Non-stationary States and Electric Dipole Transitions

Transcription

1 Pre-Lab Lecture II Non-stationary States and Electric Dipole Transitions You will recall that the wavefunction for any system is calculated in general from the time-dependent Schrödinger equation ĤΨ(x,t)=i h Ψ (1) t We will be interested in three special cases. Ĥ and Ψ independent of time, with Ψ(x,t)=ψ(x) e i het. The time-independent wavefunction ψ obeys the time-dependent wave equation Ĥ(x)ψ(x) =Eψ(x) () The wavefunction is an eigenfunction of the Hamiltonian with eigenvalue E. The probability density is independent of time: Ψ (x,t)ψ(x,t)=ψ (x)ψ(x) and the wavefunction is said to the stationary. You have seen several example which fall under this category: wavefunction for particle in a box, wavefunction for the harmonic oscillator, wavefunction for the three-dimensional rotor and the wavefunction for the hydrogen atom. Ĥ independent of time, Ψ is time-dependent. These are time-dependent or non-stationary states. Ĥ depends on time briefly. Here the wavefunction varies with time. This variation may be interpreted as a transition of the system between stationary states. Evolution in time of mixed states It is possible to prepare, by selective excitation of a stationary system, a system whose wave equation cannot be written in terms of just one stationary state (pure state) but has to be written in terms of a sum of stationary states: Ψ(x,t)=c 1 ψ(x,t)e ie1t/ h + c ψ(x,t)e iet/ h + (3) The probability density is clearly a function of time and this case falls under the second category we discussed above. Such a mixed state is also sometimes referred to as a wave packet. We wish to see how the wavefunction, and consequently the probability density, changes with time. To keep matters simple we will consider the case of a free particle in a one-dimensional box and consider only the lowest two states ψ 1 and ψ. You will recall that for this case the stationary-state solutions are ψ 1 = sin πx E 1 = h L L 8mL (4) ψ = sin πx E = 4h L L 8mL =4E 1 (5) (6) 1

2 ψ 0 vs. x ψ 1 vs. x ψ 0 + ψ 1 vs. x The solutions are sine-waves. The ground state is a half-wave, and the 1st excited state is a full wave. The probability density ψ ψ is not a function of time and is symmetric in x. Let us consider a consider normalized wavefunction that is a mixture of these two states, i.e., c 1 = c =1: Ψ(x, t) = 1 [ψ 1 e ie1t/ h + ψ e iet/ h] (7) where the factor 1/ =1/ c 1 + c is a normalization factor. We have constructed this mixed state in terms of the lowest two stationary state time-dependent wavefunction: these include the exponentials involving the time as phase factors. Using eq 7 we can calculate that value of both Ψ and the probability density Ψ (x, t)ψ(x, t) for all values of x and t. At time t =0the phase factors are zero, and Ψ is related to the sum of ψ 1 and ψ. The graph shows that the two wavefunctions add constructively in the left half while they add destructively in the right half: the wave tends to concentrate in the left half at the beginning. The wave is no longer symmetric in x. This is seen better by examining the probability density: Ψ (x, t) Ψ(x, t) = 1 [ ] (ψ 1 e ie1t/ h + ψ e iet/ h )(ψ 1 e ie1t/ h + ψ e iet/ h ) (8) = 1 ( ψ1 + ψ + ψ 1 ψ e i(e E1)t/ h + ψ 1 ψ e i(e E1)t/ h) (9) = 1 ( ψ1 + ψ +ψ 1 ψ cos (E ) E 1 )t (10) h You will see from eq 10 that the probability density varies with time in a periodic function. We can use the eq 4 and eq 5 in the above equation and see how the probability density varies with time. This is shown in the diagram below. The probability density is concentrated in some areas more than others and moves along x with time: this is very suggestive of a particle moving back and forth. The can be made more localized and concentrated in a smaller region by mixing the higher wavefunctions ψ 3,ψ 4,... The graph shows the probability density as a function of distance for t =0, 0.9/ω 1, 1.8/ω 1, and.4/ω 1, where ω 1 = E 1 / h

