What is the failure of superposition in quantum mechanics?

[Hawkingo]

In a book it says that "we know of quantum phenomena in the electromagnetic field that represents a failure of superposition,seen from the viewpoint of the classical theory."
I want to about what quantum phenomena is he talking about?

This was from the page 11 of the book Electricity And Magnetism by Edward M.Purcell and David J.Morin (3rd edition)

 

[Nugatory]

In a book it says that "we know of quantum phenomena in the electromagnetic field that represents a failure of superposition,seen from the viewpoint of the classical theory."
I want to about what quantum phenomena is he talking about?

You will get better answers if you tell us what book.

 

[phinds]

In a book it says ...

is not a sufficient citation here on PF. Best to cite a specific book since pop-science books and ACTUAL science books are considered differently.

EDIT: I see nugatory beat me to it

 

[Hawkingo]

You will get better answers if you tell us what book.

This was from the page 11 of the book Electricity And Magnetism by Edward M.Purcell and David J.Morin (3rd edition)

 

[Hawkingo]

is not a sufficient citation here on PF. Best to cite a specific book since pop-science books and ACTUAL science books are considered differently.

This was from the page 11 of the book Electricity And Magnetism by Edward M.Purcell and David J.Morin (3rd edition)

 

[stevendaryl]

First of all, there are two different notions of linearity (or superposition) involved in quantum mechanics: at the level of states, and at the level of fields.

In quantum field theory, you have a field $\psi$ (such as the electromagnetic field, or the electron field, or the Higgs field, etc.). This field (in the Heisenberg picture) obeys Heisenberg equations of motion, which looks a lot like the classical equations for fields. For example, a free spin-zero field obeys the EOM:

$(\frac{\partial^2}{\partial t^2} - \nabla^2) \psi = -m^2 \psi$

(You have to stick in $c$ and $\hbar$ in various places to make the units work out.)

This is a linear equation of motion, in the sense that if $\psi_1$ and $\psi_2$ are two solutions, then so is the superpostion, $\psi_1 + \psi_2$.

If there is a self-interaction, then this leads to a more complicated equation of motion, maybe something like:

$(\frac{\partial^2}{\partial t^2} - \nabla^2) \psi = -m^2 \psi -\lambda \psi^3$

This equation of motion is nonlinear, and does not obey the superposition principle.

Now, there is also a quantum-mechanical notion of state, $|\Psi\rangle$. In the Schrodinger picture, the state also obeys an equation of motion, something along the lines of:

$H |\Psi\rangle = i \frac{\partial}{\partial t} |\Psi\rangle$

where $H$ involves field operators such as $\psi$ above.

The Rules of Quantum Mechanics require that the equation of motion for the state is always linear, but the equations of motions for fields is not necessarily linear.

So in the case of electromagnetism, the issue is whether the equation of motion for the field are linear, or not. Classically, in the absence of charges, the electromagnetic field obeys linear equation of motion. That means that the electromagnetic field has no self-interaction.

Quantum-mechanically, it's a little more complicated. In the perturbation expansion for the electromagnetic field, there are Feynman diagrams that involve indirect interactions between two photons. The diagram can be loosely described as: "one photon produces a virtual electron-positron pair, and then the electron or positron interacts with the other photon before annihilating into a photon again". Looking at just one such diagram, there appears to be a photon-photon interaction. I am not an expert at quantum electrodynamics enough to say whether this is a real effect. Each diagram has no actual physical meaning, but only represents one term in an infinite sum describing the interaction. I don't know whether the apparent interaction persists when you sum the diagrams, or not.

But assuming that it does, then that means that light can interact with light, and such a self-interaction means that the superposition principle doesn't hold precisely (for fields).

 

[Nugatory]

In a book it says that "we know of quantum phenomena in the electromagnetic field that do represents a failure of superposition, seen from the viewpoint of the classical theory."
I want to about what quantum phenomena is he talking about?

[Note that I've corrected the wording to exactly what appears in the text]

The superposition that Purcell and Morin are referring to is not quantum superposition, but the property of classical electromagnetic fields that allows them to be added: the total field is the sum of the contributions from the individual sources.
It's not clear what phenomena they had in mind (the quoted text is something of a throwaway line, a digression for the purposes of that textbook, and clearly not something intended rigorously) but the photoelectric effect, photon antibunching, and point detection of photons could reasonably be considered failures of superposition, seen from the viewpoint of the classical theory.

 

[andresB]

In a book it says that "we know of quantum phenomena in the electromagnetic field that represents a failure of superposition,seen from the viewpoint of the classical theory."
I want to about what quantum phenomena is he talking about?

This was from the page 11 of the book Electricity And Magnetism by Edward M.Purcell and David J.Morin (3rd edition)

As Nugatory stated this is not about quantum superposition of states, but the superposition of classical electromagnetic fields.

Do note that even in the classical theory you can find cases where a superposition of solutions of the Maxwell's equations (in a medium) is not a solution itself, see non-linear optics.

Remaining within a classical framework, we can include some quantum effects via non-linear corrections to the Maxwell's equations. There you can find effects that don't exist in the classical theory, my favorite is

https://arxiv.org/abs/1101.3433
https://link.springer.com/article/10.1023/A:1019748524213

 

[vanhees71]

One more example for the prejudice against Purcell's textbook (I suppose it's the infamous vol. 2 of the Berkeley physics course): It's of course usually half correct but expresses things in a way that tends to confuse students. Even worse, it buries the beauty of relativistic physics under confusing notation and under well-meaning "didactics" :-(.