- 7.4.1 Conservation of energy
- 7.4.2 Combining angular momenta and parities
- 7.4.3 Transition types and their photons
- 7.4.4 Selection rules

7.4 Conservation Laws in Emission

Conservation laws are very useful for understanding emission or absorption of radiation of various kinds, as well as nuclear reactions, collision processes, etcetera. As an example, this section will examine what conservation laws say about the spontaneous emission of a photon of light by an excited atom. While this example is relatively simple, the concepts discussed here apply in essentially the same way to more complex systems.

Figure 7.3 gives a sketch of the emission process. The
atom is initially in an high energy, or excited, state that will be
called 2

1s

ground state

The emitted photon has an energy given by the Planck-Einstein relation

where

Key Points

- Atoms can transition to a lower electronic energy level while emitting a photon of electromagnetic radiation.

- The Planck-Einstein relation gives the energy of a photon in terms of its frequency.

7.4.1 Conservation of energy

The first conservation law that is very useful for understanding the
emission process is conservation of energy. The final atom and photon
should have the exact same energy as the initial excited atom. So the
difference between the atomic energies

It should be pointed out that the frequency of the emitted photon does have a very slight variation. The reason can be understood from the fact that the excited state decays at all. Energy eigenstates should be stationary, section 7.1.4.

The very fact that a state decays shows that it is not truly an energy eigenstate.

The big problem with the analysis of the hydrogen atom in chapter 4.3 was that it ignored any ambient radiation that the electron might be exposed to. It turns out that there is always some perturbing ambient radiation, even if the atom is inside a black box at absolute zero temperature. This is related to the fact that the electromagnetic field has quantum uncertainty. Advanced quantum analysis is needed to take that into account, {A.23}. Fortunately, the uncertainty in energy is extremely small for the typical applications considered here.

As a measure of the uncertainty in energy of a state, physicists often
use the so-called “natural width”

The claim that this width gives the uncertainty in energy of the state
is usually justified using the all-powerful energy-time uncertainty
equality (7.9). A different argument will be given at the
end of section 7.6.1. In any case, the bottom line is
that

As an example, the hydrogen atom 2

Still, since a small range of frequencies can be emitted, the observed line in the emission spectrum is not going to be a mathematically exact line, but will have a small width. Such an effect is known as “spectral line broadening.”

The natural width of a state is usually only a small part of the actual line broadening. If the atom is exposed to an incoherent ambient electromagnetic field, it will increase the uncertainty in energy. (The evolution of atoms in an incoherent electromagnetic field will be analyzed in {D.41}.) Frequent interactions with surrounding atoms or other perturbations will also increase the uncertainty in energy, in part for reasons discussed at the end of section 7.6.1. And anything else that changes the atomic energy levels will of course also change the emitted frequencies.

An important further effect that causes spectral line deviations is atom motion, either thermal motion or global gas motion. It produces a Doppler shift in the radiation. This is not necessarily bad news in astronomy; line broadening can provide an hint about the temperature of the gas you are looking at, while line displacement can provide a hint of its overall motion away from you.

It may also be mentioned that the natural width is not always small.
If you start looking at excited nuclear particles, the uncertainty in
energy can be enormous. Such particles may have an uncertainty in
energy that is of the order of 10% of their relativistic rest mass
energy. And as you might therefore guess, they are hardly stationary
states. Typically, they survive for only about 1

(To be fair, physicists do not actually manage to see these particles during their infinitesimal lifetime. Instead they infer the lifetime from the variation in energy of the resulting state.)Generally speaking, the shorter the lifetime of a state, the larger its uncertainty in energy, and vice-versa.

Key Points

- In a transition, the difference in atomic energy levels gives the energy, and so the frequency, of the emitted photon.

- Unstable states have some uncertainty in energy, but it is usually very small. For extremely unstable particles, the uncertainty can be a lot.

- The width of a state is
with the mean lifetime. It is a measure for the minimum observed variation in energy of the final state.

7.4.2 Combining angular momenta and parities

Conservation of angular momentum and parity is easily stated:

The question is however, how do you combine angular momenta and parity values? Even combining angular momenta is not trivial, because angular momenta are quantized.The angular momentum and parity of the initial atomic state must be the same as the combined angular momentum and parity of the final atomic state and photon.

To get an idea of how angular momenta combine, first consider what
would happen in classical physics. Conservation of angular momentum
would say that

Here

Now consider what possible lengths the vector

All together:

Note that the omission of a vector symbol indicates that the length of the vector is meant, rather than the vector itself. The second inequality is the famous “triangle inequality.” (The first inequality is a rewritten triangle inequality for the longer of the two vectors in the absolute value.) The bottom line is that according to classical physics, the length of the final atomic angular momentum can take any value in the range given above.