3 Ψ(x, t) vs.. x for t =(n(0.3/ω 1 ),n=0, 1,, 3 Electric Dipole Transitions In this section we will try to understand, at least qualitatively, how a molecule interacts with light (electromagnetic radiation). You will recall that electromagnetic radiation is associated with electric and magnetic fields which vary with time. If these fields are not very strong, we may treat these as perturbations; however these are time-dependent perturbations. We have therefore to use time-dependent perturbation theory. Under the effect of the perturbation, the wavefunction becomes a mixture of pure states, but the contributions of the pure states vary with time. We may describe this situation in terms of a transition of the system from the original state to other sates with varying probabilities. The probability of transition to various states depends on the relation of the frequency of the incident light to the difference in the energies of the two states involved in the transition as well as other factors like the overlap of the wavefunctions. Thus when the energy of the photon of the incident light is equal to the difference in the energy levels, the probability of transition is highest, and falls off as the frequency of light deviates from equality on either side. We may compare this to a tuning fork that puts surrounding objects into sympathetic vibration. If the natural frequency of oscillation of the object is equal to the frequency of the tuning fork, it comes into resonance and is able to absorb energy from the vibrations of the tuning fork. As the natural frequency of the object deviates from equality the object responds less to the tuning fork. Though the light will change both the electric as well as magnetic properties, the latter is relatively small and we will neglect it. Also, the interaction of the electric field with the molecule may be broken up into interactions resulting in changes to (i) the electric dipole of the molecule, (ii) the electric quadrupole of the molecule, (iii) higher multipoles. Here too, we will restrict our attention to changes in the electric dipole, since this is the most important of these. Only if this interaction is zero need we consider the others. For this reason our results refer only to the electric dipole interaction with light. We will assume that the electric field of the light is given by E(r,t)=E 0 cos (k.r ωt) (11) Here, the angular frequency ω is related to the frequency ν by ω =πν. If the molecule is at the origin, and we assume that over the small distance occupied by a molecule the variation of k.r can be neglected, i.e., k.r =0and we write E(r,t)=E(t) =E 0 cos ωt (1) The cosine variation with time with a single frequency is strictly true only if the light is on for infinite time. Since we are exposing the atom to light only briefly, we have to strictly consider a mixture of frequencies with a slight spread about the frequency ω. We have seen earlier that in the presence of an electric field there is an addition potential V = r.e. Thus the 3

4 additional energy and consequently the perturbation to the Hamiltonian operator is given by Ĥ (1) = i q i V i (13) = i = ( i q i (r i.e(t)) (14) q i r i ).E(t) (15) = ˆµ.E(t) (16) where ˆµ i q ir i is the dipole moment operator. The sum has to be carried over all the charges in the molecule. if the electric field is in the z direction; µ is the dipole moment at time t. Ĥ (1) = µ.e 0 cos ωt (17) = µe 0,z cos ωt (18) We can treat our problem in terms of the perturbation theory if the amplitude of the electromagnetic field E 0 is small. The case is very similar to the polarization of the H atom by a static electric field, except that the coefficients c i are functions of time. For simplicity we will assume that the system has only two states, and write the perturbed wavefunction as Ψ(t) =c 0 (t)ψ 1 e ie0t/ h + c 1 (t)ψ e ie1t/ h (19) We are particularly interested in c 1 (t) which gives us the probability that the system has made a transition from state 0 to state 1. We saw that according to stationary perturbation theory, the coefficient of ψ 1 in the perturbed wavefunction is qproportional to ψ 0 Ĥ(1) ψ 1. This is also valid for time-dependent perturbation theory. Here, according to eqn 18 the perturbation is equal to µe 0,z cos ωt. Thus the equation of c involves the transition dipole moment operator defined by We define the following angular frequencies which appear in our equations: µ 10 ψ 1 µ ψ 0 (0) Rabi frequency : ω R = µ 10 E 0,z / h (1) resonance frequency : ω 10 = E 1 E 0 h () detuning frequency : = ω ω 10 (3) Ω=[ωR + ] 1/ (4) The detuning frequency is 0 when the angular frequency of the light ω is equal to the resonance frequency. With these definitions, our solutions are c 1 (t) = ω R Ωt Ω sin (5) c 0 (t) =1 ω R Ωt Ω sin (6) 4

