Energies of a two particle system

Remember, the J^2 operator is defined as the sum of the squares of the individual angular momentum operators (J_x, J_y, J_z), so its eigenvalues are related to the total angular momentum j. It can be derived using the commutation relations of the individual operators.
  • #1
Sekonda
207
0
Hey,

This question is on determining the energies of a two particle system given the Hamiltonian, I believe it to be simple enough but would like you guys to check it and fill in any gaps in my reasoning

Hamiltonian.png


So I believe the eigenvalues of J^2 and J^2(z) are given by:

[tex]\hat{J}^{2}:j(j+1)\: ,\: \hat{J}^{2}_{z}:m(m+1)\Leftrightarrow \hbar=1[/tex]

('z' subscript same as '3')

and so the energy of state 1,1 is :

[tex]2(\alpha+\beta)[/tex]

The second part state that j=3, therefore m=3,2,1,0,-1,-2,-3
and so we just pop these into our eigenvalue equations above to attain the energies :

[tex]|3,3> : 12\alpha+12\beta\: ,\: |3,2>:12\alpha+6\beta[/tex]

etc.

Is this right?

Thanks for any comment/help!
SK
 
Last edited:
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  • #2
There is a mistake in your work. What is the eigenvalue of the Jz operator?
 
  • #3
The eigenvalue of the Jz operator is 'm', so does that mean the eigenvalue of the Jz^2 operator is m(m+1)?

Oh and 'z' is the same as '3' for the subscripts!
 
  • #4
Sekonda said:
The eigenvalue of the Jz operator is 'm', so does that mean the eigenvalue of the Jz^2 operator is m(m+1)?

No. If [itex]J_z|j,m>=m|j,m>[/itex] then what is

[tex]J_z^2|j,m>=J_z(J_z|j,m>)=?[/tex]
 
  • #5
Oh, m^2?
 
  • #6
Sekonda said:
Oh, m^2?

Yep. :smile:
 
  • #7
Oh ok, well I just assumed it was the same as the J^2 eigenvalue j(j+1), does the J^2 operator raise the state by 1 then?
 
  • #8
Sekonda said:
Oh ok, well I just assumed it was the same as the J^2 eigenvalue j(j+1), does the J^2 operator raise the state by 1 then?

You are confused about the J^2 operator. It is a total angular momentum operator (squared), not a ladder operator. The eigenvalues for the J^2 operator are correct as you have them: [itex] J^2 |j,m>=j(j+1)|j,m>[/itex]

If you don't understand why the J^2 operator has a different eigenvalues than the J_z operator, you should review the derivation of these eigenvalue equations in your textbook or with your instructor.
 
  • #9
It's not that I don't understand why they're different but more why the J^2 is equal to j(j+1), I'm sure we've 'shown' it somewhat before but these things are easily forgotten by myself.

Thanks anyway G01 for being patient with me and helping!
SK
 
  • #10
Sekonda said:
Thanks anyway G01 for being patient with me and helping!
SK

You're welcome!
 

Related to Energies of a two particle system

1. What is the definition of "energies of a two particle system"?

The energies of a two particle system refer to the total energy of a system consisting of two particles, taking into account their masses, positions, and interactions with each other and the surrounding environment.

2. How are the energies of a two particle system calculated?

The energies of a two particle system can be calculated using various equations, such as the gravitational potential energy equation for particles with masses and distance between them, or the kinetic energy equation for particles in motion. Additionally, the energies can also be calculated by considering the interactions between the particles and other forces in the system.

3. Can the energies of a two particle system change over time?

Yes, the energies of a two particle system can change over time due to various factors such as the particles' positions and velocities, external forces, and interactions with other particles or systems. These changes can be calculated using equations that take into account the initial and final states of the system.

4. How do the energies of a two particle system affect the overall behavior of the system?

The energies of a two particle system play a crucial role in determining the behavior of the system. The total energy of the system must remain constant, but the distribution of energy between the particles can affect their relative motions and interactions. High energy particles tend to have more erratic movements, while low energy particles tend to have more stable and predictable movements.

5. Are there any real-world examples of a two particle system with significant energy interactions?

Yes, there are many real-world examples of two particle systems with significant energy interactions, such as the Earth-Moon system, where the gravitational pull between the two bodies affects their orbits and movements. Another example is the proton-electron system, where the energy interactions between these particles determine the stability and properties of atoms and molecules.

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