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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
∂Ψ
(1)
ĤΨ(x, t) = ih̄
∂t
We will be interested in three special cases.
• Ĥ and Ψ independent of time, with Ψ(x, t) = ψ(x) × e−ih̄Et . The time-independent wavefunction ψ obeys
the time-dependent wave equation
Ĥ(x)ψ(x) = Eψ(x)
(2)
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) = c1 ψ(x, t)e−iE1 t/h̄ + c2 ψ(x, t)e−iE2 t/h̄ + · · ·
(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 ψ1 and ψ2 . You will recall that for this case the stationary-state solutions are
2
h2
πx
ψ1 =
E1 =
(4)
sin
L
L
8mL2
2
4h2
2πx
E2 =
= 4E1
(5)
ψ2 =
sin
L
L
8mL2
(6)
1
ψ0 vs. x
ψ0 + ψ1 vs. x
ψ1 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 50-50 mixture of these two states, i.e., c1 = c2 = 1:
1 (7)
Ψ(x, t) = √ ψ1 e−iE1 t/h̄ + ψ2 e−iE2 t/h̄
2
√
where the factor 1/ 2 = 1/ c21 + c22 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 = 0 the phase factors are zero, and Ψ is related to the sum of ψ1 and ψ2 . 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:
1
(ψ1 eiE1 t/h̄ + ψ2 eiE2 t/h̄ )(ψ1 e−iE1 t/h̄ + ψ2 e−iE2 t/h̄ )
2
1 2
ψ1 + ψ22 + ψ1 ψ2 e−i(E2 −E1 )t/h̄ + ψ1 ψ2 ei(E2 −E1 )t/h̄
=
2
1
(E2 − E1 )t
2
2
=
ψ1 + ψ2 + 2ψ1 ψ2 cos
2
h̄
Ψ∗ (x, t) × Ψ(x, t) =
(8)
(9)
(10)
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 ψ3 , ψ4 , . . .. The graph shows the probability density as a function of distance for
t = 0, 0.9/ω1 , 1.8/ω1 , and 2.4/ω1 , where ω1 = E1 /h̄
2
|Ψ(x, t)|2 vs.. x for t = (n(0.3/ω1 ), n = 0, 1, 2, 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 = 2πν. 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 = 0 and
we write
E(r, t) = E(t) = E 0 cos ωt
(12)
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
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additional energy and consequently the perturbation to the Hamiltonian operator is given by
Ĥ (1) =
qi V i
i
=−
qi × (ri .E(t))
(13)
(14)
i
= −(
qi ri ).E(t)
(15)
i
= −µ̂.E(t)
where µ̂ ≡
i qi ri
(16)
is the dipole moment operator. The sum has to be carried over all the charges in the molecule.
Ĥ (1) = −µ.E 0 cos ωt
= −µE0,z cos ωt
(17)
(18)
if the electric field is in the z direction; µ is the dipole moment at time t.
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 ci
are functions of time. For simplicity we will assume that the system has only two states, and write the perturbed
wavefunction as
Ψ(t) = c0 (t)ψ1 e−iE0 t/h̄ + c1 (t)ψ2 e−iE1 t/h̄
(19)
We are particularly interested in |c1 (t)|2 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 ψ1 in the perturbed wavefunction is qproportional to ψ0 |Ĥ(1)|ψ1 . This is also valid for time-dependent perturbation theory. Here, according to eqn 18
the perturbation is equal to µE0,z cos ωt. Thus the equation of c2 involves the transition dipole moment operator
defined by
µ10 ≡ ψ1 |µ|ψ0 (20)
We define the following angular frequencies which appear in our equations:
Rabi frequency : ωR = µ10 E0,z /h̄
E1 − E 0
resonance frequency : ω10 =
h̄
detuning frequency : ∆ = ω − ω10
2
Ω = [ωR
+ ∆2 ]1/2
(21)
(22)
(23)
(24)
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
2
Ωt
ωR
2
2
|c1 (t)| = 2 sin
(25)
Ω
2
Ωt
ω2
|c0 (t)|2 = 1 − R2 sin2
(26)
Ω
2
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At resonance ∆ = 0 and Ω = ωR , and in this case
ωR t
|c1 (t)| = sin
2
ωR t
ωR t
= cos2
|c0 (t)|2 = sin2
2
2
2
2
(27)
(28)
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 |a21 | 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 ω10 .
The above theory assumes (1) that the light is on for a long time and (2) 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 c0 is very
close to 1. If the light is on for a time t we get
c1 = −
ωR −i∆t
(e
− 1)
2∆
We now get for the the probability of transition from state 0 to 1
2
2
µ210 E0,z
∆t
ωR
sin2 [(ω − ω10 )t/2]
2
2
P1←0 = |c1 | = 2 sin
×
=
2
∆
2
(ω − ω10 )2
h̄
(29)
(30)
The absorption spectrum will be determined by P1←0 . We note the following points:
• The probability of transition is proportional to the square of the electric field, E0,z
• The probability of transition is directly proportional to the square of the transition moment µ10 . Transitions
for which the transition moment is zero are not allowed. This gives us the so called selection rules: if µ10
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 ω10 . 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 τsp = 1/γ (sp for spontaneous), and we include this in solving the equations for c1 (t) the normalized line function is broadened into a
function of the form
γ/(2π)
(γ/2)2 + (ω − ω10 )2
γ
L(ν) =
(γ/2)2 + (2π)2 (ν − ν10 )2
L(ω) =
(31)
(32)
Recall that ω = 2πν. 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 = ν10 , and the range of frequencies for which
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L(ν − ν10 )
4/γ
2/γ
∆ν1/2
ν10
ν
L is half this value (2/γ) is denoted by ∆ν1/2 and is referred to as the full width at half maximum (FWHM).
It is easily shown that
∆ν1/2 =
1
γ
=
2π
2πτsp
(33)
We can use these relations to check whether the broadening is due to natural lifetime, and then determine the
lifetime τsp 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 µ10 = ψ1 |µ|ψ0 , where the dipole
moment operator is i qi ri 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 Rn Pm (θ)eimφ . The transition dipole moment will factorize thus:
∞
π
2π
m1
m0
3
µ10 = −e
r Rn1 1 Rn0 0 dr
P1 (sin θ, sin θ, cos θ)P1 sin θdθ
e−im1 (cos φ, sin φ, 1)eim0 dφ (34)
0
0
0
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:
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• 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 ∆0 ,1 ±1 = 0, i.e., ∆
= ±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 is associated with an angular
momentum of h̄: the photon has an angular momentum of ±h̄. The absorption of a photon will therefore be
accompanied by a change in angular momentum of ±h̄. According to this selection rule the transition 1s to 2s
is forbidden, since ∆
= 0, but the transtion 1s to any of the 2p levels is allowed. The transition 1s to 3d is
forbidden, since ∆
= 2
• The factor involving φ gives the selection rule ∆m = 1 for the x coordinate, −1 for the y coordinate, and 0 for
the z coordinate.
• In summary, the selection rules are: ∆n = any value, ∆
= ±1, ∆m = 0, ±1
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