The total hamiltonian for this system has two parts: the atom's hamiltonian and the interaction hamiltonian. The interaction hamiltonian contains the fundamental physics of how the atom and field influence each other. The external forces on the system must be known in order to derive the interaction hamiltonian. The external force is , where e is the charge of the electron, is the position vector in three space, and is the electric field. The electric field is . Using , the external potential can be generated:
This potential is the interaction hamiltonian.
The total hamiltonian is the atom's hamiltonian plus the interaction hamiltonian
Finding the equations of motion for this system is now straightforward. Let us write down the time dependent Schrödinger equation:
Writing as and substituting this result into eq:SE
Multiplying eq:T1 by an arbitrary eigenbra <n|
eq:T2 forms two equations:
where the orthonormality condition for <n|m> has been utilized. eq:T3 and eq:T4 provide a route for finding the terms for both the ground and excited states for any time.
The only undefined term in these equations is , which is a matrix element for the interaction hamiltonian. Let us substitute the complete form of the interaction hamiltonian into this expression:
The only operator is the position vector. This expression can then be rewritten as
is the matrix element for the electric dipole moment.
The electric dipole moment determines whether or not a particular transition is allowed. or are the possible transitions. All other configurations will have an electric dipole moment equal to zero. When the dipole moment is non--zero, it could be either real or complex, depending on what particular spatial dependencies the eigenkets have. This spatial dependence does not add anything to the physics per se. Therefore, the magnitude of the dipole moment will be arbitrarily set to unity; however, the electric dipole moment is a vector and a unit vector, , will be left to hold its place in the dot product. Since the dipole moment has been defined as a real quantity, .
This new knowledge allows us to rewrite eq:T3 and eq:T4:
Let us now substitute our electric field function into the matrix element of the interaction hamiltonian as it now stands
The result that we have obtained says something rather interesting. Only the portion of the electric field which is parallel to the dipole moment of the atom will interact with the atom. A quantized atom will only oscillate if it is hit at just the right angle with an electromagnetic wave---the Lorentz Oscillator model had no such selective directionality.
Finally, the equations of motion for the probability amplitudes in their most complete form can be written as:
Dividing these equations by and collecting all of the constants into a single constant, ,