Surface chemistry of neural implants

2013-09-30T18:34:44Z

150.202.8.1: /* Conducting Polymers */

{{quantum mechanics}}

In [[quantum mechanics]] and its applications to quantum [[many body problem|many-particle systems]], notably [[quantum chemistry]], '''angular momentum diagrams''', or more accurately from a mathematical viewpoint '''angular momentum graphs''', are a diagrammatic method for representing [[angular momentum]] [[quantum state]]s of a quantum system allowing calculations to be done symbolically. More specifically, the arrows encode angular momentum states in [[bra–ket notation]] and include the abstract nature of the state, such as [[tensor product]]s and transformation rules.

The notation parallels the idea of [[Penrose graphical notation]] and [[Feynman diagram]]s. The diagrams consist of arrows and vertices with [[quantum number]]s as labels, hence the alternative term "[[graph (mathematics)|graph]]s". The sense of each arrow is related to [[Hermitian adjoint|Hermitian conjugation]], which roughly corresponds to [[T-symmetry|time reversal]] of the angular momentum states (c.f. [[Schrödinger equation]]). The diagrammatic notation is a considerably large topic in its own right with a number of specialized features – this article introduces the very basics.

They were developed primarily by [[Adolfas Jucys]] in the twentieth century.

==Equivalence between Dirac notation and Jucys diagrams==

===Angular momentum states===

The [[quantum state]] vector of a single particle with total [[angular momentum quantum number]] ''j'' and total [[magnetic quantum number]] ''m'' = ''j'', ''j'' − 1, ..., , −''j'' + 1, −''j'', is denoted as a [[Bra–ket notation|ket]] {{ket|''j'', ''m''}}. As a diagram this is a ''single''headed arrow.

Symmetrically, the corresponding bra is {{bra|''j'', ''m''}}. In diagram form this is a ''double''headed arrow, pointing in the opposite direction to the ket.

In each case;

*the quantum numbers ''j'', ''m'' are often labelled next to the arrows to refer to a specific angular momentum state,
*arrowheads are almost always placed at the middle of the line, rather than at the tip,
*equals signs "=" are placed between equivalent diagrams, exactly like for multiple algebraic expressions equal to each other.

The most basic diagrams are for kets and bras:

{{multiple image
|align = center
|width=100
|image1=AM diagrams ket.svg
|caption1='''Ket''' {{ket|''j'', ''m''}}
|image2=AM diagrams bra.svg
|caption2='''Bra''' {{bra|''j'', ''m''}}
}}

{{-}}

Arrows are directed to or from vertices, a state transforming according to:

*a [[standard representation]] is designated by an oriented line leaving a vertex,
*a [[contrastandard representation]] is depicted as a line entering a vertex.

As a general rule, the arrows follow each other in the same sense. In the contrastandard representation, the [[T-symmetry|time reversal]] operator, denoted here by ''T'', is used. It is unitary, which means the [[Hermitian conjugate]] ''T''† equals the inverse operator ''T''−1, that is ''T''† = ''T''−1. It's action on the [[position operator]] leaves it invariant:

:<math>T \hat{\mathbf{x}} T^\dagger = \hat{\mathbf{x}} </math>

but the linear [[momentum operator]] becomes negative:

:<math>T \hat{\mathbf{p}} T^\dagger = - \hat{\mathbf{p}} </math>

and the [[spin (physics)|spin]] operator becomes negative:

:<math>T \hat{\mathbf{S}} T^\dagger = - \hat{\mathbf{S}} </math>

Since the orbital [[angular momentum operator]] is '''L''' = '''x''' × '''p''', this must also become negative:

:<math>T \hat{\mathbf{L}} T^\dagger = - \hat{\mathbf{L}} </math>

and therefore the total angular momentum operator '''J''' = '''L''' + '''S''' becomes negative:

:<math>T \hat{\mathbf{J}} T^\dagger = - \hat{\mathbf{J}} </math>

Acting on an eigenstate of angular momentum {{ket|''j'', ''m''}}, it can be shown that [see for example P.E.S. Wormer and J. Paldus (2006)]:

:<math>T \left|j,m\right\rangle \equiv \left|T (j,m)\right\rangle = {(-1)}^{j-m} \left|j,-m\right\rangle </math>

The time-reversed diagrams for kets and bras are:

{{multiple image
|align = center
|width=100
|image1=AM diagrams ket time reversed.svg
|caption1=Time reversed ket {{ket|''j'', ''m''}}.
|image2=AM diagrams bra time reversed.svg
|caption2=Time reversed bra {{bra|''j'', ''m''}}.
}}

{{-}}

It is important to position the vertex correctly, as forward-time and reversed-time operators would become mixed up.

