Subsections


7.7 Absorption and Stimulated Emission

This section will address the basic physic of absorption and emission of radiation by a gas of atoms in an electromagnetic field. The next section will give practical formulae.

Figure 7.9: Emission and absorption of radiation by an atom.
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Figure 7.9 shows the three different processes of interest. The previous sections already discussed the process of spontaneous emission. Here an atom in a state $\psi_{\rm {H}}$ of high energy emits a photon of electromagnetic radiation and returns to an atomic state $\psi_{\rm {L}}$ of lower energy. For example, for a hydrogen atom the excited state $\psi_{\rm {H}}$ might be the $\psi_{210}$ 2p$_z$ state, and the lower energy state $\psi_{\rm {L}}$ the $\psi_{100}$ 1s ground state, as defined in chapter 4.3.

To a superb approximation, the photon carries off the difference in energy between the atomic states. In view of the Planck-Einstein relation, that means that its frequency $\omega$ is given by

\begin{displaymath}
\hbar\omega = E_{\rm {H}} - E_{\rm {L}}
\end{displaymath}

Unfortunately, the discussion of spontaneous emission in the previous sections had to remain incomplete. Nonrelativistic quantum mechanics as covered in this book cannot accommodate the creation of new particles like the photon in this case. The number of particles has to stay the same.

The second process of interest in figure 7.9 is absorption. Here an atom in a low energy state $\psi_{\rm {L}}$ interacts with an external electromagnetic field. The atom picks up a photon from the field, which allows it to enter an excited energy state $\psi_{\rm {H}}$. Unlike spontaneous emission, this process can reasonably be described using nonrelativistic quantum mechanics. The trick is to ignore the photon absorbed from the electromagnetic field. In that case, the electromagnetic field can be approximated as a known one, using classical electromagnetics. After all, if the field has many photons, one more or less is not going to make a difference.

The third process is stimulated emission. In this case an atom in an excited state $\psi_{\rm {H}}$ interacts with an electromagnetic field. And now the atom does not do the logical thing; it does not pick up a photon to go to a still more excited state. Instead it uses the presence of the electromagnetic field as an excuse to dump a photon and return to a lower energy state $\psi_{\rm {L}}$.

This process is the operating principle of lasers. Suppose that you bring a large number of atoms into a relatively stable excited state. Then suppose that one of the atoms performs a spontaneous emission. The photon released by that atom can stimulate another excited atom to release a photon too. Then there are two coherent photons, which can go on to stimulate still more excited atoms to release still more photons. And so on in an avalanche effect. It can produce a runaway process of photon release in which a macroscopic amount of monochromatic, coherent light is created.

Masers work on the same principle, but the radiation is of much lower energy than visible light. It is therefore usually referred to as microwaves instead of light. The ammonia molecule is one possible source of such low energy radiation, chapter 5.3.

The analysis in this section will illuminate some of the details of stimulated emission. For example, it turns out that photon absorption by the lower energy atoms, figure 7.9(b), competes on a perfectly equal footing with stimulated emission, figure 7.9(c). If you have a 50/50 mixture of atoms in the excited state $\psi_{\rm {H}}$ and the lower energy state $\psi_{\rm {L}}$, just as many photons will be created by stimulated emission as will be absorbed. So no net light will be produced. To get a laser to work, you must initially have a “population inversion;” you must have more excited atoms than lower energy ones.

(Note that the lower energy state is not necessarily the same as the ground state. All else being the same, it obviously helps to have the lower energy state itself decay rapidly to a state of still lower energy. To a considerable extent, you can pick and choose decay rates, because decay rates can vary greatly depending on the amount to which they are forbidden, section 7.4.)


Key Points
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An electromagnetic field can cause atoms to absorb photons.

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However, it can also cause excited atoms to release photons. That is called stimulated emission.

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In lasers and masers, an avalanche effect of stimulated emission produces coherent, monochromatic light.


7.7.1 The Hamiltonian

To describe the effect of an electromagnetic field on an atom using quantum mechanics, as always the Hamiltonian operator is needed.

The atom will be taken to be a hydrogen atom for simplicity. Since the proton is heavy, the electromagnetic field interacts mainly with the electron. The proton will be assumed to be at rest.

It is also necessary to simplify the electromagnetic field. That can be done by decomposing the field into separate “plane waves.” The total interaction can usually be obtained by simply summing the effects produced by the separate waves.

