scattering processes
What are the “components” of a scattering experiment and what are we measuring?
Components: initial free particles, beam, target, detector
- free particles: particles prepared in such a way that they are so far away from each other that they don’t interact (initial state of scattering)
- initial state: t = -∞, observation results: t = +∞
We’ll be counting the event number.
cross section
What’s the cross section of scattering experiments? What type of experiments do we do to measure it?
The proportinality factor to determine the number of events. It measures the effective area of the “active” region around each scatterer in the target.
- the active region describes the region around the target in which the beam has to pass through for an event to take place
- the larger the cross section the more probable the evets and the stronger the interaction
Fixed target: insert képlet
Collider experiments: insert képlet
cross section
What does the Wigner-Breit-formula tell us?
It is an approximate description for particle resonance when an unstable particle is created as an intermediate product in a scattering experiment.
- it tells us how the cross section is proportional to the energy of the particle
- when E = particle mass and the full width at half height is the decay rate of the particle, these properties can be identified
QM review
What pictures are there to formulate QM?
Schrödinger picture: states evolve with time, observables are time-indepedent
Heisenberg picture: states are fixed at their t = 0 value, observables evolve with time determined by the Hamiltonian
Dirac (interaction) picture: the Hamiltonian can be split into the free Hamiltonian and the interaction part, states evolve with the interaction part only, observables obey the free temporal evolution
Expectation values are the same in every picture though.
formal theory of scattering
What are in and out states? How do we construct them?
They describe what is means that the initial and final states of a scattering process look like freely-evolving particle states.
- physically: the state that evolves with the full Hamiltonian, is the same as the evolution of the freely-evolving asymptotic states when the system is prepared in the distant past/future
- mathematically: the norm of these states’ difference from the exact state goes to zero
They’re the states that describe the exact temporal evolution of the system with the full Hamiltonian, derived from the statements above
formal theory of scattering
What are Møller operators? What’s the S-matrix?
They’re unitary operators, another name for them is scattering operators. insert képlet
From the relevant transition amplitude at time T, inserting the Møller operators, the S-matrix can be constructed. The S-matrix encodes all the necessary information about the scattering processes.
At time T the initial state is observed in some prescribed final state.
formal theory of scattering
What are the properties of an S-matrix?
- It’s a unitary operator (it’s the product of unitary operators). From this conservation of probability follows.
- Intertwining relations. Can be derived for any s.
- It commutes with H0, P and J meaning that energy, momentum and angular momentum is conserved.
- If we build the potential V properly, Lorentz invariance can be achieved.
- It’d be unitary even if the Møller operators weren’t.
formal theory of scattering
What does Dyson’s formula tell us? What are the steps for deriving it?
What does the time ordering symbol do?
It gives a straightforward approximation scheme for the S matrix. For a small perturabtion it makes sense to expand the time ordered exponential and take the first few terms as approximation for the matrix elements of S.
Deriving the formula:
- From the definition of S we can introduce the unitary op. U(t2,t1) (we want an explicit expression for this)
- Taking the derivative of U(t2,t1) with respect to t2 (or t1, since it’s unitary): here we ca introduce V(I)(t) which is the interaction part of the Hamiltionian in the interaction picture
- Giving the solution for U(t2,t1) taking into account the prescribed initial condition: U(t,t) = 1
- For U(+∞,–∞) we get Dyson’s formula
It places the operators in descending order with respect to time.
cross sections from the S-matrix
What’s the goal of computing S-matrix elements? What is the quantity that let’s us do this?
If we have a theory from which we can compute S-matrix elements, we can predict the outcome of scattering experiments, allowing the theory to be tested.
We cannot measure the transition probability because the initial state is not known with arbitrary accuracy due to the practical processes being affected by inherent uncertainties.
The measurable quantity that is directly related to the transition probability is the cross section. Through this, what actually gets measured is the transition probability between idealized initial and final momentum eigenstates.
cross sections from the S-matrix
How can one derive the relation between the S-matrix and the cross section?
- Looking at the original formula for the differential cross section: discrete and continuous parameters
- Considering an ideal process, meaning one particle from the beam and one target. The average number of scattering event is the probability.
- Considering particles with sharply defined momenta in the initial state and measuring the momenta of the final state with infinite precision: studying the transition from one momentum eigenstate to another
- Observing the finite spacetime box in which the scattering takes place: applying the relativistic invariant normalisation of momentum eigenstates
- Substituting everything in from the previous steps and making it contuinuous + writing it in a Lorentz-invariant way
cross sections from the S-matrix
Why do we have to be careful when taking the infinite-volume and infinite-time limit?
If we take the limits too soon, we can run into problems because in an infinite spatial volume the momentum eigenstates are improper, non-normalisable eigenstates for which the definition of the transition probability makes no sense.
The square of a Dirac-delta would be present in the numerator which makes no sense.