===Inner product===

The inner product of two states {{ket|''j''1, ''m''1}} and {{ket|''j''2, ''m''2}} is:

:<math> \langle j_2 , m_2 | j_1 , m_1 \rangle = \delta_{j_1 j_2} \delta_{m_1 m_2} </math>

and the diagrams are:

{{multiple image
|align = center
|width=200
|image1=AM diagrams inner product.svg
|caption1='''Inner product''' of {{ket|''j''1, ''m''1}} and {{ket|''j''2, ''m''2}}, that is {{bra-ket|''j''2, ''m''2|''j''1, ''m''1}}.
|image2=AM diagrams inner product time reversed.svg
|caption2=Time reversed equivalent.
}}

{{-}}

For summations over the inner product, also known in this context as a contraction (c.f. [[tensor contraction]]):

:<math>\sum_m \langle j,m | j,m \rangle = 2j + 1 </math>

it is conventional to denote the result as a closed circle labelled only by ''j'', not ''m'':

:[[File:AM diagrams inner product contraction.svg|200px|center|thumb|Inner product contraction.]]

{{-}}

===Outer products===

The outer product of two states {{ket|''j''1, ''m''1}} and {{ket|''j''2, ''m''2}} is an operator:

:<math>\left| j_2 , m_2 \right\rangle \left\langle j_1 , m_1 \right|</math>

and the diagrams are:

{{multiple image
|align = center
|width=200
|image1=AM diagrams outer product.svg
|caption1='''Outer product''' of {{ket|''j''1, ''m''1}} and {{ket|''j''2, ''m''2}}, that is {{ket|''j''2, ''m''2}}{{bra|''j''1, ''m''1}}.
|image2=AM diagrams outer product time reversed.svg
|caption2=Time reversed equivalent.
}}

{{-}}

For summations over the outer product, also known in this context as a contraction (c.f. [[tensor contraction]]):

:<math>\begin{align}
\sum_m | j,m \rangle \langle j,m | & = \sum_m | j, -m \rangle \langle j, -m | \\
& = \sum_m {(-1)}^{2(j-m)}| j, -m \rangle \langle j, -m | \\
& = \sum_m {(-1)}^{j-m}| j, -m \rangle \langle j, -m |{(-1)}^{j-m} \\
& = \sum_m T| j, m \rangle \langle j, m |T^\dagger
\end{align}</math>

where the result for ''T''{{ket|''j'', ''m''}} was used, and the fact that ''m'' takes the set of values given above. There is no difference between the forward-time and reversed-time states for the outer product contraction, so here they share the same diagram, represented as one line without direction, again labelled by ''j'' only and not ''m'':

[[File:AM diagrams outer product contraction.svg|center|200px|thumb|Outer product contraction.]]

{{-}}

===Tensor products===

The tensor product &otimes; of ''n'' states {{ket|''j''1, ''m''1}}, {{ket|''j''2, ''m''2}}, ... {{ket|''j''''n'', ''m''''n''}} is written

:<math>\begin{align}
\left|j_1 , m_1 , j_2 , m_2 , ... j_n , m_n \right\rangle & \equiv \left|j_1,m_1\right\rangle\otimes\left|j_2,m_2\right\rangle\otimes\cdots\otimes\left|j_n,m_n\right\rangle \\
& \equiv \left|j_1,m_1\right\rangle \left|j_2,m_2\right\rangle \cdots \left|j_n,m_n\right\rangle
\end{align}</math>

and in diagram form, each separate state leaves or enters a common vertex creating a "fan" of arrows - ''n'' lines attached to a single vertex.

Vertices in tensor products have signs (sometimes called "node signs"), to indicate the ordering of the tensor-multiplied states:

*a ''minus'' sign '''(−)''' indicates the ordering is ''clockwise'', <math>\circlearrowright</math>, and
*a ''plus'' sign '''(+)''' for ''anticlockwise'', <math>\circlearrowleft</math>.

Signs are of course not required for just one state, diagrammatically one arrow at a vertex. Sometimes curved arrows with the signs are included to show explicitly the sense of tensor multiplication, but usually just the sign is shown with the arrows left out.

