Ch 10 567-576; 555-559

1. Lecture 26 – Quantum description of absorption Ch 10 567-576; 555-559

2. Quantum mechanical description The interaction leading to absorption of light is electromagnetic in origin. The oscillating electromagnetic field associated with the incoming photon generates a force on the charged particles in a molecule (electrons); if the interaction results in a change in electronic state, we say that a transition has occurred; this is absorption The only proper way to understand absorption of electromagnetic radiation is through quantum mechanics. Although we have only considered until now the time-independent Schrodinger equation, you should recall that the time-dependent Schroedinger equation is:

3. Quantum mechanical description If the Hamiltonian is independent of time, the wave function, i.e. the solution to the time-dependent Schroedinger equation is: where En and Yn(x,y,z) are obtained by solving the time-independent Schroedinger equation: Let us now consider the system is perturbed because electromagnetic radiation is applied. A light wave has an electric field component: n is the frequency of the radiation

4. Quantum mechanical description This electric field interacts with an atom or molecule which has an electric dipole moment m defined from the charges qi and their position in space as follows: Classical electrodynamics has provided an expression for the energy of interaction between the molecular dipole and the electric field component of the light wave: where q is the angle between the electric dipole moment vector and the electric field vector

5. Quantum mechanical description This energy of interaction between the molecular dipole and the light wave changes the Hamiltonian to a time-dependent expression: reflects all time-independent energies and interactions and U(t) reflects the interaction of the molecular dipole with a time-dependent electric field Schroedinger’s equation is now:

6. Quantum mechanical description If there is no electromagnetic field V, then Schroedinger’s equation is: The solution to this equation has the form: When the electric field is present, the Schroedinger’s equation has solutions of the form:

7. Quantum mechanical description The physical interpretation is as follows. Suppose just before the atom is exposed to light the atom is in its ground state described by the ground state wave function When the light and the atom interact, energy is absorbed from the light into the atom. The result of this absorption of energy is that the atom may undergo transitions from its ground state into higher energy states described by wave functions is telling us the condition of the system after it has absorbed radiation for a time t

8. Quantum mechanical description The probability Pn that at time t the system has made a transition If the interaction of the electric field with the electric dipole moment is weak the probability amplitude has the general form: Un,0 is the transition moment from state 0 to state n

9. Quantum mechanical description This is called transition dipole and is what allows us to understand absorption of radiation and also calculate how radiation is absorbed from atomic or molecular orbitals. The probability amplitudes have the final form: This integral is non-zero if and only if:

10. Transition dipoles The concept of transition dipole can be generalized, because the dipole itself is a vector and not a scalar operator. The transition dipole between two states m and n is defined as: Because the transition dipole is a vector, it has direction in addition to intensity. Most electronic transitions are polarized, which means that the transition dipole is not the same in all directions. The magnitude of the transition is defined as the dipole strength: Its units are called Debyes, and are obviously the same units as the square of a dipole (i.e. Cxm). 1 D=3.336x10-30Cxm

11. Transition dipoles Example - Calculation of Transition Dipole Moments Suppose the vibrational motion of a homonuclear diatomic molecule is modeled as a simple harmonic oscillation. Assume the system is initially in the ground state, described by the wave function: The system is irradiated at the fundamental frequency: Will a transition occur to the n=1 state? The answer rests on the value of the transition dipole moment

12. Transition dipoles If we define the dipole moment as qx, the transition dipole moment has the form: the integral has the form: (the function is even so the transition dipole is not zero). Because the transition dipole moment is non-zero, the transition has non-zero intensity if DE=hn

13. Transition dipoles Will a transition from n=0 to n=2 if we irradiate at (E2-E0)/h? The transition moment is now: the function is odd so the transition dipole is zero If we irradiate a harmonic oscillator exactly at we will observe a transition from n=0 to n=1 but not from n=0 to n=2. This is an example of a selection rule. For a harmonic oscillator, resonant irradiation will only induce transitions for which

14. Stimulated and Spontaneous Emission Thus far we have discussed the response of a two energy level system to a resonant field, where the system is initially in the ground state: Suppose instead that the system is initially in the excited state: A transition to the ground state from the excited state will occur when the system is irradiated with a resonant field. Because the transition is from a higher energy state to a lower energy state, energy is emitted instead of absorbed. This event is called stimulated emission.

15. Stimulated and Spontaneous Emission In fact, the probability per unit time that a stimulated emission will occur has exactly the same value as the corresponding absorption problem given above. In addition to stimulated emission, spontaneous emission may also occur, whereby in the absence of a resonant field the system spontaneously emits energy and drops from the excited state into the ground state.

16. Two Level system We can learn more about absorption by considering in greater detail a simple example of the absorption of energy by a quantized system consisting of only two energy levels: The ground state corresponds to n=0. If the system is exposed to a time dependent electric field the wave function of the system is:

17. Two Level system If the energy of interaction between the system’s electric dipole moment m and the electric field V(t) is not large, the system changes slowly, that is to say, the transition from the ground state into the excited state occurs slowly and the probability amplitudes have the form:

18. Two Level system Using the fact that: The probability of finding a particle in the excited state is:

19. Two Level system There are two interesting features of this probability. First of all, it oscillates sinusoidally as a function of time, reaching the larger maximum probability as the splitting between the energy levels DE becomes closer to hn, the frequency of the radiation times Planck’s constant. Second, the probability of a transition to the excited state approaches 100% only as the difference between DE and hn approach zero. The probability of observing a transition to the excited state increases as the frequency of the radiation approaches resonance i.e. DE=hn.

20. Two Level system The transition probability is sharply peaked around DE-hn=0 As DE approaches hn, the transition probability approaches: The probability per unit time that a transition will occur to the excited state is, on resonance: