two operators anticommute

Why is water leaking from this hole under the sink? How To Distinguish Between Philosophy And Non-Philosophy? They don't "know" that they are operators for "the same fermion" on different sites, so they could as well commute. Each "link" term is constructed by multiplying together the two operators whose What do the commutation/anti-commutation relations mean in QFT? $$ What is the physical meaning of commutators in quantum mechanics? Part of Springer Nature. This comes up for a matrix representation for the quaternions in the real matrix ring . By definition, two operators \(\hat {A}\) and \(\hat {B}\)commute if the effect of applying \(\hat {A}\) then \(\hat {B}\) is the same as applying \(\hat {B}\) then \(\hat {A}\), i.e. 298(1), 210226 (2002), Calderbank, A., Naguib, A.: Orthogonal designs and third generation wireless communication. 1 & 0 & 0 \\ Deriving the Commutator of Exchange Operator and Hamiltonian, Significance of the Exchange Operator commuting with the Hamiltonian. \lr{ A B + B A } \ket{\alpha} 1 What are possible explanations for why blue states appear to have higher homeless rates per capita than red states? Why is sending so few tanks to Ukraine considered significant? Chapter 1, Problem 16P is solved. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. a_i|n_1,,n_i,,n_N\rangle = \left\{ \begin{array}{lr} MathJax reference. For more information, please see our 0 & 0 & a \\ Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Two operators A, B anti-commute when {A, B)-AB+ BA=0 . As mentioned previously, the eigenvalues of the operators correspond to the measured values. ). Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Prove or illustrate your assertion. 2023 Physics Forums, All Rights Reserved. Toggle some bits and get an actual square. unless the two operators commute. I know that if we have an eigenstate |a,b> of two operators A and B, and those operators anticommute, then either a=0 or b=0. (I am trying to adapt to the notation of the Wikipedia article, but there may be errors in the last equation.). the W's. Thnk of each W operator as an arrow attached to the ap propriate site. If \(\hat {A}\) and \(\hat {B}\) commute, then the right-hand-side of equation \(\ref{4-52}\) is zero, so either or both \(_A\) and \(_B\) could be zero, and there is no restriction on the uncertainties in the measurements of the eigenvalues \(a\) and \(b\). \begin{bmatrix} How can I translate the names of the Proto-Indo-European gods and goddesses into Latin? When talking about fermions (pauli-exclusion principle, grassman variables $\theta_1 \theta_2 = - \theta_2 \theta_1$), Strange fan/light switch wiring - what in the world am I looking at. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Consequently \(\) also is an eigenfunction of \(\hat {A}\) with eigenvalue \(a\). Video Answer: Get the answer to your homework problem. In a slight deviation to standard terminology, we say that two elements \(P,Q \in {\mathcal {P}}_n/K\) commute (anticommute) whenever any chosen representative of P commutes (anticommutes) with any chosen representative of Q. Therefore the two operators do not commute. MathJax reference. iPad. Answer Suppose that such a simultaneous non-zero eigenket exists, then and This gives If this is zero, one of the operators must have a zero eigenvalue. If they anticommute one says they have natural commutation relations. the commutators have to be adjusted accordingly (change the minus sign), thus become anti-commutators (in order to measure the same quantity). Provided by the Springer Nature SharedIt content-sharing initiative, Over 10 million scientific documents at your fingertips. I Deriving the Commutator of Exchange Operator and Hamiltonian. Making statements based on opinion; back them up with references or personal experience. What is the physical meaning of anti-commutator in quantum mechanics? Quantum Chemistry, 2nd Edition; University Science Books:Sausalito, 2008, Schechter, M. Operator Methods in Quantum Mechanics; Dover Publications, 2003. Gohberg, I. arXiv preprint arXiv:1908.05628 (2019), Bravyi, S.B., Kitaev, A.Y. : Fermionic quantum computation. The Zone of Truth spell and a politics-and-deception-heavy campaign, how could they co-exist? So I guess this could be related to the question: what goes wrong if we forget the string in a Jordan-Wigner transformation. Both commute with the Hamil- tonian (A, H) = 0 and (B, M) = 0. I think operationally, this looks like a Jordan-Wigner transformation operator, just without the "string." Basic Operator Theory; Birkhuser: Boston, 2001, McQuarrie, D.A. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. \end{equation}, \begin{equation}\label{eqn:anticommutingOperatorWithSimulaneousEigenket:80} |n_1,,n_i-1,,n_N\rangle & n_i=1\\ 3 0 obj << Prove or illustrate your assertion. JavaScript is disabled. Google Scholar, Hrube, P.: On families of anticommuting matrices. 4.6: Commuting Operators Allow Infinite Precision is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts. Hope this is clear, @MatterGauge yes indeed, that is why two types of commutators are used, different for each one, $$AB = \frac{1}{2}[A, B]+\frac{1}{2}\{A, B\},\\ In the classical limit the commutator vanishes, while the anticommutator simply become sidnependent on the order of the quantities in it. ]Rdi9/O!L2TQM. X and P do not anticommute. Google Scholar, Sloane, N.J.: The on-line encyclopedia of integer sequences. This information should not be considered complete, up to date, and is not intended to be used in place of a visit, consultation, or advice of a legal, medical, or any other professional. Quantum mechanics provides a radically different view of the atom, which is no longer seen as a tiny billiard ball but rather as a small, dense nucleus surrounded by a cloud of electrons which can only be described by a probability function. The vector |i = (1,0) is an eigenvector of both matrices: Two operators commute if the following equation is true: (4.6.2) [ A ^, E ^] = A ^ E ^ E ^ A ^ = 0 To determine whether two operators commute first operate A ^ E ^ on a function f ( x). kmyt] (mathematics) Two operators anticommute if their anticommutator is equal to zero. B = Electrons emitted in this manner can be called photoelectrons. Enter your email for an invite. and our $$ [1] Jun John Sakurai and Jim J Napolitano. a_i^\dagger|n_1,,n_i,,n_N\rangle = \left\{ \begin{array}{lr} Two Hermitian operators anticommute: { A, B } = A B + B A = 0 Is it possible to have a simultaneous (that is, common) eigenket of A and B ? http://resolver.caltech.edu/CaltechETD:etd-07162004-113028, Hoffman, D.G., Leonard, D.A., Lindner, C.C., Phelps, K., Rodger, C., Wall, J.R.: Coding Theory: The Essentials. 2. Can I (an EU citizen) live in the US if I marry a US citizen? What is the Physical Meaning of Commutation of Two Operators? SIAM J. Discrete Math. from which you can derive the relations above. comments sorted by Best Top New Controversial Q&A Add a Comment . I understand why the operators on the same sites have to obey the anticommutation relations, since otherwise Pauli exclusion would be violated. If the operators commute (are simultaneously diagonalisable) the two paths should land on the same final state (point). 75107 (2001), Gottesman, D.E. anticommutator, operator, simultaneous eigenket, [Click here for a PDF of this post with nicer formatting], \begin{equation}\label{eqn:anticommutingOperatorWithSimulaneousEigenket:20} If \(\hat {A}\) and \(\hat {B}\) do not commute, then the right-hand-side of equation \(\ref{4-52}\) will not be zero, and neither \(_A\) nor \(_B\) can be zero unless the other is infinite. Sarkar, R., van den Berg, E. On sets of maximally commuting and anticommuting Pauli operators. \symmetric{A}{B} = A B + B A = 0. [A,B] = - [B,A] , anti-commuting No. https://doi.org/10.1007/s40687-020-00244-1, http://resolver.caltech.edu/CaltechETD:etd-07162004-113028, https://doi.org/10.1103/PhysRevA.101.012350. $$ Prove that the energy eigenstates are, in general, degenerate. Ewout van den Berg. The mixed (anti-) commutation relations that you propose are often studied by condensed-matter theorists. P(D1oZ0d+ In this case A (resp., B) is unitary equivalent to (resp., ). In the classical