A single plane wave has an electric field $\skew3\vec{\cal E}$ and a magnetic field $\skew2\vec{\cal B}$ that can be written in the form, (13.10):

\begin{displaymath}
\skew3\vec{\cal E}= {\hat k}{\cal E}_{\rm {f}} \cos\Big(\o...
...rac1c {\cal E}_{\rm {f}} \cos\Big(\omega(t - y/c)-\alpha\Big)
\end{displaymath}

For convenience the $y$-​axis was taken in the direction of propagation of the wave. Also the $z$-​axis was taken in the direction of the electric field. Since there is just a single frequency $\omega$, the wave is monochromatic; it is a single color. And because of the direction of the electric field, the wave is said to be polarized in the $z$-​direction. Note that the electric and magnetic fields for plane waves are normal to the direction of propagation and to each other. The constant $c$ is the speed of light, ${\cal E}_{\rm {f}}$ the amplitude of the electric field, and $\alpha$ is some unimportant phase angle.

Fortunately, the expression for the wave can be greatly simplified. The electron reacts primarily to the electric field, provided that its kinetic energy is small compared to its rest mass energy. That is certainly true for the electron in a hydrogen atom and for the outer electrons of atoms in general. Therefore the magnetic field can be ignored. (The error made in doing so is described more precisely in {D.39}.) Also, the wave length of the electromagnetic wave is usually much larger than the size of the atom. For example, the Lyman-transition wave lengths are of the order of a thousand Å, while the atom is about one Å. So, as far as the light wave is concerned, the atom is just a tiny speck at the origin. That means that $y$ can be put to zero in the expression for the plane wave. Then the wave simplifies to just:

\begin{displaymath}
\skew3\vec{\cal E}= {\hat k}{\cal E}_{\rm {f}} \cos(\omega t-\alpha)
\end{displaymath} (7.40)

This may not be applicable to highly energetic radiation like X-rays.

Now the question is how this field changes the Hamiltonian of the electron. Ignoring the time dependence of the electric field, that is easy. The Hamiltonian is

\begin{displaymath}
H = H_{\rm {atom}} + e {\cal E}_{\rm {f}} \cos(\omega t-\alpha) z %
\end{displaymath} (7.41)

where $H_{\rm {atom}}$ is the Hamiltonian of the hydrogen atom without the external electromagnetic field. The expression for $H_{\rm {atom}}$ was given in chapter 4.3, but it is not of any interest here.

The interesting term is the second one, the perturbation caused by the electromagnetic field. In this term $z$ is the $z$-​position of the electron. It is just like the $mgh$ potential energy of gravity, with the charge $e$ playing the part of the mass $m$, the electric field strength ${\cal E}_{\rm {f}}\cos({\omega}t-\alpha)$ that of the gravity strength $g$, and $z$ that of the height $h$.

To be sure, the electric field is time dependent. The above perturbation potential really assumes that “the electron moves so fast that the field seems steady to it.” Indeed, if an electron speed is ballparked from its kinetic energy, the electron does seem to travel through the atom relatively fast compared to the frequency of the electric field. Of course, it is much better to write the correct unsteady Hamiltonian and then show it works out pretty much the same as the quasi-steady one above. That is done in {D.39}.


Key Points
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An approximate Hamiltonian was written down for the interaction of an atom with an electromagnetic wave.

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By approximation the atom sees a uniform, quasi-steady electric field.


7.7.2 The two-state model

The big question is how the electromagnetic field affects transitions between a typical atomic state $\psi_{\rm {L}}$ of lower energy and one of higher energy $\psi_{\rm {H}}$.

The answer depends critically on various Hamiltonian coefficients. In particular, the expectation values of the energies of the two states are needed. They are

\begin{displaymath}
E_{\rm {L}} = \langle\psi_{\rm {L}}\vert H\vert\psi_{\rm {...
...{H}} = \langle\psi_{\rm {H}}\vert H\vert\psi_{\rm {H}}\rangle
\end{displaymath}

Here the Hamiltonian to use is (7.41) of the previous subsection; it includes the electric field. But it can be seen that the energies are unaffected by the electric field. They are the unperturbed atomic energies of the states. That follows from symmetry; if you write out the inner products above using (7.41), the square wave function is the same at any two positions ${\skew0\vec r}$ and $\vphantom0\raisebox{1.5pt}{$-$}$${\skew0\vec r}$, but $z$ in the electric field term changes sign. So integration values pairwise cancel each other.

