Shape of wave function (probability distribution) of a single photon

[J O Linton]

TL;DR

What is the shape of the wave function (i.e. the probability distribution) of an isolated photon?

A photon is emitted from an excited atom at the origin. I would like to know what is the shape of the wave function or rather the probability distribution after 1 second. Is it
A: a point 1 light second from the origin
B: a spherical shell of radius 1 light second
C: a spherical packet of waves of average radius 1 light second (like the ripples in a pond)
D: a fuzzy blob of probability moving in a certain direction but spreading out as it goes
or is the question defective in some way?

 

[sophiecentaur]

TL;DR Summary: What is the shape of the wave function (i.e. the probability distribution) of an isolated photon?

or is the question defective in some way?

I'd say that you should expect spherical symmetry but that's all.

 

[PeroK]

TL;DR Summary: What is the shape of the wave function (i.e. the probability distribution) of an isolated photon?

A photon is emitted from an excited atom at the origin. I would like to know what is the shape of the wave function or rather the probability distribution after 1 second. Is it
A: a point 1 light second from the origin
B: a spherical shell of radius 1 light second
C: a spherical packet of waves of average radius 1 light second (like the ripples in a pond)
D: a fuzzy blob of probability moving in a certain direction but spreading out as it goes
or is the question defective in some way?

A single photon doesn't have a position-space wave-function. Basic QM is explicitly non-relativistic, so does not describe light.

 

[J O Linton]

Golly! That is quite some cop-out!

suppose the atom is surrounded by a spherical half-silvered mirror of radius half a light second. Surely the photon is now in a superposition of two states, one transmitted, one reflected so it must have a wave function. What is the probability distribution of the photon inside and outside the mirror now?

In any case, I could have asked the same question of a particle such as an electron.

 

[PeroK]

Golly! That is quite some cop-out!

There's no cop out, as you describe it. The Schrödinger equation is manifestly not covariant. Hence QM is a non-relativistic theory. After basic QM was developed, Dirac developed a relativistic equation for the electron:

https://en.wikipedia.org/wiki/Dirac_equation

Then, QED (Quantum Electrodynamics) was developed, as a quantum theory of light:

https://en.wikipedia.org/wiki/Quantum_electrodynamics

 

[Nugatory]

In any case, I could have asked the same question of a particle such as an electron.

You could, and then you would indeed get a wave function as an answer. If the process creating the electron allows the electron to be emitted in any direction (this is a statement about its momentum, not its position) that wave function will be spherically symmetric as you would expect. (One might reasonably wonder how we get tracks in cloud chambers then - this is the “Mott problem” and the Wikipedia article is pretty good).

But the situation with photons is completely different. They have no position operator so they don’t have a wave function like particles that can be treated using non-relativistic quantum mechanics. Instead we have to use the methods of quantum electrodynamics. In that mathematical formalism there’s no such thing as “the photon”; instead we calculate the probability of finding the electromagnetic field in a particular state at each point in space.

 

[J O Linton]

This makes a lot of sense. In the case of an electron the correct answer is therefore C. Thanks. I looked up the Mott problem. That answered some other questions too.
Very interesting.

 

[J O Linton]

One more thing, if photons do not have a position operator and if non-relativistic QT does not apply to photons, why is the SWE so successful in predicting the outcome of situations like the double slit experiment?

 

[renormalize]

One more thing, if photons do not have a position operator and if non-relativistic QT does not apply to photons, why is the SWE so successful in predicting the outcome of situations like the double slit experiment?

Can you cite a textbook example where the Schrödinger Wave Equation is successfully applied to an electromagnetic wave passing through a double slit?

 

[Nugatory]

One more thing, if photons do not have a position operator and if non-relativistic QT does not apply to photons, why is the SWE so successful in predicting the outcome of situations like the double slit experiment?

It isn't.
We explain the double-slit experiment with light by doing a classical calculation of classical electromagnetic waves passing through a double-slit barrier and forming an interference pattern on the screen - the same thing that Young did a century before quantum mechanics and photons to show that light is a wave.

