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ort download as pdf printable version in other projects wikimedia commons languages on this wikipedia the language links are at the top of the page across from the article title go to top contents move to sidebar hide top 1 equation toggle equation subsection 1 1 weak magnetic fields 2 from dirac equation toggle from dirac equation subsection 2 1 derivation 2 2 from a foldy wouthuysen transformation 3 pauli coupling 4 see also 5 footnotes 6 references toggle references subsection 6 1 books toggle the table of contents toggle the table of contents pauli equation 20 languages العربية беларуская català deutsch español فارسی français հայերեն hrvatski bahasa indonesia italiano 日本語 ਪੰਜਾਬੀ polski português română русский srpskohrvatski српскохрватски українська 中文 edit links article talk english read edit view history more read edit view history from wikipedia the free encyclopedia quantum mechanical equation of motion of charged particles in magnetic field part of a series of articles about quantum mechanics i ℏ t ψ t h ψ t displaystyle i hbar frac partial partial t psi t rangle hat h psi t rangle schrödinger equation introduction glossary history background classical mechanics old quantum theory bra ket notation hamiltonian interference fundamentals complementarity decoherence entanglement energy level measurement nonlocality quantum number state superposition symmetry tunnelling uncertainty wave function collapse experiments bell s inequality davisson germer double slit elitzur vaidman franck hertz leggett garg inequality mach zehnder popper quantum eraser delayed choice schrödinger s cat stern gerlach wheeler s delayed choice formulations overview heisenberg interaction matrix phase space schrödinger sum over histories path integral equations dirac klein gordon pauli rydberg schrödinger interpretations bayesian consistent histories copenhagen de broglie bohm ensemble hidden variable local many worlds objective collapse quantum logic relational transactional advanced topics relativistic quantum mechanics quantum field theory quantum information science quantum computing quantum chaos epr paradox density matrix scattering theory quantum statistical mechanics quantum machine learning scientists aharonov bell bethe blackett bloch bohm bohr born bose de broglie compton dirac davisson debye ehrenfest einstein everett fock fermi feynman glauber gutzwiller heisenberg hilbert jordan kramers pauli lamb landau laue moseley millikan onnes planck rabi raman rydberg schrödinger simmons sommerfeld von neumann weyl wien wigner zeeman zeilinger v t e in quantum mechanics the pauli equation or schrödinger pauli equation is the formulation of the schrödinger equation for spin ½ particles which takes into account the interaction of the particle s spin with an external electromagnetic field it is the non relativistic limit of the dirac equation and can be used where particles are moving at speeds much less than the speed of light so that relativistic effects can be neglected it was formulated by wolfgang pauli in 1927 1 equation edit for a particle of mass m displaystyle m and electric charge q displaystyle q in an electromagnetic field described by the magnetic vector potential a displaystyle mathbf a and the electric scalar potential ϕ displaystyle phi the pauli equation reads pauli equation general 1 2 m σ p q a 2 q ϕ ψ i ℏ t ψ displaystyle left frac 1 2m boldsymbol sigma cdot mathbf hat p q mathbf a 2 q phi right psi rangle i hbar frac partial partial t psi rangle here σ σ x σ y σ z displaystyle boldsymbol sigma sigma _ x sigma _ y sigma _ z are the pauli operators collected into a vector for convenience and p i ℏ displaystyle mathbf hat p i hbar nabla is the