User:Maschen/Spin wave function

In physics, specifically quantum mechanics and relativistic quantum mechanics, a spin wave function is a wave function for a particle with spin. This article concentrates on the position and spin degrees of freedom a particle has, other variables can also be accounted for.

The concept of a wave function is a fundamental postulate of quantum mechanics, which can be formulated very generally in any suitable set of observable quantities. This article concentrates on wave functions for particles of any spin, in non relativistic and non relativistic quantum mechanics.

Elementary particles have the property of a "built-in" angular momentum, called spin. The wave function for a particle with spin can be expressed as a single complex number with dependence on space, time, and spin, or an array of complex numbers with dependence on space and time only. The space in either case may be referred to as "position-spin space".

The wave function has a different dependence on spin than it does for space and time. For one thing, all three position coordinates of the particle $r = (x, y, z)$ can be known simultaneously, but not all the components of the particle's spin vector $S = (S_{x}, S_{y}, S_{z})$ can be known simultaneously, only one component and the square of its magnitude $|S|^{2}$ can. (These facts follows from the commutation relations of the position and spin operators). This means all three position coordinates can be placed as variables in the wave function, but only one spin direction can be a variable. For the case of spin, we choose a direction and project the spin along that direction. A conventional, but arbitrary, choice is the z-direction, which will be considered in detail first. Other directions can be obtained from the z-component of spin if the wave function is transformed appropriately.

For another thing, unlike $r$ and $t$ which are continuous variables, spin is a discrete variable, for a particle of spin $s$ there are $2s + 1$ values along any direction. (The exception is in 2d space, where particles called anyons can have continuous spin, but these are not considered here).

Spin-1/2
The spin projection quantum number along the $z$ axis is denoted $s_{z}$, and for a particle with spin $s$ the allowed values are only $s, s − 1, ..., −s + 1, −s$, and no other values. For example, for a spin-1/2 particle, $s_{z}$ can only be the two values $+1/2$ or $−1/2$. The case of spin-1/2 will be exemplified below for simplicity, concreteness, and practical interest - all the leptons and quarks which constitute matter are elementary particles with spin-1/2.

As a single complex number, it is a linear combination of space functions and basis spin functions:


 * $$ \Psi(\mathbf{r},t,s_z) = \psi^z_{-1/2}(\mathbf{r},t)\xi^z_{-1/2}(s_z) + \psi^z_{1/2}(\mathbf{r},t)\xi^z_{1/2}(s_z) \,.$$

The space functions corresponding to individual, definite values of $s_{z}$ are $ψ_{−1/2}^{z}$ and $ψ_{1/2}^{z}$, which take in space coordinates and time and return a complex number. The basis spin functions $ξ_{−1/2}^{z}$ and $ξ_{1/2}^{z}$ take in the spin number and return a complex number.

Often, the complex values of the wave function for all the spin numbers are arranged into a column vector, in which there are as many entries in the column vector as there are allowed values of $s_{z}$. In this case, the spin dependence is placed in indexing the entries and the wave function is a complex vector-valued function of space and time only:


 * $$\Psi(\mathbf{r},t) = \begin{bmatrix} \Psi(\mathbf{r},t,1/2) \\ \Psi(\mathbf{r},t,-1/2) \end{bmatrix}$$

The connection between the "scalar-valued" and "vector-valued" wave functions is the following. As with all discrete observables in quantum mechanics, the eigenstate $Ψ(r, t)$ of the z-component spin operator can be expanded as a linear combination of the eigenvectors $χ_{s_{z}}^{z}|undefined$ of the operator, in other words the $χ_{s_{z}}^{z}|undefined$ form a basis and the functions $Ψ(r, t, s_{z})$ are the complex components of the vector:


 * $$\Psi(\mathbf{r},t) = \Psi(\mathbf{r},t,-1/2) \chi^z_{-1/2} + \Psi(\mathbf{r},t,1/2) \chi^z_{1/2}$$

where the eigenvectors have entries given by the spin basis functions above,


 * $$ \left[\chi_{s_z}\right]_{{s'}_z} = \xi_{s_z}({s'}_z) \,, $$

in full


 * $$\chi^z_{1/2} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} \xi^z_{1/2}(1/2) \\ \xi^z_{1/2}(-1/2) \end{bmatrix} $$
 * $$\chi^z_{-1/2} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} \xi^z_{-1/2}(1/2) \\ \xi^z_{-1/2}(-1/2) \end{bmatrix} $$

so for the case of the z-projection, the spin functions can be defined simply by the Kronecker delta,


 * $$\xi^z_{s_z}(s'_z) = \delta_{s_z s'_z}$$

Substitution of any allowed spin number yields the particular component of the entire wave function for that spin number. This motivates the notation:


 * $$\Psi(\mathbf{r},t,s_z) \equiv \psi_{s_z}(\mathbf{r},t)$$

which may be misleading since the spin number is a variable, not just an index.

