Spin-1/2 particle in a 1D box

See also: the infinite square well.
Consider the QM formulation of a particle in a 1-dimensional box $[0, L]$ . We have a state vector $| Ψ ⟩$ , that can be formally written in the basis of "pure positions" $| x ⟩$ as

| Ψ ⟩ = \int_{0}^{L} ψ (x) | x ⟩ d x .

The function $ψ (x)$ is telling to us the probability amplitude of the particle being located at $x$ .
Consider now that we add a new "continuous" degree of freedom, as another spatial dimension $y$ . Then we would have

| Ψ ⟩ = \int_{R} ψ (x, y) | x, y ⟩ d x d y,

where $R$ is a rectangle in $(x, y)$ .
But what if the new degree of freedom is discrete. For example, if the particle has half integer spin. Quantum mechanics naturally accommodates both continuous and discrete degrees of freedom within the formalism of Hilbert spaces. When a system has both a continuous variable $x$ and a discrete variable $s$ , the total Hilbert space is the tensor product of the corresponding Hilbert spaces for each degree of freedom.

The position degree of freedom $x$ lives in the Hilbert space $H_{x} = L^{2} ([0, L])$ , the space of square-integrable functions on the interval $[0, L]$ .
The discrete degree of freedom $s$ takes values in a finite set, here $s \in {- 1, 1}$ . This corresponds to a finite-dimensional Hilbert space $H_{s} = C^{2}$ , which is typically the space of a two-level system (such as spin- $\frac{1}{2}$ particles).
Thus, the total Hilbert space of the system is:

H = L^{2} ([0, L]) \otimes C^{2} .

This means that a general state vector $| Ψ ⟩$ in this space is written as:

| Ψ ⟩ = \sum_{s = - 1, 1} \int_{0}^{L} ψ_{s} (x) | x, s ⟩ d x,

where $ψ_{s} (x)$ are functions associated with each discrete value of $s$ , and $| x, s ⟩ = | x ⟩ \otimes | s ⟩$ forms a basis.

Wavefunction Representation

Since the total state is expanded in terms of the basis states $| x, s ⟩$ , we can define the wavefunction representation as:

ψ (x, s) = ⟨ x, s | Ψ ⟩ .

This can be interpreted as if the wavefunction now has two components, one for each value of $s$ :

ψ (x, - 1), ψ (x, 1) .

So, instead of a single wavefunction $ψ (x)$ , we now have a two-component wavefunction, often written as a column vector:

Ψ (x) = (\begin{matrix} ψ_{- 1} (x) \\ ψ_{1} (x) \end{matrix}) .

For example, if we consider a particle in a 1D box with an internal spin degree of freedom, the wavefunction satisfies:

Ψ (x) = (\begin{matrix} ψ_{- 1} (x) \\ ψ_{1} (x) \end{matrix}) .

This is precisely what it is called, in some contexts, an spinor. So all the stuff about spinors is, indeed, encoding the fact that the domain of the wavefunction of a particle with spin-1/2 is, in a sense, "semidiscrete".