momentum operator in position representation the state of the system ψ displaystyle psi rangle written in dirac notation can be considered as a two component spinor wavefunction or a column vector after choice of basis ψ ψ ψ ψ ψ displaystyle psi rangle psi _ mathord uparrow rangle psi _ mathord downarrow rangle stackrel cdot begin bmatrix psi _ psi _ end bmatrix the hamiltonian operator is a 2 2 matrix because of the pauli operators h 1 2 m σ p q a 2 q ϕ displaystyle hat h frac 1 2m left boldsymbol sigma cdot mathbf hat p q mathbf a right 2 q phi substitution into the schrödinger equation gives the pauli equation this hamiltonian is similar to the classical hamiltonian for a charged particle interacting with an electromagnetic field see lorentz force for details of this classical case the kinetic energy term for a free particle in the absence of an electromagnetic field is just p 2 2 m displaystyle frac mathbf p 2 2m where p displaystyle mathbf p is the kinetic momentum while in the presence of an electromagnetic field it involves the minimal coupling π p q a displaystyle mathbf pi mathbf p q mathbf a where now π displaystyle mathbf pi is the kinetic momentum and p displaystyle mathbf p is the canonical momentum the pauli operators can be removed from the kinetic energy term using the pauli vector identity σ a σ b a b i σ a b displaystyle boldsymbol sigma cdot mathbf a boldsymbol sigma cdot mathbf b mathbf a cdot mathbf b i boldsymbol sigma cdot left mathbf a times mathbf b right note that unlike a vector the differential operator p q a i ℏ q a displaystyle mathbf hat p q mathbf a i hbar nabla q mathbf a has non zero cross product with itself this can be seen by considering the cross product applied to a scalar function ψ displaystyle psi p q a p q a ψ q p a ψ a p ψ i q ℏ a ψ a ψ i q ℏ ψ a a ψ a ψ i q ℏ b ψ displaystyle left left mathbf hat p q mathbf a right times left mathbf hat p q mathbf a right right psi q left mathbf hat p times left mathbf a psi right mathbf a times left mathbf hat p psi right right iq hbar left nabla times left mathbf a psi right mathbf a times left nabla psi right right iq hbar left psi left nabla times mathbf a right mathbf a times left nabla psi right mathbf a times left nabla psi right right iq hbar mathbf b psi where b a displaystyle mathbf b nabla times mathbf a is the magnetic field for the full pauli equation one then obtains 2 pauli equation standard form h ψ 1 2 m p q a 2 q ℏ σ b q ϕ ψ i ℏ t ψ displaystyle hat h psi rangle left frac 1 2m left left mathbf hat p q mathbf a right 2 q hbar boldsymbol sigma cdot mathbf b right q phi right psi rangle i hbar frac partial partial t psi rangle weak magnetic fields edit for the case of where the magnetic field is constant and homogenous one may expand p q a 2 textstyle mathbf hat p q mathbf a 2 using the symmetric gauge a 1 2 b r textstyle mathbf hat a frac 1 2 mathbf b times mathbf hat r where r textstyle mathbf r is the position operator and a is now an operator we obtain p q a 2 p 2 q r p b 1 4 q 2 b 2 r 2 b r 2 p 2 q l b displaystyle mathbf hat p q mathbf hat a 2 mathbf hat p 2 q mathbf hat r times mathbf hat p cdot mathbf b frac 1 4 q 2 left mathbf b 2 mathbf hat r 2 mathbf b cdot mathbf hat r 2 right approx mathbf hat p 2 q mathbf hat l cdot mathbf b where l textstyle mathbf hat l is the particle angular momentum operator and we neglected terms in the magnetic field squared b 2 textstyle b 2 therefore we obtain pauli equation weak magnetic fields 1 2 m p 2 q l 2 s b q ϕ ψ i ℏ t ψ displaystyle left frac 1 2m left left mathbf hat p 2 q mathbf hat l 2 mathbf hat s cdot mathbf b right right q phi right psi rangle i hbar frac partial partial t psi rangle where s ℏ σ 2 textstyle mathbf s hbar boldsymbol sigma 2 is the spin