The basis spin functions are different if the spin is projected along a different direction. In the direction of the spatial unit vector, using spherical coordinates $θ$ for the polar angle from the z-axis and $φ$ for the azimuthal angle in the xy-plane from the x-axis:


 * $$\hat{\mathbf{n}} = \sin\theta(\cos\phi\mathbf{e}_x + \sin\phi\mathbf{e}_y) + \cos\theta\mathbf{e}_z$$

the wave function in scalar form reads


 * $$ \Psi(\mathbf{r},t,s_\hat{\mathbf{n}}) = \psi^\hat{\mathbf{n}}_{-1/2}(\mathbf{r},t)\xi^\hat{\mathbf{n}}_{-1/2}(s_\hat{\mathbf{n}}) + \psi^\hat{\mathbf{n}}_{1/2}(\mathbf{r},t)\xi^\hat{\mathbf{n}}_{1/2}(s_\hat{\mathbf{n}}) \,.$$

or arranging the components into the column vector form,


 * $$\Psi(\mathbf{r},t) = \begin{bmatrix} \Psi(\mathbf{r},t,1/2) \\ \Psi(\mathbf{r},t,-1/2) \end{bmatrix}$$

expanding this as a linear combination of the eigenvectors of the $n$-component spin operator:


 * $$\Psi(\mathbf{r},t) = \Psi(\mathbf{r},t,-1/2) \chi^\hat{\mathbf{n}}_{-1/2} + \Psi(\mathbf{r},t,1/2) \chi^\hat{\mathbf{n}}_{1/2}$$

where


 * $$\chi^\hat{\mathbf{n}}_{1/2} = \begin{bmatrix} \cos(\theta/2) \\ e^{i\phi}\sin(\theta/2) \end{bmatrix} = \begin{bmatrix} \xi^\hat{\mathbf{n}}_{1/2}(1/2) \\ \xi^\hat{\mathbf{n}}_{1/2}(-1/2) \end{bmatrix} $$
 * $$\chi^\hat{\mathbf{n}}_{-1/2} = \begin{bmatrix} -e^{-i\phi}\sin(\theta/2) \\ \cos(\theta/2) \end{bmatrix} = \begin{bmatrix} \xi^\hat{\mathbf{n}}_{-1/2}(1/2) \\ \xi^\hat{\mathbf{n}}_{-1/2}(-1/2) \end{bmatrix} $$

and the spin functions depend on the angles,


 * $$\xi^\hat{\mathbf{n}}_{\pm 1/2}(\pm 1/2) = \cos(\theta/2) \,,\quad \xi^\hat{\mathbf{n}}_{\pm 1/2}(\mp 1/2) = \pm e^{\pm i\phi} \sin(\theta/2) \,. $$

The relation from the z-projection basis $χ_{s_{z}}^{z}|undefined$ to the $n$-projection basis $χ_{s_{n}}^{n}|undefined$ is a change of basis.

Higher spin
The extension to the general case of a particle with higher spin is straightforward in principle, but finding the basis spin functions, for arbitrary spin, is nontrivial for directions other than the z-axis.

Spin operator (for reference)
To determine the spin functions and components of the spin operator for any $s$, first neglect the spatial dependence of the wave function (the dependence on position can be introduced later), so that the wave function depends on the spin only. Write the action of the spin operator on the wave function generally as


 * $$\varphi(\sigma) = \sum_{\sigma'=-s}^s\hat{S}_{\sigma\sigma'}\psi(\sigma') $$

where $φ$ is the new function obtained from the action of the spin operator on $ψ$, and $S_{σσ&prime;}$ are constant complex numbers which make up the spin operator. The indices $σ,σ&prime;$ each take the allowed spin values ($−s, −s + 1, ..., s − 1, s$). This is a general way to write an operator on a function of a discrete variable, and is easily put into a matrix form


 * $$\begin{bmatrix}

\varphi(s) \\ \varphi(s-1) \\ \vdots \\ \varphi(-s+1)\\ \varphi(-s)\\ \end{bmatrix} = \begin{bmatrix} S_{s,s} & S_{s,s-1} & \cdots & S_{s,-s+1} & S_{s,-s} \\ S_{s-1,s} & S_{s-1,s-1} & \cdots & S_{s-1,-s+1} & S_{s-1,-s} \\ \vdots \\ S_{-s+1,s} & S_{-s+1,s-1} & \cdots & S_{-s+1,-s+1} & S_{-s+1,-s} \\ S_{-s,s} & S_{-s,s-1} & \cdots & S_{-s,-s+1} & S_{-s,-s} \\ \end{bmatrix}\begin{bmatrix} \psi(s) \\ \psi(s-1) \\ \vdots \\ \psi(-s+1)\\ \psi(-s)\\ \end{bmatrix}$$

