Pauli equation

Pauli equation

The Pauli Equation, also known as the Schrödinger-Pauli equation, is the formulation of the Schrödinger equation for spin one-half particles which takes into account the interaction of the particle's spin with the electromagnetic field. It is the non-relativistic border case of the Dirac equation and can be used where particles are slow enough that relativistic effects can be neglected.

The Pauli equation was formulated by Wolfgang Pauli.

Details

The Pauli equation is stated as:

::left [ frac{1}{2m}(vec{sigma}cdot(vec{p} - q vec{A}))^2 + q phi ight] |psi angle = i hbar frac{partial}{partial t} |psi angle

Where:
* m is the mass of the particle.
* q is the charge of the particle.
* vec{sigma} is a three-component vector of the two-by-two Pauli matrices. This means that each component of the vector is a Pauli matrix.
* vec{p} is the three-component vector of the momentum operators. The components of this vector are - i hbar frac{partial}{partial x_n}
* vec{A} is the three-component magnetic vector potential.
* phi is the electric scalar potential.
* |psi angle is the two component spinor wavefunction, which can be represented as egin{pmatrix} psi_0 \psi_1end{pmatrix} .

Somewhat more explicitly, the Pauli equation is:

::left [ frac{1}{2m} left( sum_{n=1}^3 (sigma_n ( - i hbar frac{partial}{partial x_n} - q A_n)) ight) ^2 + q phi ight] egin{pmatrix} psi_0 \ psi_1 end{pmatrix} = i hbar egin{pmatrix} frac{ partial psi_0 }{partial t} \ frac{ partial psi_1 }{partial t} end{pmatrix}

Notice that the Hamiltonian (the expression between square brackets) is a two-by-two matrix operator, because of the Pauli sigma matrices.

Relationship to the Schrödinger Equation and the Dirac Equation

The Pauli equation is non-relativistic, but it does predict spin. As such, it can be thought of an occupying the middle ground between:
* The familiar Schrödinger Equation (on a complex scalar wavefunction), which is non-relativistic and does not predict spin.
* The Dirac Equation (on a complex four-component spinor), which is fully relativistic (with respect to special relativity) and predicts spin.

Note that because of the properties of the Pauli matrices, if the magnetic vector potential old{A} is equal to zero, then the equation reduces to the familiar Schrödinger equation for a particle in a purely electric potential phi , except that it operates on a two component spinor. Therefore, we can see that the spin of the particle only affects its motion in the presence of a magnetic field.

Special Cases

Both spinor components satisfy the Schrödinger-Equation. This means that the system is degenerated as to the additional degree of freedom.

With an external electromagnetic field the full Pauli equation reads:


underbrace{i hbar partial_t vec varphi_pm = left( frac{(underline{vec p}-q vec A)^2}{2 m} + q phi ight) hat 1 vec varphi_pm}_mathrm{Schrddot{o}dinger~equation} - underbrace{frac{q hbar}{2m}vec{hat sigma} cdot vec B vec varphi_pm}_ ext{Stern Gerlach term}.

where :: phi is the scalar electric potential :: A the electromagnetic vector potential:: vec varphi_pm, in Dirac notation |psi angle :=egin{pmatrix} |varphi_+ angle \
varphi_- angle end{pmatrix}, are the Pauli spinor components:: vec{hat sigma} are the Pauli matrices:: vec B is the external magnetic field:: hat 1 two dimensional Identity matrix

With the Stern Gerlach term it is possible to comprehend the obtaining of spin orientation of atoms with one valence electron e.g. silver atoms which flow through an inhomogeneous magnetic field.

Analogously, the term is responsible for the splitting of spectral lines (corresponding to energy levels) in a magnetic field as can be viewed in the anomalous Zeeman effect.

Derivation of the Pauli equation by Schrodinger

Starting from the Dirac equation for weak electromagnetic interactions :


i hbar partial_t left( egin{array}{c} vec varphi_1\vec varphi_2end{array} ight) = c left( egin{array}{c} vec{hat sigma} vec pi vec varphi_2\vec{hat sigma} vec pi vec varphi_1end{array} ight)+q phi left( egin{array}{c} vec varphi_1\vec varphi_2end{array} ight) + mc^2 left( egin{array}{c} vec varphi_1 \-vec varphi_2end{array} ight)
with vec pi = vec p - q vec A

using the following approximatations :

* Simplification of the equation through following ansatz ::left( egin{array}{c} vec varphi_1 \ vec varphi_2 end{array} ight) = e^{-i frac{mc^2t}{hbar left( egin{array}{c} vec{ ilde varphi_1} \ vec{ ilde varphi_2} end{array} ight)
* Eliminating the rest energy through an Ansatz with slow time dependence::partial_t vec varphi_i ll frac{mc^2}{hbar} vec varphi_i
* weak coupling of the electric potential::q phi ll mc^2

Examples

References

*cite book | author=Schwabl, Franz| title=Quantenmechanik I | publisher=Springer |year=2004 |id=ISBN 978-3540431060
*cite book | author=Schwabl, Franz| title=Quantenmechanik für Fortgeschrittene | publisher=Springer |year=2005 |id=ISBN 978-3540259046
*cite book | author=Claude Cohen-Tannoudji, Bernard Diu, Frank Laloe| title= Quantum Mechanics 2| publisher=Wiley, J |year=2006 |id=ISBN 978-0471569527

External links


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