What are L+ and L- matrices for l=3 ?

In summary, there is a formula for raising and lowering operators for l=3, and matrix elements can be calculated using the given formula.
  • #1
niloun
1
0
Hi everyone
I need raising and lowering operators for l=3 (so it should be 7 dimensional ).
is it enough to use this formula:
(J±)|j, m > =sqrt(j(j + 1) - m(m ± 1))|j, m ± 1 >
The main problem is about calculating lx=2 for a given wave function , I know L^2 and Lz but I need L+ and L- to solve the problem.
Thanks in advance
 
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  • #2
niloun said:
Hi everyone
I need raising and lowering operators for l=3 (so it should be 7 dimensional ).
is it enough to use this formula:
(J±)|j, m > =sqrt(j(j + 1) - m(m ± 1))|j, m ± 1 >
The main problem is about calculating lx=2 for a given wave function , I know L^2 and Lz but I need L+ and L- to solve the problem.
Thanks in advance
Yes, you can use that formula. Matrix elements for an operator ##\hat{A}## are simply given by
$$
A_{ij} = \langle i | \hat{A} | j \rangle
$$
 
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Related to What are L+ and L- matrices for l=3 ?

1) What is the purpose of L+ and L- matrices for l=3?

The L+ and L- matrices for l=3 are used in quantum mechanics to represent the angular momentum operators for a spin 3/2 system. They are also used to calculate the eigenvalues and eigenstates of the system.

2) How are the L+ and L- matrices related to each other?

The L+ and L- matrices are related to each other through the commutation relation [L+, L-] = 2Lz, where Lz is the third component of the angular momentum operator. This relation allows for the calculation of the eigenvalues and eigenstates of the system.

3) Can the L+ and L- matrices be used for other values of l?

Yes, the L+ and L- matrices can be used for any value of l, as long as they are properly normalized. However, they are most commonly used for l=3 in the context of spin 3/2 systems.

4) What is the physical significance of the L+ and L- matrices?

The L+ and L- matrices represent the operators for increasing and decreasing the angular momentum of a spin 3/2 system. They are important in understanding the behavior of particles with spin, such as electrons, in quantum mechanics.

5) How are the L+ and L- matrices represented mathematically?

The L+ and L- matrices are represented as square matrices, with dimensions (2l+1) x (2l+1), where l is the spin quantum number. They are typically written in terms of ladder operators, which are used to raise and lower the angular momentum states of the system.

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