Using the common convention, taking the z direction to be the direction of measuring the spin along, and introducing the spin ladder operators which are defined to obtain the spin states with higher or lower spin quantum numbers,


 * $$\hat{S}_{\pm} = \hat{S}_x \pm i \hat{S}_y $$

the nonzero elements for the Cartesian components of the spin operator are


 * $$(\hat{S}_x)_{\sigma,\sigma-1} = (\hat{S}_x)_{\sigma-1,\sigma} = \frac{1}{2}\sqrt{(s-\sigma)(s+\sigma+1)}$$
 * $$(\hat{S}_y)_{\sigma,\sigma-1} = - (\hat{S}_y)_{\sigma-1,\sigma} = -\frac{i}{2}\sqrt{(s+\sigma)(s-\sigma+1)}$$
 * $$(\hat{S}_z)_{\sigma,\sigma} = \sigma $$

or explicitly as $(2s + 1)×(2s + 1)$ matrices


 * $$\hat{S}_x = \begin{bmatrix}

0 & (S_x)_{s,s-1} & 0 & \cdots & 0 & 0 & 0 \\ (S_x)_{s-1,s} & 0 & (S_x)_{s-1,s-2} & \cdots & 0 & 0 & 0 \\ 0 & (S_x)_{s-2,s-1} & 0 & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 0 & (S_x)_{-s+2,-s+1} & 0 \\ 0 & 0 & 0 & \cdots & (S_x)_{-s+1,-s+2} & 0 & (S_x)_{-s+1,-s} \\ 0 & 0 & 0 & \cdots & 0 & (S_x)_{-s,-s+1} & 0 \end{bmatrix} $$


 * $$\hat{S}_y = \begin{bmatrix}

0 & (S_y)_{s,s-1} & 0 & \cdots & 0 & 0 & 0 \\ -(S_y)_{s-1,s} & 0 & (S_y)_{s-1,s-2} & \cdots & 0 & 0 & 0 \\ 0 & -(S_y)_{s-2,s-1} & 0 & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 0 & (S_y)_{-s+2,-s+1} & 0 \\ 0 & 0 & 0 & \cdots & -(S_y)_{-s+1,-s+2} & 0 & (S_y)_{-s+1,-s} \\ 0 & 0 & 0 & \cdots & 0 & -(S_y)_{-s,-s+1} & 0 \end{bmatrix} $$


 * $$\hat{S}_z = \begin{bmatrix}

s & 0 & 0 & \cdots & 0 & 0 & 0 \\ 0 & s-1 & 0 & \cdots & 0 & 0 & 0 \\ 0 & 0 & s-2 & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & -s+2 & 0 & 0 \\ 0 & 0 & 0 & \cdots & 0 & -s+1 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 0 & -s \end{bmatrix} $$

Then the spin operator for an arbitrary direction is easily constructed out of these and the unit direction vector n in spherical coordinates


 * $$\hat{S}_\hat{\mathbf{n}} = \hat{\mathbf{n}}\cdot\hat{\mathbf{S}} = \sin\theta\cos\phi \hat{S}_x + \sin\theta\sin\phi \hat{S}_y + \cos\theta \hat{S}_z$$

Eigenvectors and spin functions
For the z-direction, as for the spin-1/2 case, the $2s + 1$ components of the wave function can be expressed as a single complex number as a function of position, time, and spin


 * $$ \Psi(\mathbf{r},t,\sigma) = \sum_{\sigma'=-s}^s \psi_{\sigma'}(\mathbf{r},t)\xi^z_{\sigma'}(\sigma) \,, $$

where the spin functions $ξ$ are functions of spin only.