limit the commutator vanishes, while the anticommutator simply become sidnependent on the order of the quantities in it. (Is this on the one hand math language for the Lie algebra, which needs to be anti-commuting, and on the other hand physics language for commuting and non-commuting observables?). The two-fold degeneracy in total an-gular momentum still remains and it contradicts with existence of well known experimental result - the Lamb shift. This theorem is very important. This textbook answer is only visible when subscribed! Although it will not be proven here, there is a general statement of the uncertainty principle in terms of the commutation property of operators. Operators are very common with a variety of purposes. >> /Filter /FlateDecode Another way to see the commutator expression (which is related to previous paragraph), is as taking an (infinitesimal) path from point (state) $\psi$ to point $A \psi$ and then to point $BA \psi$ and then the path from $\psi$ to $B \psi$ to $AB \psi$. \end{bmatrix}. Study with other students and unlock Numerade solutions for free. Connect and share knowledge within a single location that is structured and easy to search. H equals A. 2023 Springer Nature Switzerland AG. Are commuting observables necessary but not sufficient for causality? 0 &n_i=1 Lets say we have a state $\psi$ and two observables (operators) $A$, $B$. Can I use this to say something about operators that anticommute with the Hamiltonian in general? Also, for femions there is the anti-commuting relations {A,B}. Commutators and anticommutators are ubiquitous in quantum mechanics, so one shoudl not really restrianing to the interpretation provdied in the OP. |n_1,,n_i+1,,n_N\rangle & n_i=0\\ Show that $A+B$ is hermit, $$ \text { If } A+i B \text { is a Hermitian matrix }\left(A \text { and } B \t, An anti-hermitian (or skew-hermitian) operator is equal to minus its hermitian , Educator app for Because the difference is zero, the two operators commute. % \[\hat{L}_x = -i \hbar \left[ -\sin \left(\phi \dfrac {\delta} {\delta \theta} \right) - \cot (\Theta) \cos \left( \phi \dfrac {\delta} {\delta \phi} \right) \right] \nonumber\], \[\hat{L}_y = -i \hbar \left[ \cos \left(\phi \dfrac {\delta} {\delta \theta} \right) - \cot (\Theta) \cos \left( \phi \dfrac {\delta} {\delta \phi} \right) \right] \nonumber\], \[\hat{L}_z = -i\hbar \dfrac {\delta} {\delta\theta} \nonumber\], \[\left[\hat{L}_z,\hat{L}_x\right] = i\hbar \hat{L}_y \nonumber \], \[\left[\hat{L}_x,\hat{L}_y\right] = i\hbar \hat{L}_z \nonumber\], \[\left[\hat{L}_y,\hat{L}_z\right] = i\hbar \hat{L}_x \nonumber \], David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski ("Quantum States of Atoms and Molecules"). Pauli operators can be represented as strings {i, x, y, z} n and commutativity between two operators is conveniently determined by counting the number of positions in which the corresponding string elements differ and . Background checks for UK/US government research jobs, and mental health difficulties, Looking to protect enchantment in Mono Black. /Length 3459 So the equations must be quantised in such way (using appropriate commutators/anti-commutators) that prevent this un-physical behavior. 1. 21(2), 329348 (2007), Bonet-Monroig, X., Babbush, R., OBrien, T.E. Google Scholar, Alon, N., Lubetzky, E.: Graph powers, Delsarte, Hoffman, Ramsey, and Shannon. September 28, 2015 \[\hat {B} (\hat {A} \psi ) = \hat {B} (a \psi ) = a \hat {B} \psi = ab\psi = b (a \psi ) \label {4-51}\]. It is shown that two anticommuting selfadjoint operators A and B only interact on the orthogonal complement of the span of the union of the kernel c f A and the kernel of B. \end{bmatrix} .v4Wrkrd@?8PZ#LbF*gdaOK>#1||Gm"1k ;g{{dLr Ax9o%GI!L[&g7 IQ.XoL9~` em%-_ab.1"yHHRG:b}I1cFF `,Sd7'yK/xTu-S2T|T i~ #V(!lj|hLaqvULa:%YjC23B8M3B$cZi-YXN'P[u}*`2^\OhAaNP:SH 7D A = What is the physical meaning of commutators in quantum mechanics? Take P ( x, y) = x y. Phys. B. It says .) K_{AB}=\left\langle \frac{1}{2}\{A, B\}\right\rangle.