Switch (Algebra)

Operator Switch ${\hat {A}}$ ${\ displaystyle {\ hat {A}}}$ ${\ hat {A}}$ and ${\hat {B}}$ ${\ displaystyle {\ hat {B}}}$ ${\ hat B}$ in algebra , as well as quantum mechanics is called the operator $[{\hat {A}},{\hat {B}}]={\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}}$ ${\ displaystyle [{\ hat {A}}, {\ hat {B}}] = {\ hat {A}} {\ hat {B}} - {\ hat {B}} {\ hat {A}} }$ $[{\ hat A}, {\ hat B}] = {\ hat A} {\ hat B} - {\ hat B} {\ hat A}$ . In general, it is not equal to zero. The concept of a commutator also extends to arbitrary associative algebras (not necessarily operator ones). In quantum mechanics, the name of the Poisson bracket is also attached to the commutator of operators.

If the commutator of two operators is equal to zero, then they are called commuting, otherwise noncommuting.

Switch Identities

Anti-commutativity : $[A,B]=-[B,A].$ ${\ displaystyle [A, B] = - [B, A].}$ $[A,B]=-[B,A].$ From this identity it follows that $[A,A]=0$ ${\ displaystyle [A, A] = 0}$ $[A,A]=0$ for any operator $A$ ${\ displaystyle A}$ $A$ .

The following identities are also true in associative algebra :

