Kronecker delta by using creation/annihilation operators

In summary, the conversation discusses the use of the Kronecker delta and its role in the sum over i in the expression given. Faust90 expresses confusion about the sum not being over r_i and asks for more clarification. Mr-R explains that the sum is over the subscript i and not the imaginary number, and provides a website for further understanding. Faust90 clarifies their confusion and Mr-R responds with a suggestion to assume that the r_i are a set of positions. The conversation ends with Faust90 summarizing that the expression is a product of exponentials.
  • #1
20
0
Hey all,

i've found the following expression:

yeKNHjC.png


How do they get that? They somehow used the kronecker delta Sum_k exp(i k (m-n))=delta_mn. But in the expression above, they're summing over i and not over r_i??

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  • #2
Do you need more background or is the question not precise enough? :-)
 
  • #3
Hi Faust90

The index ##i## under the sum refers to the subscript ##i## under the ##r##. The other i is the imaginary number. The Kronecker delta gets rid of the exponential and thus the sum on ##i## anyway. Try this website and see if it clears any confusion you are having. http://www.physicspages.com/2014/11/09/discrete-fourier-transforms/
 
  • #4
Hi Mr-R,

thanks for your answer. Yes, but my problem is that the sum is not running over r_i but over i.
Let's assume the r_i are an set of positions, for example always the same position, i.e. r_i={1,1,1,1,1,1,...}. Then in the end, that's just a product
prod_n=0^\infity e^{i(k-q)}

 

1. What is the Kronecker delta symbol?

The Kronecker delta symbol, denoted as δ, is a mathematical function used in linear algebra to represent the identity matrix. It is commonly used in physics and engineering to simplify calculations involving matrices and vectors.

2. What are creation and annihilation operators?

Creation and annihilation operators are mathematical operators used in quantum mechanics to describe the creation and destruction of particles. They are represented by &hat;a and &hat;a, respectively, and act on quantum states to produce new states.

3. How are creation and annihilation operators related to the Kronecker delta?

The Kronecker delta can be expressed in terms of creation and annihilation operators as δij = ⟨0&hat;ai&hat;aj⟩, where ⟨0⟩ is the vacuum or zero-particle state. This relationship allows for the manipulation and simplification of equations involving the Kronecker delta.

4. What is the significance of the Kronecker delta in quantum mechanics?

The Kronecker delta plays a crucial role in quantum mechanics as it allows for the representation and manipulation of quantum states and operators. It also helps in simplifying calculations and making predictions about the behavior of quantum systems.

5. How is the Kronecker delta used in calculations involving spin states?

The Kronecker delta is used in calculations involving spin states to represent the orthogonality of spin states. In particular, it is used to calculate the expectation value of spin operators and to determine the probability of measuring a certain spin state in a given system.

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