# Dipole moment

1. Dec 22, 2008

### KFC

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

$$|\Psi\rangle = c_1 |\psi_1\rangle + c_2 |\psi_2\rangle$$

Try to find the expectation value of electric dipole moment $$\mu$$ (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

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

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

so

$$\langle \mu \rangle = \mu|c_1|^2 + \mu|c_2|^2$$

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

Last edited: Dec 22, 2008
2. Dec 30, 2008

### 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.