5 At resonance =0and Ω=ω R, and in this case c 1 (t) =sin ωr t c 0 (t) =sin ωr t =cos ωr t (7) (8) These equations show that under the influence of the electric field the probability of the upper state varies periodically with a frequency equal to the Rabi frequency. This is superficially similar to the tuning fork setting surround objects into sympathetic oscillation; here however the frequency of oscillation is not equal to the frequency of the light. The amplitude of a 1 is greatest at resonance, and becomes progressively smaller as the detuning frequency becomes larger. The dipole moment of the system also begins to oscillate periodically with a frequency equal to the resonance frequency ω 10. The above theory assumes (1) that the light is on for a long time and () that there are no other perturbations. An important effect we have neglected is the fact that the upper state can relax to the lower state by spontaneous emission of radiation. When we take the time of exposure to light to be small, and the electric field to be small, we can assume that c 0 is very close to 1. If the light is on for a time t we get c 1 = ω R (e i t 1) (9) We now get for the the probability of transition from state 0 to 1 P 1 0 = c 1 = ω R sin t = µ 10E0,z h sin [(ω ω 10 )t/] (ω ω 10 ) (30) The absorption spectrum will be determined by P 1 0. We note the following points: The probability of transition is proportional to the square of the electric field, E 0,z The probability of transition is directly proportional to the square of the transition moment µ 10. Transitions for which the transition moment is zero are not allowed. This gives us the so called selection rules: if µ 10 is non-zero, the transition is allowed, if it is zero, it is forbidden. These selection rules are not always valid, because we have made some assumptions, e.g., we have ignored multipole transitions and neglected magnetic interactions. The factor which includes ω is called the line shape function L(ω). In our case, it is a Dirac delta function, sharply peaked at the resonance frequency ω 10. When we consider the spontaneous emission of radiation from the upper state, the line shape function is altered. If the natural lifetime of state 1 is τ sp =1/γ (sp for spontaneous), and we include this in solving the equations for c 1 (t) the normalized line function is broadened into a function of the form γ/(π) L(ω) = (γ/) +(ω ω 10 ) (31) γ L(ν) = (γ/) +(π) (ν ν 10 ) (3) Recall that ω =πν. Functions of this form are referred to as Lorentzian functions. The Lorentzian line-shape function is broader than the Dirac delta function. The broadening we are dealing with here is natural lifetime broadening. The Lorentzian function has a maximum of 4/γ at ν = ν 10, and the range of frequencies for which 5

6 4/γ L(ν ν10) /γ ν 1/ ν 10 ν L is half this value (/γ) is denoted by ν 1/ and is referred to as the full width at half maximum (FWHM). It is easily shown that ν 1/ = γ π = 1 (33) πτ sp We can use these relations to check whether the broadening is due to natural lifetime, and then determine the lifetime τ sp from the FWHM. There are other factors which contribute to broadening of the line shape, e.g., Döppler broadening in gases due to the molecules moving towards or away from the detector and pressure broadening in gases due to collisions. Döppler broadening depends on the temperature and is Gaussian in shape, rather than Lorentzian. The FWHM is proportional to T/M. In pressure broadening the FWHM is directly proportional to the pressure. Selection Rules for the Hydrogen Atom We have seen that the selection rules are governed by the transition dipole moment µ 10 = ψ 1 µ ψ 0, where the dipole moment operator is i q ir i and the summation is over all the charges. For the H atom, the nucleus is at the origin and there is only 1 electron with q = e. The dipole moment operator is therefore equal to er where r is the coordinate of the electron. This has three components, ex = er sin θ cos φ, ey = er sin θ sin φ and ez = er cos θ. The wavefunctions are of the form R nl Pl m(θ)eimφ. The transition dipole moment will factorize thus: µ 10 = e 0 r 3 R n1l 1 R n0l 0 dr π 0 P m1 l 1 (sin θ, sin θ, cos θ)p m0 l 1 sin θdθ π 0 e im1 (cos φ, sin φ, 1)e im0 dφ (34) The quantities in parenthesis, (sin θ, sin θ, cos θ) and (cos φ, sin φ, 1) are the angular contributions of x, y, and z components of the perturbation, which in this case happens to be x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. We note the following: 6

7 The value of the transition moment will depend on the overlap of the radial functions, i.e., the the extent to which both radial wavefunction have the same sign in the same region of space. When the principal quantum numbers are very different, this tends to be small. Though there is not restriction of n for an electric dipole transition, the transition moment is small when n is large. The factor involving θ gives l0,l 1±1 =0, i.e., l = ±1 for the x, y and the z components. This has to be correlated with the law of conservation of angular momentum and the fact that l is associated with an angular momentum of l h: the photon has an angular momentum of ± h. The absorption of a photon will therefore be accompanied by a change in angular momentum of ± h. According to this selection rule the transition 1s to s is forbidden, since l =0, but the transtion 1s to any of the p levels is allowed. The transition 1s to 3d is forbidden, since l = The factor involving φ gives the selection rule m =1for the x coordinate, 1 for the y coordinate, and 0 for the z coordinate. In summary, the selection rules are: n = any value, l = ±1, m =0, ±1 7