Note however that the two energies are now expectation values of energy; due to the electric field the atomic states develop uncertainty in energy. That is why they are no longer stationary states.

The other key Hamiltonian coefficient is

\begin{displaymath}
H_{\rm {HL}} = \langle\psi_{\rm {H}}\vert H\vert\psi_{\rm {L}}\rangle
\end{displaymath}

Plugging in the Hamiltonian (7.41), it is seen that the atomic part $H_{\rm {atom}}$ does not contribute. The states $\psi_{\rm {H}}$ and $\psi_{\rm {L}}$ are orthogonal, and the atomic Hamiltonian just multiplies $\psi_{\rm {L}}$ by $E_{\rm {L}}$. But the electric field gives

\begin{displaymath}
H_{\rm {HL}}
= {\cal E}_{\rm {f}} \langle\psi_{\rm {H}}\...
...^{{\rm i}(\omega t-\alpha)}+e^{-{\rm i}(\omega t-\alpha)}}{2}
\end{displaymath}

Here the cosine in (7.41) was taken apart into two exponentials using the Euler formula (2.5).

The next question is what these coeffients mean for the transitions between two atomic states $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$. First, since the atomic states are complete, the wave function can always be written as

\begin{displaymath}
\Psi = c_{\rm {L}} \psi_{\rm {L}} + c_{\rm {H}} \psi_{\rm {H}} + \ldots
\end{displaymath}

where the dots stand for other atomic states. The coefficients $c_{\rm {L}}$ and $c_{\rm {H}}$ are the key, because their square magnitudes give the probabilities of the states $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$. So they determine whether transitions occur between them.

Evolution equations for these coefficients follow from the Schrö­din­ger equation. The way to find them was described in section 7.6, with additional manipulations in derivation {D.38}. The resulting evolution equations are:

\begin{displaymath}
\fbox{$\displaystyle
{\rm i}\hbar \dot {\bar c}_{\rm{L}}...
...H}}
= \overline{H}_{\rm{HL}}\bar c_{\rm{L}} + \ldots
$} %
\end{displaymath} (7.42)

where the dots represent terms involving states other than $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$. These equations use the modified coefficients
\begin{displaymath}
\bar c_{\rm {L}} = c_{\rm {L}} e^{{\rm i}E_{\rm {L}} t/\hb...
...\bar c_{\rm {H}} = c_{\rm {H}} e^{{\rm i}E_{\rm {H}} t/\hbar}
\end{displaymath} (7.43)

The modified coefficients have the same square magnitudes as the original ones and the same values at time zero. That makes them fully equivalent to the original ones. The modified Hamiltonian coefficient in the evolution equations is
\begin{displaymath}
\overline{H}_{\rm {HL}} = \overline{H}_{\rm {LH}}^* =
{\...
...psi_{\rm {L}}\rangle
e^{{\rm i}(\omega_0+\omega)t-\alpha} %
\end{displaymath} (7.44)

where $\omega_0$ is the frequency of a photon that has the exact energy $E_{\rm {H}}-E_{\rm {L}}$.

Note that this modified Hamiltonian coefficient is responsible for the interaction between the states $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$. If this Hamiltonian coefficient is zero, the electromagnetic wave cannot cause transitions between the two states. At least not within the approximations made.

Whether this happens depends on whether the inner product $\langle\psi_{\rm {H}}\vert ez\vert\psi_{\rm {L}}\rangle$ is zero. This inner product is called the “atomic matrix element” because it depends only on the atomic states, not on the strength and frequency of the electric wave.

However, it does depend on the direction of the electric field. The assumed plane wave had its electric field in the $z$-​direction. Different waves can have their electric fields in other directions. Therefore, waves can cause transitions as long as there is at least one nonzero atomic matrix element of the form $\langle\psi_{\rm {L}}\vert er_i\vert\psi_{\rm {H}}\rangle$, with $r_i$ equal to $x$, $y$, or $z$. If there is such a nonzero matrix element, the transition is called allowed. Conversely, if all three matrix elements are zero, then transitions between the states $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$ are called forbidden.

Note however that so-called forbidden transitions often occur just fine. The derivation in the previous subsection made several approximations, including that the magnetic field can be ignored and that the electric field is independent of position. If these ignored effects are corrected for, many forbidden transitions turn out to be possible after all; they are just much slower.