Each $Ψ(r, t, σ)$ can be arranged into a column vector, and the spin dependence is placed in indexing the entries and the wave function is a complex vector-valued function of space and time only,


 * $$\underline{\Psi}(\mathbf{r},t) = \begin{bmatrix} \Psi(\mathbf{r},t,s) \\ \Psi(\mathbf{r},t,s-1) \\ \vdots \\ \Psi(\mathbf{r},t,-(s-1)) \\ \Psi(\mathbf{r},t,-s) \\ \end{bmatrix}\,. $$

Expanding as a linear combination of the eigenvectors of the z-component spin operator,


 * $$\underline{\Psi}(\mathbf{r},t) = \sum_{\sigma' =-s}^s \Psi(\mathbf{r},t,\sigma') \underline{\chi}_{\sigma'}^z $$

where the eigenvectors are explicitly


 * $$ \underline{\chi}_s^z = \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \\ 0 \\ \end{bmatrix} \,,\quad \underline{\chi_{s-1}^z} = \begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \\ 0 \\ \end{bmatrix} \,,\quad \ldots \,, \underline{\chi_{-(s-1)}^z} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \\ 0 \\ \end{bmatrix} \,,\quad \underline{\chi_{-s}^z} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \\ \end{bmatrix} \,,$$

The action of the z-component spin operator on these eigenvectors $χ_{σ}^{z}$ yields the corresponding eigenvalues $ħσ$. By inspection, the entries (spin basis functions) of the column $χ_{σ&prime;}^{z}$ are just Kronecker deltas,


 * $$ \underline{\chi}_{\sigma'}^z = \begin{bmatrix} \xi_{\sigma'}^z(s) \\ \xi_{\sigma'}^z(s-1) \\ \vdots \\ \xi_{\sigma'}^z(-(s-1)) \\ \xi_{\sigma'}^z(-s) \\ \end{bmatrix}\quad \leftrightarrow \quad [\underline{\chi_{\sigma'}^z}]_\sigma = \xi_{\sigma'}^z(\sigma) = \delta_{\sigma' \sigma} \,. $$

The process can be repeated for spin operator components along the x, y, or $n$ directions, although the spin eigenvectors and basis spin functions are less trivial to determine in the general case for any spin.

Factorization
In some situations, the wave function factors into a product of a space function and a spin function, and the time dependence could be placed in either function:


 * $$\Psi(\mathbf{r},t,s_z) = \psi(\mathbf{r},t)\xi(s_z) = \phi(\mathbf{r})\zeta(s_z, t)\,.$$

The dynamics of each factor can be studied in isolation. This factorization is always possible for potentials or interactions which do not depend on the spin of the particle, the simplest case is the free particle. This is not possible for certain interactions, when an external field or any space-dependent quantity couples to the spin. Mathematically, this may appear as the dot product of the field or quantity with the spin operator in the Hamiltonian operator of the Schrödinger equation. For example, a particle in a magnetic field $B$ is influenced by the field because of the magnetic moment corresponding to the spin, the interaction term is $B · S$. Another example is spin-orbit coupling, the orbital angular momentum $L$ couples to the spin in the term $L · S$. These terms prevent factorization because the position coordinates are mixed into the spin operators, which are matrices that multiply the column matrix wave functions above.

Spin-1/2 particle
A wave function for one spin-1/2 particle above


 * $$\psi = \begin{pmatrix}\psi^\uparrow(\mathbf{r},t) \\ \psi^\downarrow(\mathbf{r},t) \end{pmatrix}$$

is a spinor, where $ψ^{↑}$ corresponds to spin +1/2 along some direction (for concreteness the z-direction as standard), and $ψ^{↓}$ to spin −1/2 along the same direction. The complex numbers $ψ^{↑}$ and $ψ^{↓}$ are the components of the spinor.

The wavefunction transforms according to


 * $$\psi'=U\psi$$

where U is the transformation matrix, containing parameters of the transformation (for concreteness, they may be the Euler angles of a rotation of coordinate axes, in relativistic quantum mechanics not considered here, the transformation may be a Lorentz boost).

For any spin-1/2 wave functions, tensor index notation including the summation convention carries over to spinor index notation. For example, $ψ^{α}$ corresponds to the two components. Actually, the index should take numerical values and we should denote $α = 1, 2$ or $α = 0, 1$, but to account for either convention and make the link between spin and index clear, here we denote $α = ↑, ↓$. Notational conventions in summary are


 * {| class="wikitable"

! Example of usage ! Spin up index ! Spin down index
 * Example
 * Example
 * 1
 * 2
 * Example
 * 1
 * 0
 * Example
 * 1/2
 * −1/2
 * }
 * 0
 * Example
 * 1/2
 * −1/2
 * }
 * }

The tensor product of two spinors $ψ$ and $φ$ has components $ψ^{α}φ^{β}$, and two indices can be contracted $ψ^{α}φ_{α}$ (sum over $α$). As in tensor algebra, lower indices are covariant, upper indices are contravariant, and raising and lowering indices is done via the metric spinor, e.g.


