Understanding Spin: A Guide to Particles & Annihilation

[Silviu]

Hello! I am a bit confused about the idea of spin. Let's say we have spin $1/2$ particles. This means that the spin along a random axis is $\pm 1/2$, right while the values of the particle spin (i.e. the module) is actually $\sqrt{1/2(1+1/2)}$? Also I am a bit confused about combining multiple particles. If you have 2 spin $1/2$ particles, you obtain a triplet of spin $1$ or a singlet of spin $0$. I understand the math behind, but how does it physically work? You don't know the initial directions of the spins. Let's say one has $1/2$ along the z axis while the other has $1/2$ along the x-axis (as the result of a measurement). Does it mean that if i make a random measurement after, I can get +1, -1 or 0 along any axis? And what exactly do I measure i.e. how can I measure the combined spin of both at once? And based on what do I get a singlet or a triplet? Also when we have particle-antiparticle annihilation, let say $e^+e^-$, they have to be in the triplet case, so that the spin is conserved, i.e. photon with spin one can be created. I am confused about this, too, as you don't know the initial direction of the spin of the 2 fermions, neither of the outgoing photon, so on what direction do you want this conservation to take place?

 

[jedishrfu]

Try watching these videos:

Veritaseum's:

Susskind:

 

[DrClaude]

If you have 2 spin $1/2$ particles, you obtain a triplet of spin $1$ or a singlet of spin $0$.

These would correspond to the eigenstates of the total spin operator $\hat{\mathbf{J}}$ and one projection operator, e.g., $\hat{J}_z$, but the two particles are not necessarily in such an eigenstate (although their state can be written as a superposition of these eigenstates).

I understand the math behind, but how does it physically work? You don't know the initial directions of the spins. Let's say one has $1/2$ along the z axis while the other has $1/2$ along the x-axis (as the result of a measurement). Does it mean that if i make a random measurement after, I can get +1, -1 or 0 along any axis?

Being in $s_x = +1/2$ is the same as a superposition of $s_z = +1/2$ and $s_z = -1/2$, so the total spin state will look like (choosing z as the projection axis)
$$
\frac{1}{\sqrt{2}} \left[ | \uparrow \uparrow \rangle + | \uparrow \downarrow \rangle \right]
$$
From that, you can calculate the probability of measuring the different states. (You will find, for instance, that the probabilities of measuring $S_z = -1$ or $S_x = -1$ are 0).

And what exactly do I measure i.e. how can I measure the combined spin of both at once?

I don't know how to perform a direct measurement, but if the two particles are electrons in a single atom, one can look at transitions to determine the state.

And based on what do I get a singlet or a triplet?

I don't understand what you are asking for here.

Also when we have particle-antiparticle annihilation, let say $e^+e^-$, they have to be in the triplet case, so that the spin is conserved, i.e. photon with spin one can be created. I am confused about this, too, as you don't know the initial direction of the spin of the 2 fermions, neither of the outgoing photon, so on what direction do you want this conservation to take place?

The magnitude of the spin has to be conserved, which explains the triplet state.

 

[atyy]

For an example of angular momentum addition and the use of Clebsch-Gordon coefficients, you can look at the Zeeman Effect.

 

[Silviu]

These would correspond to the eigenstates of the total spin operator $\hat{\mathbf{J}}$ and one projection operator, e.g., $\hat{J}_z$, but the two particles are not necessarily in such an eigenstate (although their state can be written as a superposition of these eigenstates).Being in $s_x = +1/2$ is the same as a superposition of $s_z = +1/2$ and $s_z = -1/2$, so the total spin state will look like (choosing z as the projection axis)
$$
\frac{1}{\sqrt{2}} \left[ | \uparrow \uparrow \rangle + | \uparrow \downarrow \rangle \right]
$$
From that, you can calculate the probability of measuring the different states. (You will find, for instance, that the probabilities of measuring $S_z = -1$ or $S_x = -1$ are 0).I don't know how to perform a direct measurement, but if the two particles are electrons in a single atom, one can look at transitions to determine the state.I don't understand what you are asking for here.The magnitude of the spin has to be conserved, which explains the triplet state.

Thank you for this. I am not sure I understand the last part. The magnitude of the spin along which axis?

 

[stevendaryl]

Thank you for this. I am not sure I understand the last part. The magnitude of the spin along which axis?

The magnitude of the spin is $S = \sqrt{(S_x)^2 + (S_y)^2 + (S_z)^2}$. In quantum mechanics, this magnitude always is equal to $\sqrt{s(s+1)} \hbar$ where $s$ has possible values: $0, 1/2, 1, 3/2, 2, 5/2,...$

 

[Stephanie Dillon]

Why does the magnitude of the spin have to be preserved? What if the parameters for measurement change instead? Use the parameters of the world of the sap. Not the parameters of the large laws. Ie. travel speed, time change etc.

 

[jedishrfu]

Why does the magnitude of the spin have to be preserved? What if the parameters for measurement change instead? Use the parameters of the world of the sap. Not the parameters of the large laws. Ie. travel speed, time change etc.

Can you clarify what you mean here? World of the “sap”?

 

[bhobba]

Why does the magnitude of the spin have to be preserved? What if the parameters for measurement change instead? Use the parameters of the world of the sap. Not the parameters of the large laws. Ie. travel speed, time change etc.

One word Noether:
https://arstechnica.com/science/201...-the-course-of-physics-but-couldnt-get-a-job/

Its related to the laws of physics being the same is all directions. This leads to angular momentum being conserved - QM spin, without going into the details (see chapter 7 Ballentine - QM - A Modern Development - but its advanced) is a kind of quantized angular momentum.

I do not understand what you mean by parameters of the measurement changing etc - if the above doesn't answer you query then can you elaborate?

Thanks
Bill