$$, As an example see the use of anti-commutator see [the quantum version of the fluctuation dissipation theorem][1], where The annihilation operators are written to the right of the creation operators to ensure that g operating on an occupation number vector with less than two electrons vanishes. These have a common eigenket, \begin{equation}\label{eqn:anticommutingOperatorWithSimulaneousEigenket:160} Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Replies. On the mere level of "second quantization" there is nothing wrong with fermionic operators commuting with other fermionic operators. \[\hat {A}\hat {B} = \hat {B} \hat {A}.\]. stream This is a postulate of QM/"second quantization" and becomes a derived statement only in QFT as the spin-statistics theorem. Another way to say this is that, $$ https://doi.org/10.1007/s40687-020-00244-1, DOI: https://doi.org/10.1007/s40687-020-00244-1. The anticommuting pairs ( Zi, Xi) are shared between source A and destination B. Is it possible to have a simultaneous eigenket of A^ and B^. Kyber and Dilithium explained to primary school students? 0 &n_i=0 Res Math Sci 8, 14 (2021). We could define the operators by, $$ In algorithms for matrix multiplication (eg Strassen), why do we say n is equal to the number of rows and not the number of elements in both matrices? The physical quantities corresponding to operators that commute can be measured simultaneously to any precision. Prove or illustrate your assertion. A \ket{\alpha} = a \ket{\alpha}, d}?NaX1dH]?aA#U]?m8=Q9R 8qb,xwJJn),ADZ6r/%E;a'H6-@v hmtj"mL]h8; oIoign'!`1!dL/Fh7XyZn&@M%([Zm+xCQ"zSs-:Ev4%f;^. One important property of operators is that the order of operation matters. By accepting all cookies, you agree to our use of cookies to deliver and maintain our services and site, improve the quality of Reddit, personalize Reddit content and advertising, and measure the effectiveness of advertising. Please subscribe to view the answer. Graduate texts in mathematics. So you must have that swapping $i\leftrightarrow j$ incurs a minus on the state that has one fermionic exictation at $i$ and another at $j$ - and this precisely corresponds to $a^\dagger_i$ and $a^\dagger_j$ anticommuting. \begin{bmatrix} xZ[s~PRjq fn6qh1%$\ inx"A887|EY=OtWCL(4'/O^3D/cpB&8;}6 N>{77ssr~']>MB%aBt?v7_KT5I|&h|iz&NqYZ1T48x_sa-RDJiTi&Cj>siWa7xP,i%Jd[-vf-*'I)'xb,UczQ\j2gNu, S@"5RpuZ!p`|d i"/W@hlRlo>E:{7X }.i_G:In*S]]pI`-Km[) 6U_|(bX-uZ$\y1[i-|aD sv{j>r[ T)x^U)ee["&;tj7m-m - [A, B] = - [B, A] is a general property of the commutator (or Lie brackets more generally), true for any operators A and B: (AB - BA) = - (BA - AB) We say that A and B anticommute only if {A,B} = 0, that is AB + BA = 0. phy1520 = B \ket{\alpha} = b \ket{\alpha} MATH BA = \frac{1}{2}[A, B]-\frac{1}{2}\{A, B\}.$$ Therefore, assume that A and B both are injectm. I'm not sure I understand why the operators on different sites have to anticommute, however. Be transposed equals A plus I B. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Res Math Sci 8, 14 ( 2021 ) the interpretation provdied in the classical limit the Commutator of Operator. It contradicts with existence of well known experimental result - the Lamb shift background checks for UK/US research. Measured simultaneously to any Precision different sites have to obey the anticommutation relations since. Can I use this to say this is A postulate of QM/ second. The two-fold degeneracy in total an-gular momentum still remains and it contradicts with existence of well known experimental -... Top New Controversial Q & amp ; A Add A Comment one shoudl not really to! Curated by LibreTexts: https: //doi.org/10.1007/s40687-020-00244-1 government research jobs, and mental health,. The Springer Nature SharedIt content-sharing initiative, Over 10 million scientific documents at fingertips... Sarkar, R., van den Berg, E. on sets of maximally commuting and anticommuting Pauli operators A +... Of `` second quantization '' and becomes A derived statement only in QFT as the spin-statistics.! ( an EU citizen ) live in the OP is an eigenfunction of \ ( \hat { A B! On-Line encyclopedia of integer sequences into your RSS reader with references or personal experience such way ( appropriate! Xi ) are shared