 * $$\psi^\alpha = g^{\alpha\beta}\psi_\beta$$

but the components of the metric spinor $g$ are different to a metric tensor.

Spin-n/2 particle
A system of $n$ spin-1/2 distinguishable particles is equivalent to a spin $n/2$ particle, because the maximum possible total spin of the system, along any direction, will be $n/2$.

The wavefunction of the spin $n/2$ particle is the tensor product of n = 2s spin-1/2 wave functions,


 * $$\Psi^{\alpha_1\alpha_2 \cdots \alpha_{2s}} = \psi^{\alpha_1} \psi^{\alpha_2} \cdots \psi^{\alpha_{2s}}$$

There appears to be $2^{(2s)} = 4^{s}$ components, since there are 2s indices and each index takes just two values. However, many will coincide because of symmetry in the indices.

The component for any value of spin projection can be found by looking at the indices. In the general case, the component


 * $$\Psi^{\overbrace{\scriptstyle{\uparrow\uparrow\cdots \uparrow}}^{s+\sigma} \overbrace{\scriptstyle{\downarrow\downarrow\cdots \downarrow}}^{s-\sigma} } = \left[

\begin{align} \Psi^{\overbrace{\scriptstyle{\uparrow\uparrow\cdots \uparrow}}^{s}\overbrace{\scriptstyle{\uparrow\uparrow\cdots \uparrow}}^{\sigma} \overbrace{\scriptstyle{\downarrow\downarrow\cdots \downarrow}}^{s-\sigma} } & \quad \sigma > 0 \\ \Psi^{\overbrace{\scriptstyle{\uparrow\uparrow\cdots \uparrow}}^{s} \overbrace{\scriptstyle{\downarrow\downarrow\cdots \downarrow}}^{s} } & \quad \sigma = 0 \\ \Psi^{\overbrace{\scriptstyle{\uparrow\uparrow\cdots \uparrow}}^{s-|\sigma|}\overbrace{\scriptstyle{\downarrow\downarrow\cdots \downarrow}}^{|\sigma|} \overbrace{\scriptstyle{\downarrow\downarrow\cdots \downarrow}}^{s} } & \quad \sigma < 0 \end{align} \right. $$

corresponds to the spin projection number $σ$ taking any of the allowed spin values $−s, −s + 1, ..., s − 1, s$, because each ↑ index corresponds to +1/2, and each ↓ index corresponds to −1/2, so the total spin is


 * $$(s+\sigma)\frac{1}{2} + (s-\sigma)\left(-\frac{1}{2}\right) = \sigma $$

The extreme cases are $σ = −s$ (no indices are ↑ while all $2s$ indices are ↓) and $σ = +s$ (all $2s$ indices are ↑ while no indices are ↓).

The spinor $Ψ$ is symmetric in all indices. Out of $2s$ indices, the number of ways of choosing $s + σ$ indices to be ↑ (the remaining $s − σ$ indices will be ↓) is the binomial coefficient


 * $$\binom{2s}{s+\sigma} = \frac{(2s)!}{(s-\sigma)!(s+\sigma)!} $$

In other words, this is the number of components that correspond to the same spin projection number $σ$, which are all equal. The number of values of $σ$ is $2s + 1$, so the number of independent components the wave function $Ψ$ has is also $2s + 1$.

The connection between a wave function for a given spin projection, and the spinor components, are found from the probability density of finding the particle at position $r$ and time $t$,


 * $$ \rho(\mathbf{r},t) = \sum_\sigma |\Psi(\mathbf{r},t,\sigma)|^2 = \sum_{\alpha_1,\alpha_2,\ldots,\alpha_{2s}}|\Psi^{\alpha_1 \alpha_2 \ldots \alpha_{2s}}|^2 $$

which must be a scalar. The general result is


 * $$\Psi(\mathbf{r},t,\sigma) = \sqrt{\frac{(2s)!}{(s-\sigma)!(s+\sigma)!}} \Psi^{\overbrace{\scriptstyle{\uparrow\uparrow\cdots \uparrow}}^{s+\sigma} \overbrace{\scriptstyle{\downarrow\downarrow\cdots \downarrow}}^{s-\sigma} }$$

for each value of $σ$.

Many particle systems
The wave function of a system of N particles is a single function of all their position coordinates, spins, and time;


 * $$\Psi(\mathbf{r}_1, \mathbf{r}_2 \cdots \mathbf{r}_N, s_{z\,1}, s_{z\,2} \cdots s_{z\,N}, t) $$

If all the particles are distinguishable, then there are no requirements on the arguments of the wave function. However, for identical particles there are symmetry requirements; antisymmetry for identical bosons, and symmetry for identical bosons.