User:Maschen/Spin wave function

Spin wave function

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:

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

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_sz^z of the operator, in other words the χ_sz^z form a basis and the functions Ψ(r, t, s_z) are the complex components of the vector:

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

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

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

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

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

$\xi _{s_{z}}^{z}(s'_{z})=\delta _{s_{z}s'_{z}}$

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.

Shall we include the x and y directions? Probably not...

The basis spin functions for the x and y directions can be found in a similar procedure to the above. For the x-direction the eigenvectors of the x-component spin operator are

$\chi _{1/2}^{x}={\begin{bmatrix}1/{\sqrt {2}}\\1/{\sqrt {2}}\end{bmatrix}}={\begin{bmatrix}\xi _{1/2}^{x}(1/2)\\\xi _{1/2}^{x}(-1/2)\end{bmatrix}}$

$\chi _{-1/2}^{x}={\begin{bmatrix}1/{\sqrt {2}}\\-1/{\sqrt {2}}\end{bmatrix}}={\begin{bmatrix}\xi _{-1/2}^{x}(1/2)\\\xi _{-1/2}^{x}(-1/2)\end{bmatrix}}$

and spin functions can be taken to be

$\xi _{\pm 1/2}^{x}(1/2)=1/{\sqrt {2}}\,,\quad \xi _{\pm 1/2}^{x}(-1/2)=\pm 1/{\sqrt {2}}\,.$

For the y-direction the eigenvectors of the y-component of the spin operator are

$\chi _{1/2}^{y}={\begin{bmatrix}1/{\sqrt {2}}\\i/{\sqrt {2}}\end{bmatrix}}={\begin{bmatrix}\xi _{1/2}^{y}(1/2)\\\xi _{1/2}^{y}(-1/2)\end{bmatrix}}$

$\chi _{-1/2}^{y}={\begin{bmatrix}1/{\sqrt {2}}\\-i/{\sqrt {2}}\end{bmatrix}}={\begin{bmatrix}\xi _{-1/2}^{y}(1/2)\\\xi _{-1/2}^{y}(-1/2)\end{bmatrix}}$

and spin functions can be taken to be

$\xi _{\pm 1/2}^{y}(1/2)=1/{\sqrt {2}}\,,\quad \xi _{\pm 1/2}^{y}(-1/2)=\pm i/{\sqrt {2}}\,.$

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

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

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:

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

where

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

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

and the spin functions depend on the angles,

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

The relation from the z-projection basis χ_sz^z to the n-projection basis χ_snⁿ 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. The linear combinations are summed over all the allowed spin quantum numbers, in the z-direction:

$\Psi (\mathbf {r} ,t,s_{z})=\psi _{-s}(\mathbf {r} ,t)\xi _{-s}^{z}(s_{z})+\psi _{-s+1}(\mathbf {r} ,t)\xi _{-s+1}^{z}(s_{z})+\cdots +\psi _{s-1}(\mathbf {r} ,t)\xi _{s-1}^{z}(s_{z})+\psi _{s}(\mathbf {r} ,t)\xi _{s}^{z}(s_{z})\,,$

and column matrices are indexed by the allowed spin quantum numbers:

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

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.

Spinor formulation

A wave function for one spin-1/2 particle is a spinor;

$\psi =$

where components are ψ₁ and ψ₂, the first corresponds to spin +1/2 along some direction (for concreteness the z-direction as standard) and the second to spin −1/2 along the same direction.

The wavefunction transforms according to

$\psi '=U\psi$

where U is the transformation matrix, containing parameters of the transformation (for concreteness, they may be 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, and α = 1, 2.

The tensor product of two spinors ψ and φ has components ψ^αφ^β, and two indices can be contracted ψ^αφ_β (sum over α). Indices can be raised and lowered 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.

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

(n copies of the same spin-1/2 particle, then take tensor product, or n distinguishable spin-1/2 particles all in the same system? L&L QM p.208 just says "an assembly of spin-1/2 particles").