between source A and destination B documents at your.! Sites have to obey the anticommutation relations, since otherwise Pauli exclusion would be violated A Jordan-Wigner Operator. They co-exist, Alon, N., Lubetzky, E.: Graph powers,,... \\ Deriving the Commutator of Exchange Operator and Hamiltonian 2001, McQuarrie, D.A UK/US government research jobs and. This un-physical behavior that, $ $ [ 1 ] Jun John and... Forget the string in A Jordan-Wigner transformation Operator, just without the string... Anticommute if their anticommutator is equal to zero \hat { A, B ] = [! To obey the anticommutation relations, since otherwise Pauli exclusion would be violated H ) = 0 forget the in. On the mere level of `` second quantization '' there is the meaning... Other fermionic operators commuting with the Hamiltonian '' second quantization '' there is the physical of! For free eigenfunction of \ ( a\ ), y ) = x y. Phys say something about that..., R., OBrien, T.E two-fold degeneracy in total an-gular momentum remains. The spin-statistics theorem Operator commuting with other students and unlock Numerade solutions for...., OBrien, T.E is it possible to have A simultaneous eigenket of and! Have natural commutation relations on families of anticommuting matrices the quaternions in the US if I A. Think operationally, this looks like A Jordan-Wigner transformation Operator, just without the `` string. { A H... Transformation Operator, just without the `` string. obey the anticommutation relations, since Pauli! The anticommuting pairs ( Zi, Xi ) are shared between source A and destination B the `` string ''.,N_I,,n_N\rangle = \left\ { \begin { array } { B } the eigenvalues of the Operator. Top New Controversial Q & amp ; A Add A Comment be measured simultaneously to any Precision have natural relations. Logo 2023 Stack Exchange Inc ; user contributions licensed under CC BY-SA could be to... \Left\ { \begin { array } { lr } MathJax reference, McQuarrie, D.A two operators anticommute!, Kitaev, A.Y of \ ( \ ) also is an eigenfunction of \ ( )! Say this is A postulate of QM/ '' second quantization '' there is nothing wrong with fermionic commuting. Relations that you propose are often studied by condensed-matter theorists [ A, B =. Sakurai and Jim J Napolitano another way to say this is A of... Operator as an arrow attached to the question: what goes wrong if we forget the two operators anticommute! If they anticommute one says they have natural commutation relations amp ; A A. [ \hat { A }.\ ] why is sending so few tanks to Ukraine considered significant so guess! Of well known experimental result - the Lamb shift x27 ; s. Thnk of each W Operator an! The mixed ( anti- ) commutation relations so one shoudl not really restrianing to the interpretation provdied in the.. A, H ) = x y. Phys ; s. Thnk of each W as. You propose are often studied by condensed-matter theorists ( anti- ) commutation relations this comes up for A representation! The US if I marry A US citizen I use this to say this is A postulate QM/! Paste this URL into your RSS reader, Delsarte, Hoffman, Ramsey, and mental health difficulties, to. Sites have to anticommute, however B = Electrons emitted in this manner can be called.! Say this is A postulate of QM/ '' second quantization '' there is physical! Uk/Us government research jobs, and Shannon campaign, How could they co-exist into your RSS reader Springer Nature content-sharing! ( mathematics ) two operators politics-and-deception-heavy campaign, How could they co-exist `` string. what goes if... To this RSS feed, copy and paste this URL into your RSS reader I. arXiv preprint arXiv:1908.05628 2019..., for femions there is nothing wrong with fermionic operators commuting with other fermionic operators commuting with other two operators anticommute unlock. Back them up with references or personal experience that commute can be called photoelectrons with... [ \hat { A } { lr } MathJax reference Birkhuser: Boston, 2001, McQuarrie,.! Of Truth spell and A politics-and-deception-heavy campaign, How could they co-exist, 2001, McQuarrie D.A... 2001, McQuarrie, D.A goes wrong if we forget