Elastic reactions, ratio of cross sections

[erwinscat]

Hello everybody,

I am having some troubles with this exercise. I have attached the text and the solutions to my post. My problem is, after I have coupled the angular momenta and used the CK coefficients like in ($$|\pi ^{+}p>=|1,1>|1/2, 1/2>=|3/2,3/2>$$) (pag.432) I don't understand what follows..well, I have a problem with :

$$<\frac{3}{2},m_{j}|\hat{H}|\frac{3}{2},m_{i}>=a_{1}$$

$$<\frac{1}{2},m_{j}|\hat{H}|\frac{1}{2},m_{i}>=a_{2}$$

What exactly are coefficient $$a_1$$ and $$a_2$$ ? I assume elements of the $$\hat{H}$$ matrix...Is the matrix $$\hat{H}$$ the transition matrix of the reaction ? How can I calculate it ? This is really my problem, finding the $$\hat{H}$$ matrix. I attach the rest of the exrcise just you know, but my problem is really to understand how to calculate the $$\hat{H}$$ matrix and how to use it properly.

Any help would be very much appreciated!

Thanks a lot in advance!
Erwin

 

[AndyW]

Hello Erwin,

Thank you for sharing your problem with the exercise. It seems like you have a good understanding of the initial steps in the exercise, but are unsure about the calculation of the \hat{H} matrix. The \hat{H} matrix is indeed the transition matrix of the reaction, and it represents the energy transfer between the initial and final states of the system. In this case, it is the energy transfer between the \frac{3}{2} and \frac{1}{2} states of the system.

To calculate the \hat{H} matrix, you will need to use the Hamiltonian operator, which represents the total energy of the system. This operator can be written as a sum of the kinetic and potential energy operators, \hat{H} = \hat{T} + \hat{V}. The kinetic energy operator can be expressed as the sum of the individual angular momentum operators, \hat{T} = \hat{L}^2 + \hat{S}^2. The potential energy operator, on the other hand, will depend on the specific system and interaction being studied.

Once you have the expression for the Hamiltonian operator, you can then use it to calculate the \hat{H} matrix elements as shown in the exercise. These elements will give you the energy transfer between the different states of the system. I would recommend consulting your textbook or speaking with your instructor for more specific guidance on how to calculate the \hat{H} matrix for this particular system.

I hope this helps and good luck with your exercise! Don't hesitate to reach out if you have any further questions.

 

[sir-pinski]

Hello Erwin,

Thank you for reaching out with your question. It seems like you are working on a physics exercise involving elastic reactions and the ratio of cross sections. To answer your question, the coefficients a_1 and a_2 represent the matrix elements of the Hamiltonian operator, which is usually denoted as \hat{H}. The Hamiltonian operator is the total energy operator for a system and is used to calculate the transition probabilities between different energy states.

To calculate the \hat{H} matrix, you will need to use the appropriate equations and mathematical techniques for your specific problem. This may involve using the Clebsch-Gordan coefficients, as you mentioned in your post, along with other relevant equations and principles from quantum mechanics. I would recommend consulting your textbook or speaking with your instructor for guidance on how to approach this calculation.

I hope this helps clarify the concept of the \hat{H} matrix and how to calculate it. Best of luck with your exercise, and don't hesitate to reach out if you have any further questions. Good luck!

Best regards,