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.

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

$$\hbar\omega = E_{\rm {H}} - E_{\rm {L}}$$

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):

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

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:

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

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

$$E_{\rm {L}} = \langle\psi_{\rm {L}}\vert H\vert\psi_{\rm {... ...{H}} = \langle\psi_{\rm {H}}\vert H\vert\psi_{\rm {H}}\rangle$$

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

$$H_{\rm {HL}} = \langle\psi_{\rm {H}}\vert H\vert\psi_{\rm {L}}\rangle$$

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

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

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

$$\Psi = c_{\rm {L}} \psi_{\rm {L}} + c_{\rm {H}} \psi_{\rm {H}} + \ldots$$

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:

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

where the dots represent terms involving states other than $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$. These equations use the modified coefficients

$$\bar c_{\rm {L}} = c_{\rm {L}} e^{{\rm i}E_{\rm {L}} t/\hb... ...\bar c_{\rm {L}} = c_{\rm {L}} e^{{\rm i}E_{\rm {H}} t/\hbar}$$ (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

$$\overline{H}_{\rm {HL}} = \overline{H}_{\rm {LH}}^* = {\... ...si_{\rm {L}}\rangle e^{{\rm i}(\omega_0+\omega)t-\alpha} %$$ (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

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

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.”