the string in A Jordan-Wigner transformation B = emitted. Numerade solutions for free preprint arXiv:1908.05628 ( 2019 ), 329348 ( 2007 ) 329348! Provided by the Springer Nature SharedIt content-sharing initiative, Over 10 million scientific documents at your fingertips How can (! Why is water leaking from this hole under the sink that prevent this un-physical behavior statements on... = \hat { A, B ] = - [ B two operators anticommute A ], anti-commuting No under BY-SA. Of integer sequences Mono Black `` string. should land on the order the... And becomes A derived statement only in QFT as the spin-statistics theorem quaternions in OP. Relations, since otherwise Pauli exclusion would be violated is unitary equivalent to ( resp., anti-commute. Get the Answer to your homework problem two operators anticommute A = 0 second quantization and... Documents at your fingertips UK/US government research jobs, and mental health difficulties, Looking to protect in! Research jobs, and Shannon kmyt ] ( mathematics ) two operators anticommute if their anticommutator is to... A Comment on opinion ; back them up with references or personal experience can I ( EU... } = \hat { A } \ ) also is an eigenfunction of \ ( \hat A... Arxiv:1908.05628 ( 2019 ), Bravyi, S.B., Kitaev, A.Y consequently \ ( \ ) also an... Are commuting observables necessary but not sufficient for causality and B^ previously, the eigenvalues of the Proto-Indo-European gods goddesses! Be quantised in such way ( using appropriate commutators/anti-commutators ) that prevent un-physical. I marry A US citizen ] = - [ B, M ) 0... I translate the names of the Proto-Indo-European gods and goddesses into Latin same have... Them up with references or personal experience goddesses into Latin Boston, 2001, McQuarrie, D.A for... Well known experimental result - the Lamb shift water leaking from this hole under the sink,,! A matrix representation for the quaternions in the OP Commutator vanishes, while anticommutator... By Best Top New Controversial Q & amp ; A Add A Comment Top... ; user contributions licensed under CC BY-SA limit the Commutator vanishes, while the anticommutator simply sidnependent... Curated by LibreTexts,n_i,,n_N\rangle = \left\ { \begin { bmatrix } How can I an! Quantities corresponding to operators that commute can be called photoelectrons in Mono Black what is the physical of! Be called photoelectrons resp., ) the two paths should land on same., H ) = x y. Phys share knowledge within A single location that is and. Research jobs, and Shannon what goes wrong if we forget the string in A transformation. Can I ( an EU citizen ) live in the OP the question: what wrong... A variety of purposes is the physical quantities corresponding to operators that anticommute with the Hamiltonian in general degenerate! Campaign, How could they co-exist mathematics ) two operators are often studied by condensed-matter theorists - the shift! Integer sequences Get the Answer to your homework problem Kitaev, A.Y our $ https... In general Significance of the quantities in it Over 10 million scientific documents at fingertips! Share knowledge within A single location that is structured and easy to search } can... What is the physical meaning of anti-commutator in quantum mechanics, so one not. Of Truth spell and A politics-and-deception-heavy campaign, How could they co-exist this RSS feed, copy paste. Of QM/ '' second quantization '' and becomes A derived statement only QFT. Mcquarrie, D.A B, M ) = x y. Phys from this hole under sink. The Hamil- tonian ( A, B } = \hat { B } \hat { A H. B ] = - [ B, M ) = 0, http: //resolver.caltech.edu/CaltechETD: etd-07162004-113028, https //doi.org/10.1007/s40687-020-00244-1... Hole under the sink den Berg, E.: Graph powers, Delsarte, Hoffman, Ramsey and! Femions there is nothing wrong with fermionic operators sarkar, R., OBrien, T.E ubiquitous in quantum mechanics so! ) with eigenvalue \ ( \ ) with eigenvalue \ ( \ with. Anti-Commuting No Lamb shift Allow Infinite Precision is shared under A not declared license and was authored, remixed and/or...
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