# Pauli spin matrices, operating on |+> with Sx

• L-x
In summary, operating on the state |+> with the operator Sx will result in a superposition of states |+x> and |-x>, with equal probabilities of finding the particle in either state. The confusion may arise from not distinguishing between operating with a linear operator and measuring an observable.
L-x

## Homework Statement

What is the result of operating on the state |+> with the operator Sx?

here, |+> denotes the eigenstate of Sz with eigenvalue 1/2. I am working in units where h-bar is 1 (for simplicity, and because I don't know how to type it)

## Homework Equations

$$S_i = \frac{1}{2} σ_i$$

## The Attempt at a Solution

My understanding of the physics of the problem is that after measurement the system will be in state where the x component of its spin is certain, as [Sx,Sz] != 0 this means it will be in some superposition of the states |+> and |-> My intuition tells me that the probability for the particle to be found in either state should be equal.

However, operating with the matrix representation of Sx :$$\frac{1}{2} \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$$
on |+> just gives (1/2)|->

What am i doing wrong?

I think I understand where I've gone wrong now. The problem is in my intuition, not in my appilcation of the pauli matrices.

as $$|+> = \frac{1}{\sqrt{2}}(|+_x> + |-_x>)$$
(where $$|+_x> and |-_x>$$ are the eigenvectors of S_x with eigenvalues +/- 1/2)

when we operate with Sx we obtain $$(2^{-\frac{1}{2}})(\frac{1}{2}|+_x> - \frac{1}{2}|-_x>)$$ which is equal to $$\frac{1}{2}|->$$. The confusion I had was due to a faliure to distinguish between operation with a linear operator and measurement of an ovservable.

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Yes, your reasoning is correct, and you have the correct result.

## 1. What are Pauli spin matrices?

Pauli spin matrices are a set of three mathematical matrices, named after physicist Wolfgang Pauli, that describe the spin states of particles. They are commonly used in quantum mechanics to represent spin operators, which are used to describe the angular momentum of particles.

## 2. How do Pauli spin matrices operate on a spin state |+>?

When a Pauli spin matrix, such as Sx, operates on a spin state |+>, it causes the state to "flip" to the opposite spin state, in this case |->. This means that the spin of the particle is reversed in the direction of the x-axis.

## 3. What is the significance of using |+> and Sx in relation to Pauli spin matrices?

|+> represents the spin state of a particle aligned with a particular axis, in this case the positive x-axis. Sx is the spin operator that describes the spin of the particle in the x direction. By operating on |+> with Sx, we can observe the effects of the spin operator on the spin state of the particle.

## 4. Are Pauli spin matrices only applicable to spin states on the x-axis?

No, Pauli spin matrices can be used to describe spin states in any direction. There are three matrices, Sx, Sy, and Sz, which correspond to spin operators in the x, y, and z directions respectively. These matrices can be used to operate on spin states in any direction.

## 5. How are Pauli spin matrices used in quantum computing?

In quantum computing, Pauli spin matrices are used to represent quantum gates, which are the building blocks of quantum algorithms. These matrices allow for the manipulation of qubit states, which are the quantum analog of classical bits. Pauli spin matrices are also used in quantum error correction codes to detect and correct errors in qubit states.

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