A.24 Quantum spontaneous emission

Chapter 7.8 explained the general interaction between atoms and electromagnetic fields. However, spontaneous emission of radiation was found using a dirty trick due to Einstein. He peeked at the solution for blackbody radiation. This addendum will give a proper quantum description. Warning: while this addendum tries to be reasonably self-contained, to really appreciate the details you may have to read some other addenda too.

The problem with the descriptions of emission and absorption of radiation in chapter 7.7 and 7.8 is that they assume that the electromagnetic field is given. The electromagnetic field is not given; it changes by one photon. That is rather important for spontaneous emission, where it changes from no photons to one photon. To account for that correctly requires that the electromagnetic field is properly quantized. That is done in this note.

To keep it simple, it will be assumed that the atom is a hydrogen one.
Then there is just one electron to worry about. (The general analysis
can be found in {A.25}). The hydrogen atom is
initially in some high energy state

Recall that the photon energy is given in terms of its frequency

Only a single photon energy state needs to be considered at a time.
At the end of the story, the results can be summed over all possible
photon states. To allow for stimulated emission, it will be assumed
that initially there may already be

Here the so-called Fock space ket

In the final state the atom has decayed to a lower energy state

The key to the emission process is now the set of Hamiltonian
coefficients, chapter 7.6,

Here

To identify the Hamiltonian coefficients, first the Hamiltonian must
be identified. Recall that the Hamiltonian is the operator of the
total energy of the system. It will take the form

The first term in the right hand side is the inherent energy of the hydrogen atom. This Hamiltonian was written down way back in chapter 4.3. However, its precise form is of no interest here. The second term in the right hand side is the energy in the electromagnetic field. Electromagnetic fields too have inherent energy, about

Unlike the first term in the Hamiltonian, the other two are inherently
relativistic: the number of photons is hardly a conserved quantity.
Photons are readily created or absorbed by a charged particle, like
the electron here. And it turns out that Hamiltonians for photons are
intrinsically linked to operators that annihilate and create photons.
Mathematically, at least. These operators are defined by the
relations

The Hamiltonian that describes the inherent energy in the electromagnetic
field turns out to be, {A.23},

As a sanity check, this Hamiltonian can be applied on a state of

The factor in front of the final ket is the energy eigenvalue. So the energy in the field increases by one unit

Finally the energy of the interaction between the electron and
electromagnetic field is needed. This third part of the total
Hamiltonian is the messiest. To keep it as simple as possible, it
will assumed that the transition is of the normal electric dipole
type. In such transitions the electron interacts only with the
electric part of the electromagnetic field. In addition, just like in
the analysis of chapter 7.7.1 using a classical
electromagnetic field, it will be assumed that the electric field is
in the

Now recall that in quantum mechanics, observable properties of
particles are the eigenvalues of Hermitian operators, chapter
3.3. For example, the observable values of linear momentum
of an electron in the

Similarly, the electric field

In the analysis using a classical electromagnetic field, the energy of
interaction between the electron and the electromagnetic field was
taken to be approximately

The operator

The electric field operator

Here

The rules to get the operator of the observable electric field were
discussed in addendum {A.23}. First the unobservable
electric field above is multiplied by the annihilation operator, then
the Hermitian conjugate of that product is added, and the sum is
divided by

(Note that for the usual Schrödinger approach followed here, time dependence is described by the wave function. Most sources switch here to a Heisenberg approach where the time-dependence is pushed into the operators. There is however no particular need to do so.)

In the electric dipole approximation, it is assumed that the atom is
so small compared to the wave length of the photon that

The combined Hamiltonian is then

with the first two terms as described earlier.

Next the Hamiltonian matrix coefficients are needed. The first one is

Now the atomic part of the Hamiltonian produces a mere factor

All together then

The same way

Finally the matrix element:

In this case the atomic part of the Hamiltonian produces zero. The reason is that this Hamiltonian produces a simple scalar factor

The reason is that the creation operator

atomic matrix element,as it only depends on what the atomic states are.

The task laid out in chapter 7.6.1 has been accomplished: the relativistic matrix element has been found. A final expression for the spontaneous emission rate can now be determined.

Before doing so, however, it is good to first compare the obtained
result with that of chapter 7.7.1. That section used a
classical given electromagnetic field, not a quantized one. So the
comparison will show up the effect of the quantization of the
electromagnetic field. The section defined a modified matrix element

This matrix element determined the entire evolution of the system. For the quantized electric field discussed here, this coefficient works out to be

(A.167) |

That is essentially the same form as for the classical field. Recall
that the second term in (7.44) for the classical field can
be ignored. The first term is the same as above, within a constant.
To see the real difference in the constants, note that the transition
probability is proportional to the square magnitude of the matrix
element. The square magnitudes are:

Now if there is a large number

But for spontaneous emission there is a big difference. In that case,
classical physics would take the initial electromagnetic field

Instead quantum mechanics takes the initial field to have

That can also be seen without detouring through the messy analysis of
chapter 7.7 and 7.8. To find the spontaneous
emission rate directly, the matrix element above can be plugged into
Fermi’s Golden Rule (7.38) of chapter
7.6.1. The density of states needed in it was given
earlier in chapter 6.3 (6.7) and
6.19. Do note that these modes include all directions of
the electric field, not just the

The above result is the same as Einstein’s, (7.47) and (7.48). (To see why a simple average works in the final term, first note that it is obviously the right average for photons with axial linear momenta and fields. Then note that the average is independent of the angular orientation of the axis system in which the photons are described. So it also works for photons that are axial in any rotated coordinate system. To verify that the average is independent of angular orientation does not really require linear algebra; it suffices to show that it is true for rotation about one axis, say the

Some additional observations may be interesting. You might think of
the spontaneous emission as caused by excitation from the ground state
electromagnetic field. But as seen earlier, the actual energy of
the ground state is half a photon, not one photon. And the zero level
of energy should not affect the dynamics anyway. According to the
analysis here, spontaneous emission is a twilight effect, chapter
5.3. The Hamiltonian coefficient