$[A,BC]=[A,B]C+B[A,C]$ ${\ displaystyle [A, BC] = [A, B] C + B [A, C]}$ $[A,BC]=[A,B]C+B[A,C]$ . This identity is the Leibniz rule for the operator $D_{A}=[A,\cdot ].$ ${\ displaystyle D_ {A} = [A, \ cdot].}$ $D_{A}=[A,\cdot ].$ For this reason, the operator $D_{A}$ ${\ displaystyle D_ {A}}$ $D_{A}$ called internal differentiation in algebra. The operator has a similar property ${\tilde {D}}_{A}=[\cdot ,A].$ ${\ displaystyle {\ tilde {D}} _ {A} = [\ cdot, A].}$ ${\tilde D}_{A}=[\cdot ,A].$
Jacobi Identity : $[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0.$ ${\ displaystyle [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0.}$ $[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0.$ An algebra satisfying the Jacobi identity is called a Lie algebra . Thus, from any associative algebra one can obtain a Lie algebra if one defines multiplication in a new algebra as a commutator of elements of an old algebra.
$[AB,C]+[BC,A]+[CA,B]=0$ ${\ displaystyle [AB, C] + [BC, A] + [CA, B] = 0}$ $[AB,C]+[BC,A]+[CA,B]=0$ This identity is another record of Jacobi's identity.
$[AB,C]=A[B,C]+[A,C]B$ ${\ displaystyle [AB, C] = A [B, C] + [A, C] B}$ $[AB,C]=A[B,C]+[A,C]B$
$[ABC,D]=AB[C,D]+A[B,D]C+[A,D]BC$ ${\ displaystyle [ABC, D] = AB [C, D] + A [B, D] C + [A, D] BC}$ $[ABC,D]=AB[C,D]+A[B,D]C+[A,D]BC$
$[AB,CD]=A[B,CD]+[A,CD]B=A[B,C]D+AC[B,D]+[A,C]DB+C[A,D]B$ ${\ displaystyle [AB, CD] = A [B, CD] + [A, CD] B = A [B, C] D + AC [B, D] + [A, C] DB + C [A, D ] B}$ $[AB,CD]=A[B,CD]+[A,CD]B=A[B,C]D+AC[B,D]+[A,C]DB+C[A,D]B$
$[A,BCD]=[A,B]CD+B[A,C]D+BC[A,D]$ ${\ displaystyle [A, BCD] = [A, B] CD + B [A, C] D + BC [A, D]}$ $[A,BCD]=[A,B]CD+B[A,C]D+BC[A,D]$
$[A,BCDE]=[A,B]CDE+B[A,C]DE+BC[A,D]E+BCD[A,E]$ ${\ displaystyle [A, BCDE] = [A, B] CDE + B [A, C] DE + BC [A, D] E + BCD [A, E]}$ $[A,BCDE]=[A,B]CDE+B[A,C]DE+BC[A,D]E+BCD[A,E]$
$[ABCD,E]=ABC[D,E]+AB[C,E]D+A[B,E]CD+[A,E]BCD$ ${\ displaystyle [ABCD, E] = ABC [D, E] + AB [C, E] D + A [B, E] CD + [A, E] BCD}$ $[ABCD,E]=ABC[D,E]+AB[C,E]D+A[B,E]CD+[A,E]BCD$
$[[[A,B],C],D]+[[[B,C],D],A]+[[[C,D],A],B]+[[[D,A],B],C]=[[A,C],[B,D]]$ ${\ displaystyle [[[[A, B], C], D] + [[[B, C], D], A] + [[[C, D], A], B] + [[[D, A], B], C] = [[A, C], [B, D]]}$ $[[[A,B],C],D]+[[[B,C],D],A]+[[[C,D],A],B]+[[[D,A],B],C]=[[A,C],[B,D]]$
$e^{A}Be^{-A}=B+[A,B]+{\frac {1}{2!}}[A,[A,B]]+{\frac {1}{3!}}[A,[A,[A,B]]]+\cdots \equiv e^{\operatorname {ad} (A)}B.$ ${\ displaystyle e ^ {A} Be ^ {- A} = B + [A, B] + {\ frac {1} {2!}} [A, [A, B]] + {\ frac {1} { 3!}} [A, [A, [A, B]]] + \ cdots \ equiv e ^ {\ operatorname {ad} (A)} B.}$ $e^{A}Be^{-A}=B+[A,B]+{\frac {1}{2!}}[A,[A,B]]+{\frac {1}{3!}}[A,[A,[A,B]]]+\cdots \equiv e^{\operatorname {ad} (A)}B.$ This formula is valid in algebras where a matrix exponent can be defined, for example, in a Banach algebra or in a ring of formal power series . It also plays a crucial role in quantum mechanics and quantum field theory in constructing perturbation theory for operators in the Heisenberg representation and the interaction representation .
$\ln \left(e^{A}e^{B}e^{-A}e^{-B}\right)=[A,B]+{\frac {1}{2!}}[(A+B),[A,B]]+{\frac {1}{3!}}\left([A,[B,[B,A]]]/2+[(A+B),[(A+B),[A,B]]]\right)+\cdots .$ ${\ displaystyle \ ln \ left (e ^ {A} e ^ {B} e ^ {- A} e ^ {- B} \ right) = [A, B] + {\ frac {1} {2!} } [(A + B), [A, B]] + {\ frac {1} {3!}} \ Left ([A, [B, [B, A]]] / 2 + [(A + B ), [(A + B), [A, B]]] \ right) + \ cdots.}$

Switch in quantum mechanics

As you know, the physical measurement in quantum mechanics corresponds to the action of the operator ${\hat {F}}$ ${\ displaystyle {\ hat {F}}}$ physical quantity $f$ ${\ displaystyle f}$ to the system state vector . The so-called pure states in which a physical quantity has a strictly defined value correspond to eigenvectors ${\hat {F}}$ ${\ displaystyle {\ hat {F}}}$ , while the value of the quantity in this state is the eigenvalue of the vector of the pure state:

{\hat {F}}\psi _{i}=f\psi _{i}

{\ displaystyle {\ hat {F}} \ psi _ {i} = f \ psi _ {i}}

If two quantum-mechanical quantities are simultaneously measurable, then in pure states they will both have a certain value, that is, the sets of eigenvectors of the quantity operators coincide. But then they will commute:

