2. Introducing exchange
photons and some problems
We ended the first section with Heisenberg's Uncertainty Principle for time and energy:
We are going to use this equation to explain how to briefly violate the conservation of energy!
Let's look at a process that violates the conservation of energy:
The system starts with energy Ebefore but jumps up to a higher energy state Eafterwards.
For us to be able to tell the difference between the two energy levels we must be able to measure energy to an accuracy DE smaller than the difference between Ea and Eb.
and to be able to measure to a precision DE the energy state must last for
.Let's change the picture to one where the high energy state reverts back to the original one.
If energy state Ea lasts for a short enough period of time there will not be long enough to measure it accurately enough to be able to establish that the conservation of energy has been violated. It will not be detected as a violation if it lasts for less than
So what can we do with this energy before it has to be paid back? Imagine a stationary, free electron in empty space. According to Einstein's Special Theory of Relativity it has energy because it has mass. This is energy state Eb
Suppose that the electron emits a photon of energy Eg and recoils in the opposite direction. The energy of this new situation is now
If we are to get away with this photon being created it needs to be absorbed again before it extra energy can be measured. Because this photon can never be measured it is called a virtual photon as opposed to the real photons that you are using to see this. With just one electron, it will have to absorb the photon and stop moving - a pointless exercise? Perhaps, but it happens all the time and has a measurable effect called the Lamb shift. Can we do anything useful with our virtual photon? Yes!
We will use this virtual photon to exchange momentum with another electron. The first electron recoils because the photon carries momentum. If another electron absorbs the photon it will also recoil. Imagine two electrons coming towards each other.
The electrons have repelled each other by exchanging a virtual photon. So long as the total energy of the electrons before = total energy of electrons afterwards and so long as t2-t1 is less than the time allowed by the Uncertainty Principle then this process is not forbidden by quantum mechanics even if it is impossible classically.
Problems?
1. How can you do an attractive force using this sort of picture?
2. This is a very jerky picture - forces like this are smooth.
3. The electromagnetic force (Coulomb's law) is infinite - how can we borrow energy if the photon has to travel an infinite distance?
4. Very clever but does it work for the other forces?