The approximations made to arrive at the atomic matrix element $\langle\psi_{\rm {H}}\vert ez\vert\psi_{\rm {L}}\rangle$ are known as the electric dipole approximation. The corresponding transitions are called electric dipole transitions. If you want to know where the term comes from, why? Anyway, in that case note first that if the electron charge distribution is symmetric around the proton, the expectation value of $ez$ will be zero by symmetry. Negative $z$ values will cancel positive ones. But the electron charge distribution might get somewhat shifted to the positive $z$ side, say. The total atom is then still electrically neutral, but it behaves a bit like a combination of a negative charge at a positive value of $z$ and an equal and opposite positive charge at a negative value of $z$. Such a combination of two opposite charges is called a dipole in classical electromagnetics, chapter 13.3. So in quantum mechanics the operator $ez$ gives the dipole strength in the $z$-​direction. And if the above atomic matrix element is nonzero, it can be seen that nontrivial combinations of $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$ have a nonzero expectation dipole strength. So the name electric dipole transitions is justified, especially since basic electric transitions would be understandable by far too many nonexperts.

Allowed and forbidden transitions were discussed earlier in section 7.4. However, that was based on assumed properties of the emitted photon. The allowed atomic matrix elements above, and similar forbidden ones, make it possible to check the various most important results directly from the governing equations. That is done in derivation {D.39}.

There is another requirement to get a decent transition probability. The exponentials in the modified Hamiltonian coefficient (7.44) must not oscillate too rapidly in time. Otherwise opposite values of the exponentials will average away against each other. So no significant transition probability can build up. (This is similar to the cancelation that gives rise to the adiabatic theorem, {D.34}.) Now under real-life conditions, the second exponential in (7.44) will always oscillate rapidly. Normal electromagnetic frequencies are very high. Therefore the second term in (7.44) can normally be ignored.

And in order for the first exponential not too oscillate too rapidly requires a pretty good match between the frequencies $\omega$ and $\omega_0$. Recall that $\omega$ is the frequency of the electromagnetic wave, while $\omega_0$ is the frequency of a photon whose energy is the difference between the atomic energies $E_{\rm {H}}$ and $E_{\rm {L}}$. If the electric field does not match the frequency of that photon, it will not do much. Using the Planck-Einstein relation, that means that

\begin{displaymath}
\omega \approx \omega_0 \equiv (E_{\rm {H}}-E_{\rm {L}})/\hbar
\end{displaymath}

One consequence is that in transitions between two atomic states $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$, other states usually do not need to be considered. Unless an other state matches either the energy $E_{\rm {H}}$ or $E_{\rm {L}}$, it will give rise to rapidly oscillating exponentials that can be ignored.

In addition, the interest is often in the so-called collision-dominated regime in which the atom evolves for only a short time before being disturbed by collisions with its surroundings. In that case, the short evolution time prevents nontrivial interactions between different transition processes to build up. Transition rates for the individual transition processes can be found separately and simply added together.

The obtained evolution equations (7.42) can explain why absorption and stimulated emission compete on an equal footing in the operation of lasers. The reason is that the equations have a remarkable symmetry: for every solution $\bar{c}_{\rm {L}}$, $\bar{c}_{\rm {H}}$ there is a second solution $\bar{c}_{\rm {L,2}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\bar{c}_{\rm {H}}^{\,*}$, $\bar{c}_{\rm {H,2}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $-\bar{c}_{\rm {L}}^{\,*}$ that has the probabilities of the low and high energy states exactly reversed. It means that

An electromagnetic field that takes an atom out of the low energy state $\psi_{\rm {L}}$ towards the high energy state $\psi_{\rm {H}}$ will equally take that atom out of the high energy state $\psi_{\rm {H}}$ towards the low energy state $\psi_{\rm {L}}$.
It is a consequence of the Hermitian nature of the Hamiltonian; it would not apply if ${\overline{H}}_{\rm {LH}}$ was not equal to ${\overline{H}}_{\rm {HL}}^*$.


Key Points
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The governing evolution equations for the probabilities of two atomic states $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$ in an electromagnetic wave have been found.

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The equations have a symmetry property that makes electromagnetic waves equally effective for absorption and stimulated emission.

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Normally the electromagnetic field has no significant effect on transitions between the states unless its frequency $\omega$ closely matches the frequency $\omega_0$ of a photon with energy $E_{\rm {H}}-E_{\rm {L}}$.

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The governing equations can explain why some transitions are allowed and others are forbidden. The key are so-called atomic matrix elements.