{\hat {F}}{\hat {G}}\psi _{i}=g{\hat {F}}\psi _{i}=gf\psi _{i}={\hat {G}}{\hat {F}}\psi _{i}

{\ displaystyle {\ hat {F}} {\ hat {G}} \ psi _ {i} = g {\ hat {F}} \ psi _ {i} = gf \ psi _ {i} = {\ hat {G}} {\ hat {F}} \ psi _ {i}}

Accordingly, non-commuting operators correspond to physical quantities that do not have a certain value at the same time. A typical example is momentum operators (momentum components) ${\hat {p}}_{x}=-i\hbar {\frac {\partial }{\partial x}}$ ${\ displaystyle {\ hat {p}} _ {x} = - i \ hbar {\ frac {\ partial} {\ partial x}}}$ and corresponding coordinate ${\hat {x}}=x$ ${\ displaystyle {\ hat {x}} = x}$ (see uncertainty relation ).

Conservation Laws

The eigenvalues of the Hamiltonian of a quantum system are energy values in stationary states. The obvious consequence of the foregoing is that a physical quantity whose operator commutes with the Hamiltonian can be measured simultaneously with the energy of the system. However, in quantum mechanics, energy takes on a special role. From the Schrödinger equation

\imath \hbar {\frac {\partial \psi }{\partial t}}={\hat {H}}\psi

{\ displaystyle \ imath \ hbar {\ frac {\ partial \ psi} {\ partial t}} = {\ hat {H}} \ psi}

and determining the total time derivative of the operator

{\dot {\hat {f}}}={\hat {\dot {f}}}

{\ displaystyle {\ dot {\ hat {f}}} = {\ hat {\ dot {f}}}}

you can get an expression for the total time derivative of a physical quantity, namely:

{\dot {\hat {f}}}={\imath  \over \hbar }[{\hat {H}},{\hat {f}}]+{\frac {\partial {\hat {f}}}{\partial t}}

{\ displaystyle {\ dot {\ hat {f}}} = {\ imath \ over \ hbar} [{\ hat {H}}, {\ hat {f}}] + {\ frac {\ partial {\ hat {f}}} {\ partial t}}}

Consequently, if the operator of a physical quantity commutes with the Hamiltonian, then this quantity does not change over time . This relation is a quantum analogue of the identity

{\dot {f}}={\mathcal {\{}}H,f{\mathcal {\}}}+{\frac {\partial f}{\partial t}}

{\ displaystyle {\ dot {f}} = {\ mathcal {\ {}} H, f {\ mathcal {\}}} + {\ frac {\ partial f} {\ partial t}}}

from classical mechanics, where {,} is the Poisson bracket of functions. Similar to the classical case, it expresses the presence of certain symmetries in the system that generate integrals of motion . It is the conservation property with certain symmetries of space that underlies the definition of many quantum analogs of classical quantities, for example, the momentum is defined as the quantity that remains during all translations of the system, and the angular momentum is determined as the quantity that remains during rotations.

Some switching relationships

We indicate the values of some commonly encountered switches.

{\hat {r}}_{i},{\hat {p}}_{i},{\hat {L}}_{i}

{\ displaystyle {\ hat {r}} _ {i}, {\ hat {p}} _ {i}, {\ hat {L}} _ {i}}

- the operator of the i-th component, respectively, of the radius vector, momentum and angular momentum ;

\delta _{ij}

{\ displaystyle \ delta _ {ij}}

- Kronecker delta ;

e_{ijk}

{\ displaystyle e_ {ijk}}

- absolutely antisymmetric 3rd rank pseudo-tensor .