The quantum mechanical phenomenon here is not the interference pattern, it is that if we use a single-photon source we find that each photon produces a dot on the screen as it is detected at a single point.

 

[J O Linton]

OK So if we technically have to use Diracs relativistic equation to predict the behaviour of single photons in a double slit experiment but we can use SWE to predict he behaviour of electrons in a similar situation it is surely not a coincidence that they predict essentially the same result. It is this partial isomorphism between these two theories which enables us to talk about 'single photons' in some situations without going too far astray.

 

[sophiecentaur]

surely not a coincidence

No coincidence if the same Maths is used but is that the same thing as reality?

 

[J O Linton]

But my point is that the maths is different. Several people have already pointed out that you cannot apply SWE to photons.

 

[PeroK]

OK So if we technically have to use Diracs relativistic equation to predict the behaviour of single photons in a double slit experiment but we can use SWE to predict he behaviour of electrons in a similar situation it is surely not a coincidence that they predict essentially the same result. It is this partial isomorphism between these two theories which enables us to talk about 'single photons' in some situations without going too far astray.

Dirac's equation is for the electron. You need QED (Quantum Electrodynamics) for light. Feynman wrote a book on this:

https://en.wikipedia.org/wiki/QED:_The_Strange_Theory_of_Light_and_Matter

Light was explained by classical electromagnetism, where a light beam is an EM wave. This theory explains the double-slit interference pattern. Non-relativistic electrons are described by QM and the SWE. This produces a very similar double-slit pattern. The important point here is that two different mathematical frameworks can make approximately the same predictions. Think of Newton's laws and Lagrangian mechanics. They look like different theories (although they are actually equivalent).

This gives us two different mathematical ways to explain double-slit interference. Classical EM and QM.

Finally, when we do an experiment with a very low intensity source, we see the double-slit pattern build up dot by dot. This means that classical EM is an approximation for light, where the beam is of sufficiently large intensity. Classical EM cannot explain a very low intensity double-slit experiment. This is where we need a quantum theory of light.

Note that the quantum theory of light must predict the double-slit pattern. This is not a coincidence! This is a requirement of the new theory; and, a the test that you have the right quantized theory.

Now we have three different ways to explain the double-slit: Classical EM (waves); QM (SWE); QED (Quantized EM field).

 

[J O Linton]

I completely accept that. Thanks.

 

[bhobba]

You may be interested in the following

https://arxiv.org/abs/quant-ph/0703126

https://arxiv.org/abs/1902.00015

https://www.physics.usu.edu/torre/3700_Spring_2015/What_is_a_photon.pdf

Thanks
Bill

 

[A. Neumaier]

TL;DR Summary: What is the shape of the wave function (i.e. the probability distribution) of an isolated photon?

A photon is emitted from an excited atom at the origin. I would like to know what is the shape of the wave function or rather the probability distribution after 1 second. Is it
A: a point 1 light second from the origin
B: a spherical shell of radius 1 light second
C: a spherical packet of waves of average radius 1 light second (like the ripples in a pond)
D: a fuzzy blob of probability moving in a certain direction but spreading out as it goes
or is the question defective in some way?

B (idealized), C (realistically)

A single photon doesn't have a position-space wave-function. Basic QM is explicitly non-relativistic, so does not describe light.

While there is no position operator (and hence no Schrödinger picture), photons still have (relativistic) wave functions. See What is a photon? in Chapter B2: Photons and Electrons of my Theoretical Physics FAQ. The wave function of a photon in vacuum is in the most general case a solution of the classical Maxwell equation in vacuum. Emitted from a point source it is spherically symmetric (apart where it is blocked through the shape of the emitting source). As it moves with the speed of light, its support is a spherical wave in a spherical shell as in B. Though taking into account that an atom is not a perfect point source, emission is not instantaneous. Thus this shell is smeared in spacetime and hence in space, leading to C.