en>WOSlinker: /* Generalization */ class="wikitable"

2013-09-28T22:01:55Z

Generalization: class="wikitable"

← Older revision		Revision as of 00:01, 29 September 2013
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	~~The title~~ of the ~~author~~ is ~~Figures~~. ~~California is our beginning location~~. ~~Since she was 18 she~~'s ~~been working as~~ a ~~meter reader but she's usually wanted her own company~~. The ~~favorite pastime for my children~~ and me is to ~~play baseball but I haven't produced~~ a ~~dime with it~~.<br><br>~~My blog~~: ~~[http~~://~~www~~.~~hotporn123~~.~~com~~/~~user~~/~~KGottscha at home std testing~~]		In [[quantum mechanics]], '''eigenspinors''' are thought of as [[basis vector]]s representing the general spin state of a particle. Strictly speaking, they are not vectors at all, but in fact [[spinors]]. For a single spin 1/2 particle, they can be defined as the [[eigenvectors]] of the [[Pauli matrices]].

			== General eigenspinors ==
			In quantum mechanics, the [[Spin (physics)\|spin]] of a particle or collection of particles is [[quantization (physics)\|quantized]]. In particular, all particles have either half integer or integer spin. In the most general case, the eigenspinors for a system can be quite complicated. If you have a collection of [[Avogadro's number]] of particles, each one with two (or more) possible spin states, there would be no hope of writing down a complete set of eigenspinors. However, eigenspinors are very useful when dealing with the spins of a very small number of particles.

			== The spin 1/2 particle ==
			The simplest and most illuminating example of eigenspinors is for a single spin 1/2 particle. A particle's spin has three components, corresponding to the three [[Three-dimensional space\|spatial]] dimensions: <math>S_x</math>, <math>S_y</math>, and <math>S_z</math>. For a spin 1/2 particle, there are only two possible [[eigenstates]] of spin: spin up, and spin down. Spin up is denoted as the column matrix:
			<math>\chi_+ = \begin{bmatrix}
			1\\
			0\\
			\end{bmatrix}
			</math>
			and spin down is
			<math>\chi_- = \begin{bmatrix}
			0\\
			1\\
			\end{bmatrix}
			</math>.

			Each component of the angular momentum thus has two eigenspinors. By convention, the z direction is chosen as having the <math>\chi_+</math> and <math>\chi_-</math> states as its eigenspinors. The eigenspinors for the other two directions follow from this convention:

			<math>S_z</math>:
			:<math>\chi_+^z = \begin{bmatrix}
			1\\
			0\\
			\end{bmatrix}
			</math>
			:<math>\chi_-^z = \begin{bmatrix}
			0\\
			1\\
			\end{bmatrix}
			</math>

			<math>S_x</math>:
			:<math>\chi_+^x = {1 \over \sqrt{2}} \begin{bmatrix}
			1\\
			1\\
			\end{bmatrix}
			</math>
			:<math>\chi_-^x = {1 \over \sqrt{2}} \begin{bmatrix}
			1\\
			-1\\
			\end{bmatrix}
			</math>
			<math>S_y</math>:
			:<math>\chi_+^y = {1 \over \sqrt{2}} \begin{bmatrix}
			1\\
			i\\
			\end{bmatrix}
			</math>

			:<math>\chi_-^y = {1 \over \sqrt{2}} \begin{bmatrix}
			1\\
			-i\\
			\end{bmatrix}
			</math>

			==Example usage==
			Suppose there is a spin 1/2 particle in a state <math>\chi = {1 \over \sqrt{5}} \begin{bmatrix}
			1\\
			2\\
			\end{bmatrix}
			</math>. To determine the probability of finding the particle in a spin up state, we simply multiply the state of the particle by the adjoint of the eigenspinor matrix representing spin up, and square the result. Thus, the eigenspinor allows us to sample the part of the particle's state that is in the same direction as the eigenspinor. First we multiply:

			<math>c_+ = \begin{bmatrix}
			1\ 0\\

			\end{bmatrix}
			*\chi = {1 \over \sqrt{5}}
			</math>.

			Now, we simply square this value to obtain the probability of the particle being found in a spin up state:

			<math>P_+ = {1 \over 5}</math>

			==Properties==
			Each set of eigenspinors forms a [[complete space\|complete]], [[orthonormal basis\|orthonormal]] basis. This means that any state can be written as a [[linear combination]] of the [[basis (linear algebra)\|basis]] spinors.

			The eigenspinors are eigenvectors of the Pauli matrices in the case of a single spin 1/2 particle.

			==See also==
			*[[Spin (physics)\|Spin]]
			*[[Spinor]]
			*[[Eigenvector]]
			*[[Pauli matrices]]

			==References==
			Griffiths, David J. (2005) Introduction to Quantum Mechanics(2nd ed.). Upper Saddle River, NJ: Pearson Prentice Hall. ISBN 0-13-111892-7.

			[[Category:Quantum mechanics]]

en>Dr.enh: /* Examples */ fix link to redirect page

2012-06-10T19:43:53Z

Examples: fix link to redirect page

New page

The title of the author is Figures. California is our beginning location. Since she was 18 she's been working as a meter reader but she's usually wanted her own company. The favorite pastime for my children and me is to play baseball but I haven't produced a dime with it.<br><br>My blog: [http://www.hotporn123.com/user/KGottscha at home std testing]

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en>WOSlinker: /* Generalization */ class="wikitable"

en>Dr.enh: /* Examples */ fix link to redirect page