Please help; calculation phase space -> how do I use the Delta functions?

AI Thread Summary
The discussion focuses on understanding the use of delta functions in calculating the two-particle phase space for interactions involving spin-1 particles. The formula presented, psi = (2pi)^2 delta(Pa+Pb-Pc-Pd) d3Pc d3Pd / 4EcEd, highlights the importance of the delta function in ensuring conservation of momentum and energy during particle interactions. Participants express confusion over the implications of the delta function being zero when Pa + Pb does not equal Pc + Pd, and how this affects integration limits. The conversation also touches on the relationship between four-momentum conservation and the simplification of delta functions in the center of mass frame. Overall, the thread seeks clarity on applying these concepts in practical calculations.
marlon1
Messages
5
Reaction score
0
Can somebody help me out? I'm reading about formulas for cross sections for spin1 particles but I don't understand the delta functions, in calculating the 2particle pahse space psi

For example the interaction; A+B -> C+D has the formula;

psi= (2pi)^2 delta(Pa+Pb-Pc-Pd) d3Pc d3Pd / 4EcEd

then all the books say the same;

delta(Pa+Pb-Pc-Pd) d3Pd = delta (Ea+Eb-Ec-Ed)
:confused:

can somebody explain this to me?

the left side delta is zero everywhere pa+pb =not pc+pd.

the integral over the delta function d3Pd is;
1 for pd<pa+pb-pc and
0 for pd>pa+pb-pc

? how do I use this?
 
Last edited:
Physics news on Phys.org
Vergeet niet dat de je p3's en p4's hebt! (p4=(E,p)

\delta^{4}(p1+p2-p3-p4) d^{3}p4
=\delta(E1+E2-E3-E4) <br /> \delta^{3}(p1+p2-p3-p4)d^{3}p4
=\delta(E1+E2-E3-E4) \delta^{3}(-p3-p4) d^{3}p4 (CM frame: p1=-p2)
=\delta(E1+E2-E3-E4) \delta(x) dx (x= -p3-p4 = 0 CM frame)
= \delta(E1+E2-E3-E4) * 1 = \delta(E1+E2-E3-E4)
QED
 
At first, I derived that: $$\nabla \frac 1{\mu}=-\frac 1{{\mu}^3}\left((1-\beta^2)+\frac{\dot{\vec\beta}\cdot\vec R}c\right)\vec R$$ (dot means differentiation with respect to ##t'##). I assume this result is true because it gives valid result for magnetic field. To find electric field one should also derive partial derivative of ##\vec A## with respect to ##t##. I've used chain rule, substituted ##\vec A## and used derivative of product formula. $$\frac {\partial \vec A}{\partial t}=\frac...
Thread 'Help with Time-Independent Perturbation Theory "Good" States Proof'
(Disclaimer: this is not a HW question. I am self-studying, and this felt like the type of question I've seen in this forum. If there is somewhere better for me to share this doubt, please let me know and I'll transfer it right away.) I am currently reviewing Chapter 7 of Introduction to QM by Griffiths. I have been stuck for an hour or so trying to understand the last paragraph of this proof (pls check the attached file). It claims that we can express Ψ_{γ}(0) as a linear combination of...
Thread 'Conducting Sphere and Dipole Problem'
Hi, I'm stuck at this question, please help. Attempt to the Conducting Sphere and Dipole Problem (a) Electric Field and Potential at O due to Induced Charges $$V_O = 0$$ This potential is the sum of the potentials due to the real charges (##+q, -q##) and the induced charges on the sphere. $$V_O = V_{\text{real}} + V_{\text{induced}} = 0$$ - Electric Field at O, ##\vec{E}_O##: Since point O is inside a conductor in electrostatic equilibrium, the electric field there must be zero...
Back
Top