# Spin = -1/2

Why there can exist some particles having spin = 1/2? I understand, the postive numbers but what do the negative number mean?

Also, I would like to ask why some photons can be detected when it comes to light, but photon is not detected when it comes to charge replusion? What is the difference between these two kinds of protons?

Cyosis
Homework Helper
A particle has spin 1/2 (non negative), but it can 'rotate' in two directions so one is called 1/2 and the opposite direction -1/2. As for the photon question, I'm not sure what you mean.

jtbell
Mentor
What is the difference between these two kinds of protons?

Fredrik
Staff Emeritus
Gold Member
There are two quantum numbers related to spin. The book I studied called them j and m. (j is usually called j, but m is sometimes called s or $\sigma$). j is one of the properties (along with mass and charge) that tell us what particle species we're dealing with (electrons, photons, etc.). m is one of the properties that define what state the particle is in.

j is always a non-negative integer or half-integer. (j=n/2 where n is an integer satisfying n≥0). m is also an integer or a half-integer. It satisfies -j ≤ m ≤ j. It can only be changed in integer steps, and j is always one of the possible values of m. So if j=1/2 (e.g. an electron), the possible results of a measurement of m are -1/2 or +1/2. If j=1 (e.g. a photon), the possible results are -1, 0 or 1.

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Why there can exist some particles having spin = 1/2? I understand, the postive numbers but what do the negative number mean?

Do you mean
$$j=-\frac{1}{2}$$
or
$$m=-\frac{1}{2}$$ ?

I know spin = 1/2 meaning that the particle rotates 720 degree than looks the same as the original one.

But, -1/2, what does that imply? j and m? What do they mean?

jtbell
Mentor
Note that intrinsic angular momentum ("spin") $\vec S$ is a vector: a quantity that has both magnitude and direction.

"spin 1/2" normally refers to the quantum number that's associated with the magnitude of $\vec S$. Most of my books call this quantum number s. Other books, and Fredrik and Mathematikawan, call it j.

$$S = \sqrt{s(s+1)} \hbar = \frac{\sqrt{3}}{2} \hbar$$

Be careful of notation here: Upper-case S is the magnitude of the vector $\vec S$. Lower-case s is the quantum number.

Where you're seeing "-1/2" it is surely referring to the quantum number that's associated with the component of $\vec S$ along a particular direction. Usually we call it the z-direction, so this component is called $S_z$. Most of my books call this quantum number $m_s$. Other books, and Fredrik and Mathematikawan, call it m.

$$S_z = m_s \hbar$$

When s = 1/2, $m_s$ can have the values -1/2 or +1/2, and $S_z$ correspondingly can have the values $- \hbar / 2$ or $+ \hbar / 2$.

When s = 1, $m_s$ can have the values -1, 0 or +1. In this case, $S = \sqrt{2} \hbar$ and $S_z$ can have the values $-\hbar$, 0 or $+\hbar$.

When s = 3/2, $m_s$ can have the values -3/2, -1/2, +1/2 or +3/2. I leave it to you to write the corresponding values of S and $S_z$.

When s = 2, $m_s$ can have the values -2, -1, 0, +1 or +2.

A positive value for $m_s$ means that the vector $\vec S$ points more or less in the +z direction. A negative value indicates that $\vec S$ points more or less in the -z direction.

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I think jtbell has done excellent job in explaining the notations.

Sorry that I have been looking the concept of spin from the mathematical view rather than from the physical view. The spin quantum number j or s is just a label for the irreducible representation of su(2). So it can be positive or negative integer or half-integer value, as long as we can construct the representation.

So when I saw the title of the thread Spin = -1/2, hei may be this forum can enlighten me something. There are speculation that there may be such thing as physical negative spin j. I have came across the following papers (there may be others)

1. Andre van Tonder, Ghosts as Negative Spinors, Nuc. Phys. B 645(2002) pp 371-386.
2. Keshav N.Shrivastava, Negative-spin Quasiparticles in Quantum Hall Effect, Physics Letters A 326(2004) pp 469-472.

I manage to understand only a little bit from those papers. I even started a thread at this forum to make a sense from those papers.