Lindblad superoperator: Difference between revisions

Latest revision as of 06:39, 6 December 2013

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In Yr 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebs of the property market. Units are snapped up in document time and at document breaking prices. Developers are having fun with overwhelming demand and consumers want more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp duty imposed on residential properties, traders switch their focus to business and industrial properties. I consider every property purchasers need their property investment to appreciate in worth. In physics, particularly quantum field theory, the Weyl Equation is a relativistic wave equation for describing massless spin-1/2 particles. It is named after the German physicist Hermann Weyl.

Equation

The general equation can be written: ^[1]^[2]

σ^{μ} \partial_{μ} ψ = 0

explicitly in SI units:

I_{2} \frac{1}{c} \frac{\partial ψ}{\partial t} + σ_{x} \frac{\partial ψ}{\partial x} + σ_{y} \frac{\partial ψ}{\partial y} + σ_{z} \frac{\partial ψ}{\partial z} = 0

where

σ_{μ} = (σ_{0}, σ_{1}, σ_{2}, σ_{3}) = (I_{2}, σ_{x}, σ_{y}, σ_{z})

is a vector whose components is the 2 × 2 identity matrix for μ = 0 and the Pauli matrices for μ = 1,2,3, and ψ is the wavefunction - one of the Weyl spinors.

Weyl spinors

The elements ψ_L and ψ_R are respectively the left and right handed Weyl spinors, each with two components. Both have the form

ψ = (\begin{matrix} ψ_{1} \\ ψ_{2} \end{matrix}) = χ e^{- i (k \cdot r - ω t)} = χ e^{- i (p \cdot r - E t) / ℏ}

where

χ = (\begin{matrix} χ_{1} \\ χ_{2} \end{matrix})

is a constant two-component spinor.

Since the particles are massless, i.e. m = 0, the magnitude of momentum p relates directly to the wave-vector k by the De Broglie relations as:

| p | = ℏ | k | = ℏ ω / c \to | k | = ω / c

The equation can be written in terms of left and right handed spinors as:

\begin{aligned} σ^{μ} \partial_{μ} ψ_{R} = 0 \\ {\bar{σ}}^{μ} \partial_{μ} ψ_{L} = 0 \end{aligned}

Helicity

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The left and right components correspond to the helicity λ of the particles, the projection of angular momentum operator J onto the linear momentum p:

p \cdot J | p, λ ⟩ = λ | p | | p, λ ⟩

Here $λ = \pm 1 / 2$ .

Derivation

The equations are obtained from the Lagrangian densities

ℒ = i ψ_{R}^{†} σ^{μ} \partial_{μ} ψ_{R}

ℒ = i ψ_{L}^{†} {\bar{σ}}^{μ} \partial_{μ} ψ_{L}

By treating the spinor and its conjugate (denoted by $†$ ) as independent variables, the relevant Weyl equation is obtained.

References

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External links

↑ Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0
↑ The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.

[1] Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0

[2] The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.

[1]

[2]

@@ Line 1: / Line 1: @@
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+{{quantum field theory}}
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+In [[physics]], particularly [[quantum field theory]], the '''Weyl Equation''' is a [[relativistic wave equation]] for describing massless [[spin-1/2]] particles. It is named after the [[German people|German]] physicist [[Hermann Weyl]].
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+==Equation==
+The general equation can be written: <ref>Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0</ref><ref>The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.</ref>
+:<math> \sigma^\mu\partial_\mu \psi=0</math>
+explicitly in [[SI units]]:
+:<math> I_2 \frac{1}{c}\frac{\partial \psi}{\partial t} + \sigma_x\frac{\partial \psi}{\partial x} + \sigma_y\frac{\partial \psi}{\partial y} + \sigma_z\frac{\partial \psi}{\partial z}=0</math>
+where
+:<math> \sigma_\mu = (\sigma_0,\sigma_1,\sigma_2,\sigma_3)= (I_2,\sigma_x,\sigma_y,\sigma_z)</math>
+is a [[vector (mathematics and physics)|vector]] whose components is the 2 × 2 [[identity matrix]] for μ = 0 and the [[Pauli matrices]] for μ = 1,2,3, and ψ is the [[wavefunction]] - one of the Weyl spinors.
+===Weyl spinors===
+The elements ψ<sub>''L''</sub> and ψ<sub>''R''</sub> are respectively the left and right handed Weyl [[spinors]], each with two components. Both have the form
+:<math> \psi = \begin{pmatrix}
+\psi_1 \\
+\psi_2 \\
+\end{pmatrix} = \chi e^{-i(\mathbf{k}\cdot\mathbf{r}-\omega t)}= \chi e^{-i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar}</math>
+where
+:<math> \chi = \begin{pmatrix}
+\chi_1 \\
+\chi_2 \\
+\end{pmatrix} </math>
+is a constant two-component spinor.
+Since the particles are [[mass]]less, i.e. ''m'' = 0, the magnitude of [[momentum]] '''p''' relates directly to the [[wave-vector]] '''k''' by the [[De Broglie relations]] as:
+:<math> |\mathbf{p}| = \hbar |\mathbf{k}| = \hbar \omega /c \, \rightarrow \, |\mathbf{k}| = \omega /c </math>
+The equation can be written in terms of left and right handed spinors as:
+:<math>\begin{align} & \sigma^\mu \partial_\mu \psi_R = 0 \\
+& \bar{\sigma}^\mu \partial_\mu \psi_L = 0
+\end{align}</math>
+===Helicity===
+{{main|Helicity (particle physics)}}
+The left and right components correspond to the helicity λ of the particles, the projection of [[angular momentum operator]] '''J''' onto the linear momentum '''p''':
+:<math>\mathbf{p}\cdot\mathbf{J}\left|\mathbf{p},\lambda\right\rangle=\lambda |\mathbf{p}|\left|\mathbf{p},\lambda\right\rangle</math>
+Here <math>\lambda=\pm 1/2</math>.
+==Derivation==
+The equations are obtained from the [[lagrangian density|Lagrangian densities]]
+:<math> \mathcal L = i \psi_R^\dagger \sigma^\mu \partial_\mu \psi_R </math>
+:<math> \mathcal L = i \psi_L^\dagger \bar\sigma^\mu \partial_\mu \psi_L </math>
+By treating the spinor and its [[Hermitian adjoint|conjugate]] (denoted by <math> \dagger </math>) as independent variables, the relevant Weyl equation is obtained.
+==See also==
+* [[Dirac equation]] (which describes massive spin-1/2 particles)
+* [[Angular momentum operator]]
+* [[Momentum operator]]
+* [[Spin (physics)]]
+==References==
+{{reflist}}
+==Further reading==
+* Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008, ISBN 978-0-07-154382-8
+* Particle Physics (2nd Edition), B.R. Martin, G. Shaw, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-470-03294-7
+* Supersymmetry P. Labelle, Demystified, McGraw-Hill (USA), 2010, ISBN 978-0-07-163641-4
+* The Road to Reality, Roger Penrose, Vintage books, 2007, ISBN 0-679-77631-1
+==External links==
+* [http://aesop.phys.utk.edu/qft/2004-5/2-2.pdf]
+* [http://www.nbi.dk/~kleppe/random/ll/l2.html]
+* [http://www.tfkp.physik.uni-erlangen.de/download/research/DW-derivation.pdf]
+* [http://www.weylmann.com/weyldirac.pdf]
+[[Category:Quantum mechanics]]

Lindblad superoperator: Difference between revisions

Latest revision as of 06:39, 6 December 2013

Contents

Equation

Weyl spinors

Helicity

Derivation

See also

References

Further reading

External links

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