However, in quantum mechanics angular momentum is quantized. The
length of an angular momentum vector

Classical physics also says that components of vectors can be added
and subtracted as ordinary numbers. Quantum physics agrees, but adds
that for nonzero angular momentum only one component can be certain at
a time. That is usually taken to be the

If you can add and subtract components of angular momentum, then you
can also add and subtract magnetic quantum numbers. After all, they
are only different from components by a factor

Putting in the possible values of the magnetic quantum number of the photon gives for the final atomic magnetic quantum number:

To be sure,

Next consider conservation of parity. Recall from section
7.3 that parity is the factor by which the wave function
changes when the positive direction of all three coordinate axes is
inverted. That replaces every position vector p.

Parity starts with a p and may well be Greek.
Also, the symbol avoids confusion, assuming that

Conservation of parity means that the initial and final parities must
be equal. The parity of the initial high energy atom must be the same
as the combined parity of the final low energy atom and photon:

Note that parity is a multiplicative quantity. You get the combined parity of the final state by multiplying the parities of atom and photon; you do not add them.

(Just think of the simplest possible wave function of two particles,
natural

conserved quantity. For details, see
addendum {A.19}.)

The parity of the atom is related to the orbital angular momentum of
the electron, and in particular to its azimuthal quantum number

The parity can therefore be written for any value of

It follows that parity conservation in the emission process
can be written as

To apply the obtained conservation laws, the next step must be to figure out the angular momentum and parity of the photon.

Key Points

- The rules for combining angular momenta and parities were discussed.

- Angular momentum and parity conservation lead to constraints on the atomic emission process given by (7.11), (7.12), and (7.14).

7.4.3 Transition types and their photons

The conservation laws of angular momentum and parity restrict the emission of a photon by an excited atom. But for these laws to be useful, there must be information about the spin and parity of the photon.

This section will just state various needed photon properties. Derivations are given in {A.21.7} for the brave. In any case, the main conclusions reached about the photons associated with atomic transitions will be verified by more detailed analysis of transitions in later sections.

There are two types of transitions, electric ones and magnetic ones. In electric transitions, the electromagnetic field produced by the photon at the atom is primarily electric, {A.21.7}. In magnetic transitions, it is primarily magnetic. Electric transitions are easiest to understand physically, and will be discussed first.

A photon is a particle with spin

The normal, efficient kind of atomic transition does in fact produce a
photon like that. Since the term normal

is too
normal, such a transition is called “allowed.” For reasons that will eventually be excused for in
section 7.7.2, allowed transitions are also more
technically called “electric dipole” transitions. According to the above, then,
the photon net angular momentum and parity are:

Transitions that cannot happen according to the electric dipole mechanism are called “forbidden.” That does not mean that these transitions cannot occur at all; just forbid your kids something. But they are much more awkward, and therefore normally very much slower, than allowed transitions.

One important case of a forbidden transition is one in which the
atomic angular momentum changes by 2 or more units. Since the photon
has only 1 unit of spin, in such a transition the photon must have
nonzero orbital angular momentum. Transitions in which the photon has
more than 1 unit of net angular momentum are called “multipole transitions.” For example, in a “quadrupole” transition, the net angular momentum of the photon
octupole

transition,

To roughly understand how orbital angular momentum arises, reconsider
the sketch of the emission process in figure 7.3. As
shown, the photon has no orbital angular momentum around the center of
the atom, classically speaking. But the photon does not have to come
from exactly the center of the atom. If the atom has a typical radius

The fraction is typically small. For example, the wave length

But according to quantum mechanics, the orbital angular momentum
cannot be a small fraction of

(If the above random mixture of unjustified classical and quantum arguments is too unconvincing, there is a quantum argument in {N.10} that may be more believable. If you are brave, see {A.21.7} for a precise analysis of the relevant photon momenta and their probabilities in an interaction with an atom or nucleus. But the bottom line is that the above ideas do describe what happens in transition processes. That follows from a complete analysis of the transition process, as discussed in later sections and notes like {A.25} and {D.39}.)

So far, only electric multipole transitions have been discussed, in
which the electromagnetic field at the atom is primarily electric. In
magnetic multipole transitions however, it is primarily magnetic. In
a “magnetic dipole” transition, the photon comes out with one unit
of net angular momentum just like in an electric dipole one. However,
the parity of the photon is now even:

You might wonder how the positive parity is possible if the photon has negative intrinsic parity and no orbital angular momentum. The reason is that in a magnetic dipole transition, the photon does have a unit of orbital angular momentum. Recall from the previous subsection that it is quite possible for one unit of spin and one unit of orbital angular momentum to combine into still only one unit of net angular momentum.