{{multiple image
|align = center
|width=200
|image1=AM diagrams tensor product kets.svg
|caption1='''Tensor product''' of {{ket|''j''1, ''m''1}}, {{ket|''j''2, ''m''2}}, {{ket|''j''3, ''m''3}}, that is {{ket|''j''1, ''m''1}}{{ket|''j''2, ''m''2}}{{ket|''j''3, ''m''3}} = {{ket|''j''1, ''m''1, ''j''2, ''m''2, ''j''3, ''m''3}}. Similarly for more than three angular momenta.
|image2=AM diagrams tensor product kets time reversed.svg
|caption2=Time reversed equivalent.
}}

{{-}}

For the inner product of two tensor product states:

:<math>\begin{align}
& \left\langle j'_n , m'_n , ... , j'_2 , m'_2 , j'_1 , m'_1 |j_1 , m_1 , j_2 , m_2 , ... j_n , m_n \right\rangle \\
= & \langle j'_n , m'_n | ... \langle j'_2 , m'_2| \langle j'_1 , m'_1 | | j_1 , m_1 \rangle | j_2 , m_2 \rangle ... | j_n , m_n \rangle \\
= & \prod_{k=1}^n \left\langle j'_k , m'_k | j_k , m_k \right\rangle
\end{align}</math>

there are ''n'' lots of inner product arrows:

{{multiple image
|align = center
|width=300
|image1=AM diagrams inner product of tensor product.svg
|caption1='''Inner product''' of {{ket|''j''′1, ''m''′1, ''j''′2, ''m''′2, ''j''′3, ''m''′3}} and {{ket|''j''1, ''m''1, ''j''2, ''m''2, ''j''3, ''m''3}}, that is {{bra-ket|''j''′3, ''m''′3, ''j''′2, ''m''′2, ''j''′1, ''m''′1|''j''1, ''m''1, ''j''2, ''m''2, ''j''3, ''m''3}}. Similarly for more than three pairs of angular momenta.
|image2=AM diagrams inner product of tensor product time reversed.svg
|caption2=Time reversed equivalent.
}}

{{-}}

==Examples and applications==



*The diagrams are well-suited for [[Clebsch–Gordan coefficients]].
*Calculations with real quantum systems, such as [[multielectron atom]]s and [[molecule|molecular]] systems.

==See also==

*[[Vector model of the atom]]
*[[Ladder operator]]
*[[Fock space]]
*[[Feynman diagrams]]

==References==

===Notes===

*{{cite article|title=Angular Momentum Diagrams|author=P.E.S. Wormer, J. Paldus|journal=Advances In Quantum Chemistry|publisher=Elsevier|volume=51|page=59-124|year=2006|issn=0065-3276|doi=10.1016/S0065-3276(06)51002-0|url=http://www.sciencedirect.com/science/article/pii/S0065327606510020}} These authors use the theta variant ''ϑ'' for the time reversal operator, here we use ''T''.

*{{cite book|title=Atomic Many-Body Theory|author=I. Lindgren, J. Morrison|publisher=Springer-Verlag|series=Chemical Physics|volume=13|edition=2nd|year=1986|isbn=3-540-166-491|url=http://books.google.co.uk/books?id=aQjwAAAAMAAJ&q=Jucys+angular+momentum+diagrams&dq=Jucys+angular+momentum+diagrams&hl=en&sa=X&ei=iBWrUa6DI-qq0QXu24HoDg&ved=0CDsQ6AEwATgK}}

===Further reading===

*{{cite book|title=Springer Handbook of Atomic, Molecular, and Optical Physics|author=G.W.F. Drake|publisher=springer|year=2006|edition=2nd|page=60|isbn=0-3872-6308-X|url=http://books.google.co.uk/books?id=Jj-ad_2aNOAC&pg=PA60&dq=Jucys+angular+momentum+diagrams&hl=en&sa=X&ei=iBWrUa6DI-qq0QXu24HoDg&ved=0CDYQ6AEwADgK#v=onepage&q=Jucys%20angular%20momentum%20diagrams&f=false}}

*{{cite book|title=Theoretical Chemistry and Physics of Heavy and Superheavy Elements|author=U. Kaldor, S. Wilson|publisher=springer|year=2003|volume=11|page=183|series=Progress in Theoretical Chemistry and Physics|isbn=1-4020-1371-X|url=http://books.google.co.uk/books?id=0xcAM5BzS-wC&pg=PA183&dq=Jucys+angular+momentum+diagrams&hl=en&sa=X&ei=fgqrUZXYMcPH0QXDhIGQDg&ved=0CGEQ6AEwCQ#v=onepage&q=Jucys%20angular%20momentum%20diagrams&f=false}}

