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

In summary, The formula for calculating the phase space for spin1 particles involves a delta function and the integral over the delta function is equal to 1 for pd<pa+pb-pc and 0 for pd>pa+pb-pc. In the CM frame, the integral simplifies to \delta(E1+E2-E3-E4), which is used to calculate the 2particle phase space psi for the interaction A+B -> C+D. This is also known as the QED formula.
  • #1
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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?
 
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  • #2
Vergeet niet dat de je p3's en p4's hebt! (p4=(E,p)

[tex]\delta^{4}(p1+p2-p3-p4) d^{3}[/tex]p4
=[tex]\delta(E1+E2-E3-E4)
\delta^{3}[/tex](p1+p2-p3-p4)[tex]d^{3}[/tex]p4
=[tex] \delta(E1+E2-E3-E4) \delta^{3}[/tex](-p3-p4) [tex]d^{3} [/tex]p4 (CM frame: p1=-p2)
=[tex]\delta(E1+E2-E3-E4) \delta[/tex](x) [tex]d[/tex]x (x= -p3-p4 = 0 CM frame)
= [tex] \delta(E1+E2-E3-E4) [/tex] * [tex]1[/tex] = [tex]\delta(E1+E2-E3-E4) [/tex]
QED
 

1. What is phase space in science?

Phase space is a mathematical concept used in physics to describe the possible states of a physical system. It is represented by a multidimensional space, with each dimension corresponding to a different variable of the system. The points in phase space represent all the possible combinations of these variables at a given time.

2. What is the calculation phase space?

The calculation phase space refers to the process of using mathematical equations and models to determine the possible states of a physical system. This involves using known information about the system, such as initial conditions and physical laws, to make predictions about its behavior over time.

3. How do I use Delta functions in phase space calculations?

Delta functions are mathematical tools used to represent a single point or event in phase space. They are commonly used in physics to describe the behavior of particles and their interactions. In phase space calculations, Delta functions can be used to simplify equations and make predictions about the behavior of a system at specific points in time.

4. What is the importance of phase space in scientific research?

Phase space is a critical concept in scientific research, particularly in physics and engineering. It allows scientists to model and predict the behavior of complex physical systems, such as particles or fluids, and make quantitative predictions about their behavior. This helps to advance our understanding of the natural world and can have practical applications in fields such as space exploration and energy production.

5. How can I improve my understanding of phase space calculations?

The best way to improve your understanding of phase space calculations is to practice using them in different scenarios and to study the underlying mathematical principles. It can also be helpful to consult with other scientists or attend workshops and seminars on the subject. Additionally, keeping up with current research and developments in the field can help deepen your understanding of phase space and its applications.

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