu as a number.