In one of my posts on the rules of quantum math, I introduced the *propagator *function, which gives us the amplitude for a particle to go from one place to another. It looks like this:

The **r _{1 }**and

**r**vectors are, obviously, position vectors describing (1) where the particle is right now, so the initial

_{2}*state*is written as |

**r**〉, and (2) where it

_{1}*might*go, so the final state is |

**r**〉. Now we can combine this with the analysis in my previous post to think about what might happen when an electron sort of ‘jumps’ from one state to another. It’s a rather funny analysis, but it will give you some feel of what these so-called ‘virtual’ particles might represent.

_{2}Let’s first look at the shape of that function. The *e*^{(i/ħ)·(p∙r12) }function in the *numerator *is now familiar to you. Note the **r _{12 }**in the argument, i.e. the

*vector*pointing from

**r**to

_{1}**r**. The

_{2}**p**∙

**r**dot product equals |

_{12}**p**|∙|

**r**|·cosθ = p∙r

_{12}_{12}·cosθ, with θ the angle between

**p**and

**r**. If the angle is the same, then cosθ is equal to 1. If the angle is π/2, then it’s 0, and the function reduces to 1/r

_{12}_{12}. So the angle θ, through the cosθ factor, sort of

*scales*the spatial frequency. Let me try to give you some idea of how this looks like by assuming the angle between

**p**and

**r**is the same, so we’re looking at the space in the direction of the momentum only and |

_{12}**p**|∙|

**r**|·cosθ = p∙r

_{12}_{12}. Now, we can look at the p/ħ factor as a

*scaling*factor, and measure the

*distance x*in units defined by that scale, so we write:

*x*= p∙r

_{12}/ħ. The whole function, including the denominator, then reduces to (ħ/p)·

*e*

^{i∙x}/

*x*= (ħ/p)·cos(

*x*)/

*x*+

*i*·(ħ/p)·sin(

*x*)/

*x*, and we just need to square this to get the probability. All of the graphs are drawn hereunder: I’ll let you analyze them. [Note that the graphs do not include the ħ/p factor, which you may look at as yet another scaling factor.] You’ll see – I hope! – that it all makes perfect sense: the probability quickly drops off with distance, both in the positive as well as in the negative x-direction, while going to infinity when

*very*near, i.e. for very small

*x*. [Note that the absolute square, using cos(

*x*)/

*x*and sin(

*x*)/

*x*

_{ }yields the same graph as squaring 1/

*x—obviously!*]

Now, this propagator function is *not *dependent on time: it’s only the momentum that enters the argument. Of course, we assume p to be some *positive real number*. Of course?

This is where Feynman starts an interesting conversation. In the previous post, we studied a model in which we had two protons, and one electron jumping from one to another, as shown below.

This model told us the equilibrium state is a stable ionized hydrogen molecule (so that’s an H_{2}^{+ }*molecule*), with an interproton distance that’s equal to 1 Ångstrom – so that’s like *twice *the size of a hydrogen *atom* (which we simply write as H) – and an energy that’s 2.72 eV *less *than the energy of a hydrogen atom and a proton (so that’s *not *an H_{2}^{+ }*molecule *but a system consisting of a separate hydrogen atom and a proton). The why and how of that equilibrium state is illustrated below. [For more details, see my previous post.]

Now, the model implies there is a sort of attractive force *pulling the two protons together even when the protons are at larger distances than 1 Å.* One can see that from the graph indeed. Now, we would

*not*associate any molecular orbital with those distances, as the system is, quite simply, not a molecule but a separate hydrogen atom and a proton. Nevertheless, the amplitude A is non-zero, and so we have an electron jumping back and forth.

We know how that works from our post on tunneling: particles can cross an energy barrier and tunnel through. One of the weird things we had to consider when a particle crosses such potential barrier, is that the *momentum *factor p in its wavefunction was some pure imaginary number, which we wrote as p = *i*·p’. We then re-wrote that wavefunction as *a·e ^{−i}*

^{θ}

*=*

*a·e*

^{−i}^{[}

^{(E/ħ)∙t − (i·p’/ħ)x]}= a·e

^{−i}

^{(E/ħ)∙t}·e

^{i2·p’·x/ħ}= a·e

^{−i}

^{(E/ħ)∙t}

*·e*

^{−p’·x/ħ}. The

*e*

^{−p’·x/ħ}factor in this formula is a

*real-valued*exponential function, that sort of ‘kills’ our wavefunction as we move across the potential barrier, which is what is illustrated below: if the

*distance*is too large, then the amplitude for tunneling goes to zero.

From a mathematical point of view, the analysis of our electron jumping back and forth is *very *similar. However, there are differences too. We can’t really analyze this in terms of a potential barrier *in space*. The barrier is the potential energy of the electron itself: it’s happy when it’s bound, because its energy then contributes to a *reduction *of the total energy of the hydrogen atomic system that is equal to the ionization energy, or the *Rydberg *energy as it’s called, which is equal to not less than 13.6 eV (which, as mentioned, is pretty big at the atomic level). Well… We can take that propagator function (1/*r*)·*e*^{(i/ħ)·p∙r} (note the argument has no minus sign: it can be quite tricky!), and just fill in the value for the momentum of the electron.