of the particle the factor 2 in front of the spin is known as the dirac g factor the term in b textstyle mathbf b is of the form μ b textstyle boldsymbol mu cdot mathbf b which is the usual interaction between a magnetic moment μ textstyle boldsymbol mu and a magnetic field like in the zeeman effect for an electron of charge e textstyle e in an isotropic constant magnetic field one can further reduce the equation using the total angular momentum j l s textstyle mathbf j mathbf l mathbf s and wigner eckart theorem thus we find p 2 2 m μ b g j m j b e ϕ ψ i ℏ t ψ displaystyle left frac mathbf p 2 2m mu _ rm b g_ j m_ j mathbf b e phi right psi rangle i hbar frac partial partial t psi rangle where μ b e ℏ 2 m textstyle mu _ rm b frac e hbar 2m is the bohr magneton and m j textstyle m_ j is the magnetic quantum number related to j textstyle mathbf j the term g j textstyle g_ j is known as the landé g factor and is given here by g j 3 2 3 4 ℓ ℓ 1 2 j j 1 displaystyle g_ j frac 3 2 frac frac 3 4 ell ell 1 2j j 1 a where ℓ displaystyle ell is the orbital quantum number related to l 2 displaystyle l 2 and j displaystyle j is the total orbital quantum number related to j 2 displaystyle j 2 from dirac equation edit the pauli equation is the non relativistic limit of dirac equation the relativistic quantum equation of motion for particles spin ½ 3 derivation edit dirac equation can be written as i ℏ t ψ 1 ψ 2 c σ π ψ 2 σ π ψ 1 q ϕ ψ 1 ψ 2 m c 2 ψ 1 ψ 2 displaystyle i hbar partial _ t begin pmatrix psi _ 1 psi _ 2 end pmatrix c begin pmatrix boldsymbol sigma cdot boldsymbol pi psi _ 2 boldsymbol sigma cdot boldsymbol pi psi _ 1 end pmatrix q phi begin pmatrix psi _ 1 psi _ 2 end pmatrix mc 2 begin pmatrix psi _ 1 psi _ 2 end pmatrix where t t textstyle partial _ t frac partial partial t and ψ 1 ψ 2 displaystyle psi _ 1 psi _ 2 are two component spinor forming a bispinor using the following ansatz ψ 1 ψ 2 e i m c 2 t ℏ ψ χ displaystyle begin pmatrix psi _ 1 psi _ 2 end pmatrix e i tfrac mc 2 t hbar begin pmatrix psi chi end pmatrix with two new spinors ψ χ displaystyle psi chi the equation becomes i ℏ t ψ χ c σ π χ σ π ψ q ϕ ψ χ 0 2 m c 2 χ displaystyle i hbar partial _ t begin pmatrix psi chi end pmatrix c begin pmatrix boldsymbol sigma cdot boldsymbol pi chi boldsymbol sigma cdot boldsymbol pi psi end pmatrix q phi begin pmatrix psi chi end pmatrix begin pmatrix 0 2 mc 2 chi end pmatrix in the non relativistic limit t χ displaystyle partial _ t chi and the kinetic and electrostatic energies are small with respect to the rest energy m c 2 displaystyle mc 2 thus χ σ π ψ 2 m c displaystyle chi approx frac boldsymbol sigma cdot boldsymbol pi psi 2 mc inserted in the upper component of dirac equation we find pauli equation general form i ℏ t ψ σ π 2 2 m q ϕ ψ displaystyle i hbar partial _ t psi left frac boldsymbol sigma cdot boldsymbol pi 2 2 m q phi right psi from a foldy wouthuysen transformation edit one can also rigorously derive pauli equation starting from dirac equation in an external field and performing a foldy wouthuysen transformation 3 pauli coupling edit pauli s equation is derived by requiring minimal coupling which provides a g factor g 2 most elementary particles have anomalous g factors different from 2 in the domain of relativistic quantum field theory one defines a non minimal coupling sometimes called pauli coupling in order to add an anomalous factor γ μ p μ γ μ p μ q γ μ a μ a σ μ ν f μ ν displaystyle gamma mu p_ mu to gamma mu p_ mu q gamma mu a_ mu a sigma _ mu nu f mu nu where p μ displaystyle p_ mu is the four momentum operator a μ displaystyle a_ mu is the electromagnetic four potential a displaystyle a is proportional to the anomalous magnetic dipole moment f μ ν μ a ν ν a μ displaystyle