In view of the crude discussion of orbital angular momentum given
above, this may still seem weird. How come that an atom of vanishing
size does suddenly manage to readily produce a unit of orbital angular
momentum in a magnetic dipole transition? The basic reason is that
the magnetic field acts in some way as if it has one unit of orbital
angular momentum less than the photon, {A.21.7}. It is
unpexpectedly strong at the atom. This allows a magnetic atom state
to get a solid grip

on a photon state of unit orbital
angular momentum. It is somewhat like hitting a rapidly spinning ball
with a bat in baseball; the resulting motion of the ball can be weird.
And in a sense the orbital angular momentum comes at the expense of
the spin; the net angular momentum

Certainly this sort of complications would not arise if the photon had
no spin. Without discussion, the photon is one of the most basic
particles in physics. But it is surprisingly complex for such an
elementary particle. This also seems the right place to confess to
the fact that electric multipole photons have uncertainty in orbital
angular momentum. For example, an electric dipole photon has a
probability for

All else being the same, the probability of a magnetic dipole
transition is normally much smaller than an electric dipole one. The
principal reason is that the magnetic field is really a relativistic
effect. That can be understood, for example, from how the magnetic
field popped up in the description of the relativistic motion of
charged particles, chapter 1.3.2. So you would expect the
effect of the magnetic field to be minor unless the atomic electron or
nucleon involved in the transition has a kinetic energy comparable to
its rest mass energy. Indeed, it turns out that the probability of a
magnetic transition is smaller than an electric one by a factor of
order

In magnetic multipole transitions, the photon receives additional angular momentum. Like for electric multipole transitions, there is one additional unit of angular momentum for each additional multipole order. And there is a corresponding slow down of the transitions.

Table 7.1 gives a summary of the photon properties in
multipole transitions. It is conventional to write electric multipole
transitions as

The column slow down

gives an order of magnitude
estimate by what factor a transition is slower than an electric dipole
one, all else being equal. Note however that all else is definitely
not equal, so these factors should not be used even for ballparks.

There are some official ballparks for atomic nuclei based on a more detailed analysis. These are called the Weisskopf and Moszkowski estimates, chapter 14.20.4 and in particular addendum {A.25.8}. But even there you should not be surprised if the ballpark is off by orders of magnitude. These estimates do happen to work fine for the nonrelativistic hydrogen atom, with appropriate adjustments, {A.25.8}.

The slow down factors

where the

Note from the table that electric quadrupole and magnetic dipole transitions have the same parity. That means that they may compete directly with each other on the same transition, provided that the atomic angular momentum does not change more than one unit in that transition.

For nuclei, the photon energy tends to be significantly less than the nucleon kinetic energy. That is one reason that the Weisskopf estimates have the electric quadrupole transitions a lot slower than magnetic dipole ones for typical transitions. Also note that the kinetic energy estimate above does not include the effect of the exclusion principle. Exclusion raises the true kinetic energy if there are multiple identical particles in a given volume.

There is another issue that should be mentioned here. Magnetic transitions have a tendency to underperform for simple systems like the hydrogen atom. For these systems, the magnetic field has difficulty making effective use of spin in changing the atomic or nuclear structure. That is discussed in more detail in the next subsection.

One very important additional property must still be mentioned. The
photon cannot have zero net angular momentum. Normally it is
certainly possible for a particle with spin

There are some effects in classical physics that are related to this
limitation. First of all, consider a photon with definite linear
momentum. That corresponds to a light wave propagating in a
particular direction. Now linear and angular momentum do not commute,
so such a photon will not have definite angular momentum. However,
the angular momentum component in the direction of motion is still
well defined. The limitation on photons is in this case that the
photon must either have angular momentum

Second, for the same type of photon, there are two equivalent states that have definite directions of the electric and magnetic fields. These states have uncertainty in angular momentum in the direction of motion. They are called “linearly polarized” light. These states illustrate that there cannot be an electric or magnetic field component in the direction of motion. The electric and magnetic fields are normal to the direction of motion, and to each other.

More general photons of definite linear momentum may have uncertainty in both of the mentioned properties. But still there is zero probability for zero angular momentum in the direction of motion, and zero probability for a field in the direction of motion.

