# Spin-1/2 Particle Transformation: Spin-Z to Spin-X

• RedX
In summary, the conversation discusses a question about the transformation matrix from the spin-z basis to the spin-x basis for spin-1/2 particles. The matrix provided by one person includes phase factors and was marked wrong, with the correct answer being the same matrix with all angles set to zero. It is noted that choosing different phase factors does not affect the physical results. The conversation also mentions that the matrix provided does not work for all calculations and that the phase factor is important in converting the Pauli matrices from the z-basis to the x-basis.
RedX
I got a quick question about the transformation matrix from the spin-z basis to the spin-x basis for spin-1/2 particles.

Would the matrix be:

$$\left(\begin{array}{ccc} \frac{e^{i\theta}}{\sqrt{2}} &\frac{e^{i\delta}}{\sqrt{2}} \\ \frac{e^{i\theta}}{\sqrt{2}} & -\frac{e^{i\delta}}{\sqrt{2}} \end{array}\right)$$

I put that down as an answer and got it marked wrong, and what the grader wrote down as the answer is what you get when you set all angles in the matrix to zero.

Do you get different physical results if you choose different phase factors?

Did you try your matrix out and see what you get?

If you do, you'll find that it only works for $\delta=\theta$, which then just becomes an overall phase for the entire matrix. An overall phase has no physical relevance.

Avodyne said:
Did you try your matrix out and see what you get?

If you do, you'll find that it only works for $\delta=\theta$, which then just becomes an overall phase for the entire matrix. An overall phase has no physical relevance.

Yeah I guess it doesn't work. The matrix I wrote down does diagonalize the spin operator in the x-direction (since the eigenvectors of that operator are in the columns of the transformation matrix, but with arbitrary phase factor which shouldn't matter), but the phase factor matters in calculations of other quantities.

Something that's interesting is that the Pauli matrices P are usually written in the z-basis. If you want to convert them to the x-basis via $$U^{\dagger}PU$$, then if you use the matrix I wrote down but with the angle in the first column equal to zero, and the second angle equal to $$\frac{-\pi}{2}$$, then the new set of matrices have the same entries as the old set of matrices. If instead you set all angles equal to 0, then you get 3 new matrices which don't look like the original Pauli matrices at all.

## 1. What is a spin-1/2 particle?

A spin-1/2 particle is a type of elementary particle, such as an electron or proton, that has a spin quantum number of 1/2. This means that the particle has an intrinsic angular momentum and can be described as either spinning clockwise or counterclockwise.

## 2. What is the difference between spin-Z and spin-X?

Spin-Z and spin-X refer to different orientations of a spin-1/2 particle's spin angular momentum. Spin-Z refers to the component of the spin angular momentum along the z-axis, while spin-X refers to the component along the x-axis.

## 3. How does a spin-1/2 particle transform from spin-Z to spin-X?

A spin-1/2 particle can transform from spin-Z to spin-X through a process called rotation. This involves applying a rotation operator to the particle's spin state, which changes the orientation of the spin angular momentum from the z-axis to the x-axis.

## 4. What are the applications of spin-1/2 particle transformation?

Spin-1/2 particle transformation has various applications in fields such as quantum computing and nuclear magnetic resonance spectroscopy. It allows for the manipulation and control of a particle's spin state, which can be used for information storage and processing.

## 5. How is spin-1/2 particle transformation related to quantum mechanics?

In quantum mechanics, particles are described by wavefunctions that contain information about their spin. Spin-1/2 particle transformation is a manifestation of the principles of quantum mechanics, as it involves changing the state of a particle's spin through the application of mathematical operators.

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