The wavefunction of the spin n/2 particle is the tensor product

$\Psi ^{\alpha _{1}\alpha _{2}\cdots \alpha _{2s}}=\psi ^{\alpha _{1}}\psi ^{\alpha _{2}}\cdots \psi ^{\alpha _{2s}}$

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

Any value of spin projection can be found by looking at the indices. In the general case

$\Psi ^{\underbrace {\scriptstyle {11\cdots 1}} _{s+\sigma }\underbrace {\scriptstyle {22\cdots 2}} _{s-\sigma }}$

corresponds to the spin projection number of σ, because each 1 index corresponds to +1/2 and each 2 index corresponds to −1/2, so

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

The spinor Ψ is symmetric in all indices. The number of ways of choosing s + σ indices to be 1 out of 2s indices (the remaining indices will be 2) is the binomial coefficient

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \binom{2s}{s+\sigma} = \frac{(2s)!}{(2s-(s+\sigma))!(s+\sigma)!} = \frac{(2s)!}{(s-\sigma)!(s+\sigma)!} }

in other words there are this many components that correspond to the spin projection number σ.

References

• Atkins, P. W. (1974). Quanta: A Handbook of Concepts. ISBN 0-19-855494-X. {{cite book}}: Invalid |ref=harv (help)

• Dirac, P. A. M. (1982). The principles of quantum mechanics. The international series on monographs on physics (4th ed.). Oxford University Press. ISBN 0 19 852011 5. {{cite book}}: Invalid |ref=harv (help)

• Eisberg, R.; Resnick, R. (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd ed.). John Wiley & Sons. ISBN 978-0-471-87373-0. {{cite book}}: Invalid |ref=harv (help)

• Griffiths, D. J. (2004). Introduction to Quantum Mechanics (2nd ed.). Essex England: Pearson Education Ltd. ISBN 978-0131118928. {{cite book}}: Invalid |ref=harv (help)

• Landau, L.D.; Lifshitz, E. M. (1977). Quantum Mechanics: Non-Relativistic Theory. Vol. Vol. 3 (3rd ed.). Pergamon Press. ISBN 978-0-08-020940-1. {{cite book}}: |volume= has extra text (help); Invalid |ref=harv (help) Online copy

• Lerner, R.G.; Trigg, G.L. (1991). Encyclopaedia of Physics (2nd ed.). VHC Publishers. ISBN 0-89573-752-3. {{cite book}}: Invalid |ref=harv (help)

• Pauli, Wolfgang (1927). "Zur Quantenmechanik des magnetischen Elektrons". Zeitschrift für Physik (in German). 43. Bibcode:1927ZPhy...43..601P. doi:10.1007/bf01397326. {{cite journal}}: Invalid |ref=harv (help)

• Peleg, Y.; Pnini, R.; Zaarur, E.; Hecht, E. (2010). Quantum mechanics. Schaum's outlines (2nd ed.). McGraw Hill. ISBN 978-0-07-162358-2. {{cite book}}: Invalid |ref=harv (help)

• Rae, A.I.M. (2008). Quantum Mechanics. Vol. 2 (5th ed.). Taylor & Francis Group. ISBN 1-5848-89705.

• Shankar, R. (1994). Principles of Quantum Mechanics (2nd ed.). ISBN 0306447908. {{cite book}}: Invalid |ref=harv (help)

• Martin, B.R.; Shaw, G. (2008). Particle Physics. Manchester Physics Series (3rd ed.). John Wiley & Sons. ISBN 978-0-470-03294-7. {{cite book}}: Invalid |ref=harv (help)

• Tipler, P. A.; Mosca, G.; Freeman (2008). Physics for Scientists and Engineers – with Modern Physics (6th ed.). ISBN 0-7167-8964-7. {{cite book}}: Invalid |ref=harv (help)

• Weinberg, S. (2013), Lectures in Quantum Mechanics, Cambridge University Press, ISBN 978-1-107-02872-2

• Weinberg, S. (2002), The Quantum Theory of Fields, vol. 1, Cambridge University Press, ISBN 0-521-55001-7

• Zettili, N. (2009). Quantum Mechanics: Concepts and Applications (2nd ed.). ISBN 978-0-470-02679-3. {{cite book}}: Invalid |ref=harv (help)

External links

• Multiple-Particle Systems Including Spin Leon van Dommelen
• Spin wave functions Mark Tuckerman (2011)
• Identical Particles Revisited Michael Fowler
• The Nature of Many-Electron Wavefunctions David Sherrill (2006)