Third, directly related to the previous case. Suppose you have a
charge distribution that is spherically symmetric, but pulsating in
the radial direction. You would expect that you would get a
fluctuating radial electrical field outside this pulsating charge.
But you do not, it does not radiate energy. Such radiation would have
the electric field in the direction of motion, and that does not
happen. Now consider the transition from the spherically symmetric
2s

state of a hydrogen atom to the spherical
symmetric 1s

state. Because of the lack of
spherically symmetric radiation, you might guess that this transition
is in trouble. And it is; that is discussed in the next subsection.

In fact, the last example is directly related to the missing state of zero angular momentum of the photon. Recall from section 7.3 that angular momentum is related to angular symmetry. In particular, a state of zero angular momentum (if exact to quantum accuracy) looks the same when seen from all directions. The fact that there is no spherically symmetric radiation is then just another way of saying that the photon cannot have zero angular momentum.

Key Points

- Normal atomic transitions are called allowed or electric dipole ones. All others are called forbidden but can occur just fine.

- In electric dipole transitions the emitted photon has angular momentum quantum number
1 and negative parity 1 .

- In the slower magnetic dipole transitions the photon parity is positive,
1.

- Each higher multipole order adds a unit to the photon angular momentum quantum number
and flips over the parity .

- The higher the multipole order, the slower the transition will be.

7.4.4 Selection rules

As discussed, a given excited atomic state may be able to transition
to a lower energy state by emitting a photon. But many transitions
from a higher energy state to a lower one simply do not happen. There
are so-called selection rules

that predict whether or
not a given transition process is possible. This subsection gives a
brief introduction to these rules.

The primary considered system will be the hydrogen atom. However, some generally valid rules are given at the end. It will usually be assumed that the effect of the spin of the electron on its motion can be ignored. That is the same approximation as used in chapter 4.3, and it is quite accurate. Basically, the model system studied is a spinless charged electron going around a stationary proton. Spin will be tacked on after the fact.

The selection rules result from the conservation laws and photon
properties as discussed in the previous two subsections. Since the
conservation laws are applied to a spinless electron, the angular
momentum of the electron is simply its orbital angular momentum. That
means that for the atomic states, the angular momentum quantum number

Now suppose that the initial high-energy atomic state has an orbital
angular momentum quantum number

That leads immediately to a stunning conclusion for the decay of the
hydrogen 2s

state. This state has
angular momentum 1s

ground state. It
has

Never say never, of course. It turns out that if left alone, the 2s state will eventually decay through the emission of two photons, rather than a single one. This takes forever on quantum scales; the 2s state survives for about a tenth of a second rather than maybe a nanosecond for a normal transition. Also, to actually observe the two-photon emission process, the atom must be in high vacuum. Otherwise the 2s state would be messed up by collisions with other particles long before it could decay. Now you see why the introduction to this section gave a 2p state, and not the seemingly more simple 2s one, as a simple example of an atomic state that decays by emitting a photon.

Based on the previous subsection, you might wonder why a second photon
can succeed where a unit of photon orbital angular momentum cannot.
After all, photons have only two independent spin states, while a unit
of orbital angular momentum has the full set of three. The
explanation is that in reality you cannot add a suitable unit
of orbital angular momentum to a photon; the orbital and spin angular
momentum of a photon are intrinsically linked. But photons do have
complete sets of states with angular momentum

It is customary to explain

photons in terms of states
of definite linear momentum. That is in fact what was done in the
final paragraphs of the previous subsection. But it is simplistic.
It is definitely impossible to understand how two photons, each
missing the state of zero angular momentum along their direction of
motion, could combine into a state of zero net angular momentum. In
fact, they simply cannot. Linear and orbital angular momentum do not
commute. But photons do not have to be in quantum states of definite
linear momentum. They can be, and often are, in quantum
superpositions of such states. The states of definite angular
momentum are quantum superpositions of infinitely many states of
linear momentum in all directions. To make sense out of that, you
need to switch to a description in terms of photon states of definite
angular, rather than linear, momentum. Those states are listed in
{A.21.7}. Unfortunately, they are much more difficult to
describe physically than states of definite linear momentum.

It should also be noted that if you include relativistic effects, the
2s state can actually decay to the 2p state that has net angular
momentum (spin plus orbital) Lamb shift,

{A.39.4}. But
because of the negligible difference in energy, such a transition is
even slower than two-photon emission. It takes over 100 years to have
a 50/50 probability for the transition.

