Problem with spin interprestation related to matrics

[Cyrus_101]

context: teacher introduced sigma matrics, which have eigenvectors "a", to calculate the proability the spin switching state from "b" to "a" P=<a|b><a|b>conjugate

qustion: |b>= (1
0) ===>a vector, old state


and the sigma matric for example can be (0 1
1 0)===> which the eigenvector would be "a"
then the teacher said P=<a|b><a|b>conjugate will meature the proability the spin going from "y" axis to "x" axis. Matric are operators in linear transformation, My perception of matrics is that it does transform things in real space...but it does not indicate cordinate information...how come that the matrics indicate the spin state will switch to "x" axix in real space? appreciate for help!

 

[eljose79]

I can understand your confusion about how matrices can indicate a spin state switching in real space. Let me explain it in more detail.

First, let's define what a matrix is. A matrix is a rectangular array of numbers or symbols that can represent a linear transformation. In the context of quantum mechanics, matrices are used to represent operators that act on quantum states.

In this case, the sigma matrix that your teacher introduced is a representation of the spin operator. The spin operator is a mathematical object that describes the spin of a particle. It has different eigenstates, which are represented by the eigenvectors "a" in your example.

Now, let's look at the expression P=<a|b><a|b>conjugate. This is known as the transition probability, which represents the probability of a quantum state transitioning from state |b> to state |a>. The conjugate in this expression simply means that we are taking the complex conjugate of the expression.

So, how does this relate to the spin state switching from the "y" axis to the "x" axis? The sigma matrix is a representation of the spin operator, which acts on the spin state of the particle. By using the transition probability expression, we can calculate the probability of the spin state transitioning from |b> to |a>. In this case, the eigenvector "a" represents the state along the "x" axis, and the eigenvector "b" represents the state along the "y" axis. Therefore, the expression P=<a|b><a|b>conjugate is telling us the probability of the spin state transitioning from the "y" axis to the "x" axis.

In summary, matrices in quantum mechanics can represent operators that act on quantum states. In this case, the sigma matrix is a representation of the spin operator, and the expression P=<a|b><a|b>conjugate is used to calculate the probability of a spin state transitioning from one state to another. I hope this helps clarify your understanding.

 

[denian]

I understand your confusion about the interpretation of spin in relation to matrices. Matrices are indeed mathematical operators that represent linear transformations and do not have direct physical meaning in terms of coordinates in real space. However, in quantum mechanics, matrices are used to represent observable quantities, such as spin, and their corresponding probabilities.

In the context you provided, the teacher introduced the sigma matrix, which has eigenvectors "a", to calculate the probability of spin switching from "b" to "a". This is a common approach in quantum mechanics, where matrices are used to represent the different possible states of a system and their probabilities.

In this case, the sigma matrix represents the observable quantity of spin, and its eigenvectors represent the different possible states of spin. The calculation of P=<a|b><a|b>conjugate is a mathematical representation of the probability of the spin switching from "b" to "a", which is a valid interpretation within the framework of quantum mechanics.

I understand that this may be a difficult concept to grasp, but it is important to remember that quantum mechanics operates on a different level than classical mechanics and may require a different way of thinking. I would suggest discussing your concerns with your teacher or seeking additional resources to further your understanding of this topic.