How many degrees of freedom for the spin 1/2?

I count four but most of the documentation I read, says just one.

1) One degree of freedom for the spin measurement outcome, either up or down.

3) three degrees of freedom for the unit quaternion representing the orientation of the spin in space.

Is this correct, or have I made a mistake?

• Number of components (degrees of freedom) for a particle with spin j = 2j+1. Put j=1/2 – Avantgarde May 13 '17 at 16:51
• @Avantgarde But then, how do you explain that the spin can be oriented differently in space? – Alexandre H. Tremblay May 13 '17 at 16:53
• Yes, spin can be oriented in an arbitrary direction, say, z direction. Now, the number of independent states that you can get in that one z direction, is 2j+1. For a spin half particle, for example, you get two independent states: $m=+\frac{1}{2}$ and $m=-\frac{1}{2}$. Coordinate system is just a labelling system. What we really want to know (degrees of freedom) comes about from calculating the number of independent bases of the underlying abstract vector space of the state – Avantgarde May 13 '17 at 16:59
• @Avantgarde Consider the case of a system with two spins. The orientation of the first one defines the axis. Then the second spin can be oriented in any direction. Surely, the additional rotational possibilities of the second spin are added as additional degrees of freedom to the system? A system where each spins are equi-oriented must be different that one where each spins have a different orientation? In other words, you need more independent numbers to define differently orientated spins, then equi-oriented ones? – Alexandre H. Tremblay May 13 '17 at 17:07
• But how can one system have two spins? For a state, $J^2$ and $J_z$ commute, and therefore characterize the state completely. Are you talking about adding the momenta of 2 separate systems and obtaining the momentum of the resulting composite system? – Avantgarde May 13 '17 at 17:15

Using the strict definition for 'number of degrees of freedom' you've given in the comments,

the quantity of real numbers required to describe the system,

then a spin-1/2 particle has two degrees of freedom if it is in a pure state. The general state of such a particle can be written as $$|\psi⟩ = \cos\left(\frac\theta2\right)\left|↑ \right>+e^{i\varphi}\sin\left(\frac\theta2\right)\left|↓\right>,$$ where $\left|↑ \right>$ and $\left|↓\right>$ are eigenstates of the $z$ component of the spin vector. Here the state is parametrized in terms of the two real parameters $\theta$ and $\varphi$, which essentially parametrize the Bloch sphere - the natural manifold representation of the state space of the particle.

More importantly, this number is derived as the manifold dimension of $\mathbb C\mathrm P^2$, the complex projective space of dimension $2$, which is essentially the complex vector space $\mathbb C^2$ (the two-complex-dimensional vector space of all the complex linear combinations of $\left|↑ \right>$ and $\left|↓\right>$, so four real dimensions), with one taken out by the fixed normalization and one taken out by the irrelevance of the global phase. More generally, a particle of spin $j$ has a $(2j+1)$-complex-dimensional state space, and therefore can be described using $4j$ real numbers.

That said, the definition you've given is essentially meaningless, and it is essentially incapable of describing systems as simple as a massive spinless particle on a 1D line; there the Hilbert state space is infinite-dimensional (spanned e.g. by the harmonic-oscillator basis) and you absolutely need an infinite number of real numbers to describe any state of the system. Moreover, the same is true in Liouvillian classical mechanics for the same situation.

What that really means is that the term "degree of freedom" is not a particularly useful concept in a quantum mechanical situation, particularly where you might have a finite-dimensional system coupled to an infinite-dimensional one (as you might e.g. in the hugely popular Jaynes-Cummings model and its plethora of generalizations). In the situations where you do need to keep track of such things, you end up needing to specify the types of systems (i.e. the dimension of their state spaces) and their number, and there just isn't a single bag where you can chuck everything in and get a meaningful number.

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