* Huh? What momentum? *It’s got no momentum to spare. On the contrary, it wants to stay with the proton, so it has no energy whatsoever to escape. Well… Not in quantum mechanics. In quantum mechanics it can use

*all*its potential energy and convert it into kinetic energy, so it can get away from its proton and convert the energy that’s being released into kinetic energy.

*But there is no release of energy! The energy is negative!*

Exactly! You’re right. So we boldly write: K.E. = m·v^{2}/2 = p^{2}/(2m) = −13.6 eV, and, because we’re working with complex numbers, we can take a square root of negative number, using the *definition* of the imaginary unit: *i* = √(−1), so we get a *purely imaginary* *value *for the* *momentum p, which we write as:

p = ±*i*·√(2m·E_{H})

The sign of p is chosen so it makes sense: our electron should go in one direction only. It’s going to be the *plus *sign. [If you’d take the *negative *root, you’d get a nonsensical propagator function.] To make a long story short, our propagator function becomes:

(1/*r*)·*e*^{(i/ħ)·i·√(2m·EH)∙r} = (1/*r*)·*e*^{(i/ħ)·i·√(2m·EH)∙r} = (1/*r*)·*e*^{i2/ħ·√(2m·EH)∙r} = (1/r)·e^{−√(2m·EH)/ħ∙r}

Of course, from a mathematical point of view, that’s the same function as* e*^{−p’·x/ħ}: it’s a *real-valued *exponential function that quickly dies. But it’s an amplitude alright, and it’s just like an amplitude for tunneling indeed: if the *distance *is too large, then the amplitude goes to zero. The final cherry on the cake, of course, is to write:

A ∼ (1/r)·e^{−√(2m·EH)/ħ∙r}

Well… No. It gets better. This amplitude is an amplitude for *an electron bond *between the two protons which, as we know, *lowers *the energy of the system. By how much? Well… By A itself. Now we know that work or energy is an integral or *antiderivative* of *force *over distance, so force is the *derivative *of energy with respect to the distance. So we can just take the derivative of the expression above to get the force. I’ll leave that you as an exercise: don’t forget to use the product rule! 🙂

So are we done? No. First, we didn’t talk about virtual particles yet! Let me do that now. However, first note that we should add one more effect in our two-proton-one-electron system: the *coulomb *field (ε) caused by the bare proton will cause the hydrogen molecule to take on an induced electric dipole moment (μ), so we should integrate that in our energy equation. Feynman shows how, but I won’t bother you with that here. Let’s talk about those virtual particles. What *are* they?

Well… There’s various definitions, but Feynman’s definition is this one:

“There is an exchange of a virtual electron when–as here–the electron has to jump across a space where it would have a negative energy. More specifically, **a ‘virtual exchange’ means that the phenomenon involves a quantum-mechanical interference between an exchanged state and a non-exchanged state**.”

You’ll say: what’s virtual about it? The electron does go from one place to another, doesn’t it? Well… Yes and no. We can’t *observe *it while it’s supposed to be doing that. Our analysis just tells us it seems to be useful to distinguish two different states and analyze all in terms of those differential equations. Who knows what’s *really *going on? What’s *actual *and what’s *virtual*? We just have some ‘model’ here: a model for* *the *interaction *between a hydrogen atom and a proton. It explains the attraction between them in terms of a sort of continuous exchange of an electron, but is it *real*?

The point is: in physics, it’s assumed that the *coulomb *interaction, i.e. all of electrostatics really, comes from the exchange of *virtual photons*: one electron, or proton, emits a photon, and then another absorbs it in the reverse of the same reaction. Furthermore, it is assumed that the amplitude for doing so is like that formula we found for the amplitude to exchange a virtual electron, except that the *rest *mass of a photon is zero, and so the formula reduces to 1/*r*. Such simple relationship makes sense, of course, because that’s how the electrostatic potential varies in space!

*That*, in essence, is all what there is to the quantum-mechanical theory of electromagnetism, which Feynman refers to as the ‘particle point of view’.

So… Yes. It’s that simple. Yes! *For a change! *🙂

**Post scriptum**: Feynman’s *Lecture *on virtual particles is actually focused on a model for the *nuclear forces*. Most of it is devoted to a discussion of the virtual ‘pion’, or π-meson, which was then, when Feynman wrote his *Lectures*, supposed to mediate the force between two nucleons. However, this theory is clearly outdated: nuclear forces are described by quantum *chromo*dynamics. So I’ll just skip the Yukawa theory here. It’s actually kinda strange his theory, which he proposed in 1935, was *the *theory for nuclear forces for such a long time. Hence, it’s surely all very interesting from a historical point of view.