How Do You Apply the Spin Operator S² to the Product of Two Spin Functions?

[greisen]

Hi,

I am looking at the product of two spin function (a_1,a_2) where I would like to apply the spin operator S² to them.

1. To express the S² operator I have seen different expressions either expressed as linear combinations of Pauli matrices or expressed as follows
S²=S_1² + S_2²+2*S_z1*S_z2 + S_+S_- + S_-S_+
using the lowering and raising operator as well. So using S² on a single spin function I get

S²|a> = \hbar^2*3/4|a>

my problem how to do it on the product of two spin functions (a_1,a_2)

Any help or advise appreciated.
Thanks in advance

 

[Nino MMP]

.2. To apply the S² operator to a product of two spin functions (a_1,a_2), you will need to first expand your product in terms of the tensor product. The tensor product is defined as: A ⊗ B = A x B where A and B are two vectors in a vector space. So, for example, if your two spin functions are |a_1> and |a_2>, then the expansion would be: |a_1> ⊗ |a_2> = |a_1> x |a_2> = |a_1a_2> Once you have expanded the product, you can then apply the S² operator to it by using the same expressions for S² as mentioned in 1. For example, S²|a_1a_2> = \hbar^2*3/4|a_1a_2>. I hope this helps.

 

[Blueshift5]

Hello,

Thank you for your question. The product of two spin functions can be expressed as follows:

(a_1,a_2) = (a_1)(a_2)

To apply the spin operator S² to this product, you can use the following formula:

S²(a_1,a_2) = S²(a_1)(a_2) = (S²(a_1))(a_2)

This means that you first apply the S² operator to one of the spin functions (a_1) and then multiply the result by the other spin function (a_2). This can also be written as:

S²(a_1,a_2) = (S²(a_1))(S²(a_2))

Now, to calculate the result, you can use the expressions for S² that you have mentioned in your question. For example, if you use the expression S² = S_1² + S_2²+2*S_z1*S_z2 + S_+S_- + S_-S_+, you can calculate S²(a_1) and S²(a_2) separately and then multiply them together to get the final result for S²(a_1,a_2).

I hope this helps. Let me know if you have any further questions. Good luck with your research!