Phase space element calculation

In summary, the conversation discusses a calculation of a 2 to 2 process using the phase space method. The integral is simplified using a substitution and the final answer is corrected to take into account the dependency of E_2 on E_1.
  • #1
kaksmet
83
0
Can anyone see what's not right?

In the phase space calculation of a 2 to 2 process I get to
[tex]I=\int dp_1d\Omega \frac{1}{(2\pi)^2}\frac{p_1^2}{2E_12E_2}\delta(E_1+E_2-E)[/tex]

then I use
[tex]p_1=\sqrt{E_1^2+m_1^2} \Rightarrow dp_1=\frac{E_1}{\sqrt{E_1^2+m_1^2}}dE_1[/tex]

thus
[tex]I = \int dE_1d\Omega \frac{1}{(2\pi)^2}\frac{E_1^2-m_1^2}{2E_12E_2}\frac{E_1}{\sqrt{E_1^2-m_1^2}}\delta(E_1+E_2-E) [/tex]
[tex]=\int d\Omega \frac{\sqrt{E_1^2-m_1^2}}{16\pi^2E_2}[/tex]

[tex]=\int d\Omega \frac{|p_1|}{16\pi^2E_2}[/tex]

However, this is the right answer, which should be
[tex]\int d\Omega \frac{|p_1|}{16\pi^2E_{CM}}[/tex]All ideas greatly appreciated
 
Last edited:
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  • #2
Problem sovled. Misstake was that E_2 is a function of E_1 so I cannot directly use the delta function. Taking this into account the correct answer is obtained.
 

FAQ: Phase space element calculation

1. What is phase space element calculation?

Phase space element calculation is a mathematical technique used in physics to describe the state of a physical system. It involves calculating the size of a small region in the space of all possible states that the system can occupy.

2. Why is phase space element calculation important in physics?

Phase space element calculation is important because it allows us to understand and predict the behavior of physical systems. It helps us analyze and model complex systems, such as gases, liquids, and even subatomic particles.

3. How is phase space element calculated?

The phase space element is calculated by multiplying the infinitesimal change in position and momentum of a particle. This can be represented mathematically as dxdp, where dx is the change in position and dp is the change in momentum.

4. What are the applications of phase space element calculation?

Phase space element calculation has a wide range of applications in physics, including statistical mechanics, thermodynamics, and quantum mechanics. It is also used in fields such as astrophysics, particle physics, and fluid dynamics.

5. Are there any limitations to phase space element calculation?

One limitation of phase space element calculation is that it assumes a classical, deterministic view of physics. It also does not take into account the uncertainty principle in quantum mechanics. Additionally, it may become increasingly complex and difficult to calculate in systems with a high number of particles.

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