f mu nu partial mu a nu partial nu a mu is the electromagnetic tensor and σ μ ν i 2 γ μ γ ν textstyle sigma _ mu nu frac i 2 gamma _ mu gamma _ nu are the lorentzian spin matrices and the commutator of the gamma matrices γ μ displaystyle gamma mu 4 5 in the context of non relativistic quantum mechanics instead of working with the schrödinger equation pauli coupling is equivalent to using the pauli equation or postulating zeeman energy for an arbitrary g factor see also edit semiclassical physics atomic molecular and optical physics group contraction gordon decomposition footnotes edit the formula used here is for a particle with spin ½ with a g factor g s 2 textstyle g_ s 2 and orbital g factor g l 1 textstyle g_ l 1 more generally it is given by g j 3 2 m s m s 1 ℓ ℓ 1 2 j j 1 displaystyle g_ j frac 3 2 frac m_ s m_ s 1 ell ell 1 2j j 1 where m s displaystyle m_ s is the spin quantum number related to s displaystyle hat s references edit pauli wolfgang 1927 zur quantenmechanik des magnetischen elektrons zeitschrift für physik in german 43 9 10 601 623 bibcode 1927zphy 43 601p doi 10 1007 bf01397326 issn 0044 3328 s2cid 128228729 bransden bh joachain cj 1983 physics of atoms and molecules 1st ed prentice hall p 638 638 isbn 0 582 44401 2 a b greiner walter 2012 12 06 relativistic quantum mechanics wave equations springer isbn 978 3 642 88082 7 das ashok 2008 lectures on quantum field theory world scientific isbn 978 981 283 287 0 barut a o mcewan j january 1986 the four states of the massless neutrino with pauli coupling by spin gauge invariance letters in mathematical physics 11 1 67 72 bibcode 1986lmaph 11 67b doi 10 1007 bf00417466 issn 0377 9017 s2cid 120901078 books edit schwabl franz 2004 quantenmechanik i springer isbn 978 3540431060 schwabl franz 2005 quantenmechanik für fortgeschrittene springer isbn 978 3540259046 claude cohen tannoudji bernard diu frank laloe 2006 quantum mechanics 2 wiley j isbn 978 0471569527 v t e quantum mechanics background introduction history timeline classical mechanics old quantum theory glossary fundamentals born rule bra ket notation complementarity density matrix energy level ground state excited state degenerate levels zero point energy entanglement hamiltonian interference decoherence measurement nonlocality quantum state superposition tunnelling scattering theory symmetry in quantum mechanics uncertainty wave function collapse wave particle duality formulations formulations heisenberg interaction matrix mechanics schrödinger path integral formulation phase space equations dirac klein gordon pauli rydberg schrödinger interpretations bayesian consistent histories copenhagen de broglie bohm ensemble hidden variable local many worlds objective collapse quantum logic relational transactional von neumann wigner experiments bell s inequality davisson germer delayed choice quantum eraser double slit franck hertz mach zehnder interferometer elitzur vaidman popper quantum eraser stern gerlach wheeler s delayed choice science quantum biology quantum chemistry quantum chaos quantum cosmology quantum differential calculus quantum dynamics quantum geometry quantum measurement problem quantum stochastic calculus quantum spacetime technology quantum algorithms quantum amplifier quantum bus quantum cellular automata quantum finite automata quantum channel quantum circuit quantum complexity theory quantum computing timeline quantum cryptography quantum electronics quantum error correction quantum imaging quantum image processing quantum information quantum key distribution quantum logic quantum logic gates quantum machine quantum machine learning quantum metamaterial quantum metrology quantum network quantum neural network quantum optics quantum programming quan...