*{{cite book|title=Fundamental World of Quantum Chemistry: A Tribute to the Memory of Per-Olov Löwdin|author=E.J. Brändas, P.O. Löwdin, E. Brändas, E.S. Kryachko|publisher=Springer|page=385|year=2004|volume=3|isbn=1-402-025-831|url=http://books.google.co.uk/books?id=w8RWWZBMlLsC&pg=PA385&dq=Jucys+angular+momentum+diagrams&hl=en&sa=X&ei=fgqrUZXYMcPH0QXDhIGQDg&ved=0CFwQ6AEwCA#v=onepage&q=Jucys%20angular%20momentum%20diagrams&f=false}}

*{{cite book|title=Relativistic Electronic Structure Theory: Part 2. Applications|author=P. Schwerdtfeger|publisher=Elsevier|page=97|year=2004|volume=14|series=Theoretical and Computational Chemistry|isbn=008-054-047-3|url=http://books.google.co.uk/books?id=VEKdnHFK3J8C&pg=PA97&dq=Jucys+angular+momentum+diagrams&hl=en&sa=X&ei=fgqrUZXYMcPH0QXDhIGQDg&ved=0CFIQ6AEwBg#v=onepage&q=Jucys%20angular%20momentum%20diagrams&f=false}}

*{{cite book|title=Relativistic Methods for Chemists|author=M. Barysz, Y. Ishikawa|publisher=Springer|page=311|year=2010|
volume=10|series=Challenges and advances in computational chemistry and physics|isbn=1-402-099-754|url=http://books.google.co.uk/books?id=QbDEC3oL7uAC&pg=PA311&dq=Jucys+angular+momentum+diagrams&hl=en&sa=X&ei=fgqrUZXYMcPH0QXDhIGQDg&ved=0CE0Q6AEwBQ#v=onepage&q=Jucys%20angular%20momentum%20diagrams&f=false}}

*{{cite book|title=Methods in Computational Molecular Physics|author=G.H.F. Diercksen, S. Wilson|publisher=Springer|year=1983|volume=113|series=Nato Science Series C|isbn=9-027-716-382|url=http://books.google.co.uk/books?id=d1cwnu-rRBMC&pg=PA158&dq=Jucys+angular+momentum+diagrams&hl=en&sa=X&ei=fgqrUZXYMcPH0QXDhIGQDg&ved=0CEkQ6AEwBA#v=onepage&q=Jucys%20angular%20momentum%20diagrams&f=false}}

*{{cite book|title=Theoretical Atomic Spectroscopy|author=Zenonas Rudzikas|chapter=8|publisher=Cambridge University Press|year=2007|isbn=0-521-026-229|location=University of Chicago|volume=7|series=Cambridge Monographs on Atomic, Molecular and Chemical Physics|url=http://books.google.co.uk/books?id=oPfRs_SQ6HoC&pg=PR8&dq=Jucys+angular+momentum+diagrams&hl=en&sa=X&ei=fgqrUZXYMcPH0QXDhIGQDg&ved=0CEQQ6AEwAw#v=onepage&q=Jucys%20angular%20momentum%20diagrams&f=false}}

*{{cite book|title=Lietuvos fizikos žurnalas|author=Lietuvos Fizikų draugija|publisher=Draugija|year=2004|location=University of Chicago|volume=44|url=http://books.google.co.uk/books?id=0hUVAQAAMAAJ&q=Jucys+angular+momentum+diagrams&dq=Jucys+angular+momentum+diagrams&hl=en&sa=X&ei=fgqrUZXYMcPH0QXDhIGQDg&ved=0CDwQ6AEwAQ}}

*{{cite book|title=Operators and Representation Theory: Canonical Models for Algebras of Operators Arising in Quantum Mechanics|author=P.E.T. Jorgensen|publisher=Elsevier|year=1987|location=University of Chicago|isbn=008-087-258-1|url=http://books.google.co.uk/books?id=dsfNGMkMHawC&pg=PA311&dq=Jucys+angular+momentum+diagrams&hl=en&sa=X&ei=iBWrUa6DI-qq0QXu24HoDg&ved=0CEEQ6AEwAjgK#v=onepage&q=Jucys%20angular%20momentum%20diagrams&f=false}}

==External links==

[[Category:Quantum mechanics]]
[[Category:Graph theory]]

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Surface chemistry of neural implants