Also, including relativistic efects, a magnetic dipole transition is
possible. An atomic state with net angular momentum

Relativistic effects remove these obstacles. But since these effects are very small, the one-photon transition does take several days, so it is again much slower than two-photon emission. In this case, it may be useful to think in terms of the complete atom, including the proton spin. The electron and proton can combine their spins into a singlet state with zero net angular momentum or a triplet state with one unit of net momentum. The photon takes one unit of angular momentum away, turning a triplet state into a singlet state or vice-versa. If the atom ends up in a 1s triplet state, it will take another 10 million year or so to decay to the singlet state, the true ground state.

For excited atomic states in general, different types of transitions
may be possible. As discussed in the previous subsection, the normal
type is called an allowed,

“electric
dipole,” or

Yes, every one of these three names is confusing. Nonallowed
transitions, called forbidden

transitions, are
perfectly allowed and they do occur. They are typically just a lot
slower. The atomic states between which the transitions occur do not
have electric dipole moments. And how many people really know what an
electric dipole is? And

The one good thing that can be said is that in the electric dipole
approximation, the atom does indeed respond to the electric part of
the electromagnetic field. In such transitions the photon comes out
with one unit of angular momentum, i.e.

(7.18) |

The second rule gives the possible magnetic quantum numbers. Recall
that these are a direct measure for the angular momentum in the chosen

The final selection rule says that the electron spin in the

Note that ignoring relativistic effects in transitions is a tricky business. Even a small effect, given enough time to build up, might produce a transition where one was not possible before. In a more sophisticated analysis of the hydrogen atom, addendum {A.39}, there is a slight interaction between the orbital angular momentum of the electron and its spin. That is known as spin-orbit interaction. Note that the s states have no orbital angular momentum for the spin to interact with.

As a result of spin-orbit interaction the correct energy
eigenfunctions, except the s states, develop uncertainty in the values
of both

(7.19) |

If the selection rules are not satisfied, the transition is called
forbidden. However, the transition may still occur through a
different mechanism. One possibility is a slower magnetic dipole
transition, in which the electron interacts with the magnetic part of
the electromagnetic field. That interaction occurs because an
electron has spin and orbital angular momentum. A charged particle
with angular momentum behaves like a little electromagnet and wants to
align itself with an ambient magnetic field, chapter
13.4. The selection rules in this case are

(7.20) |

It must be pointed out that an

Relativistic effects can change this. In particular, in the presence
of spin-orbit coupling, the selection rules become

(7.21) |

In higher-order multipole transitions the photon comes out with
angular momentum

(7.22) |

Magnetic transitions at higher multipole orders have similar problems
as the magnetic dipole one. In particular, consider the orbital
angular momentum selection rule (7.17) above. The
lowest possible multipole order in the nonrelativistic case is

Because of parity, that is always an electric multipole transition. (This excludes the case that the orbital angular momenta are equal, in which case the lowest transition is the already discussed trivial magnetic dipole one.)

The bottom line is that magnetic transitions simply cannot compete. Of course, conservation of net angular momentum might forbid the electric transition to a given final state. But in that case there will be an equivalent state that differs only in spin to which the electric transition can proceed just fine.

However, for a multi-electron atom or nucleus in an independent-particle model, that equivalent state might already be occupied by another particle. Or there may be enough spin-orbit interaction to raise the energy of the equivalent state to a level that transition to it becomes impossible. In that case, the lowest possible transition will be a magnetic one.

Consider now more general systems than hydrogen atoms. General
selection rules for electric

These rules rely only on the spin and parity of the emitted photon. So they are quite generally valid for one-photon emission.

If a normal electric dipole transition is possible for an atomic or nuclear state, it will most likely decay that way before any other type of transition can occur. But if an electric dipole transition is forbidden, other types of transitions may appear in significant amounts. If both electric quadrupole and magnetic dipole transitions are possible, they may be competitive. And electric quadrupole transitions can produce two units of change in the atomic angular momentum, rather than just one like the magnetic dipole ones.

Given the initial state, often the question is not what final states
are possible, but what transition types are possible given the final
state. In that case, the general selection rules can be written as

Since transition rates decrease rapidly with increasing multipole
order

Key Points

- Normal atomic transitions are called electric dipole ones, or allowed ones, or
ones. Unfortunately.

- The quantum numbers of the initial and final atomic states in transitions must satisfy certain selection rules in order for transitions of a given type to be possible.

- If a transition does not satisfy the rules of electric dipole transitions, it will have to proceed by a slower mechanism. That could be a magnetic dipole transition or an electric or magnetic multipole transition.

- A state of zero angular momentum cannot decay to another state of zero angular momentum through any of these mechanisms. For such transitions, two-photon emission is an option.