[{\hat {r}}_{i},{\hat {p}}_{j}]=\imath \hbar \delta _{ij}

{\ displaystyle [{\ hat {r}} _ {i}, {\ hat {p}} _ {j}] = \ imath \ hbar \ delta _ {ij}}

[{\hat {p}},f({\vec {r}})]=-\imath \hbar \nabla f

{\ displaystyle [{\ hat {p}}, f ({\ vec {r}})] = - \ imath \ hbar \ nabla f}

[{\hat {L}}_{i},{\hat {r}}_{j}]=\imath \hbar e_{ijk}{\hat {r}}_{k}

{\ displaystyle [{\ hat {L}} _ {i}, {\ hat {r}} _ {j}] = \ imath \ hbar e_ {ijk} {\ hat {r}} _ {k}}

[{\hat {L}}_{i},{\hat {p}}_{j}]=\imath \hbar e_{ijk}{\hat {p}}_{k}

{\ displaystyle [{\ hat {L}} _ {i}, {\ hat {p}} _ {j}] = \ imath \ hbar e_ {ijk} {\ hat {p}} _ {k}}

[{\hat {L}}_{i},{\hat {L}}_{j}]=\imath \hbar e_{ijk}{\hat {L}}_{k}

{\ displaystyle [{\ hat {L}} _ {i}, {\ hat {L}} _ {j}] = \ imath \ hbar e_ {ijk} {\ hat {L}} _ {k}}

[{\hat {L}}^{2},{\hat {L}}_{i}]=0

{\ displaystyle [{\ hat {L}} ^ {2}, {\ hat {L}} _ {i}] = 0}

As a rule, relations are necessary for the normalized moment: $\ {\hat {L}}_{j}=\hbar {\hat {l}}_{j}$ ${\ displaystyle \ {\ hat {L}} _ {j} = \ hbar {\ hat {l}} _ {j}}$

[{\hat {l}}_{i},{\hat {r}}_{j}]=\imath e_{ijk}{\hat {r}}_{k}

{\ displaystyle [{\ hat {l}} _ {i}, {\ hat {r}} _ {j}] = \ imath e_ {ijk} {\ hat {r}} _ {k}}

[{\hat {l}}_{i},{\hat {p}}_{j}]=\imath e_{ijk}{\hat {p}}_{k}

{\ displaystyle [{\ hat {l}} _ {i}, {\ hat {p}} _ {j}] = \ imath e_ {ijk} {\ hat {p}} _ {k}}

[{\hat {l}}_{i},{\hat {l}}_{j}]=\imath e_{ijk}{\hat {l}}_{k}

{\ displaystyle [{\ hat {l}} _ {i}, {\ hat {l}} _ {j}] = \ imath e_ {ijk} {\ hat {l}} _ {k}}

[{\hat {l}}^{2},{\hat {l}}_{i}]=0

{\ displaystyle [{\ hat {l}} ^ {2}, {\ hat {l}} _ {i}] = 0}

From these relations it is clear that the angular momentum of a particle is not measurable simultaneously with its coordinates or momentum. Moreover, except when the moment is zero, its various components are not measurable at the same time. This makes the angular momentum fundamentally different from the momentum and the radius vector, in which all three components can be simultaneously determined. For the angular momentum, only its projection onto some axis (usually z ) and the square of its length can be measured.

Lie Algebra of Physical Quantities

The switch is a quantum analogue of the Poisson bracket in classical mechanics . A commutator operation introduces the structure of a Lie algebra on operators (or elements of an algebra); therefore, anticommutative multiplication in a Lie algebra is also called a commutator.

Non-Switching Values

Non-commuting quantities A and B are quantities whose commutator $[A,B]=AB-BA\neq 0$ ${\ displaystyle [A, B] = AB-BA \ neq 0}$ .

Two physical quantities are simultaneously measurable if and only if their operators commute ^[1] .

Anti-

An anticommutator is a symmetrizing operator over elements of a ring that determines the degree of “anticommutativity” of multiplication in a ring:

[x,y]_{+}:=xy+yx

{\ displaystyle [x, y] _ {+}: = xy + yx}

A commutative " Jordan multiplication " is introduced through the anticommutator. Clifford’s algebra always naturally connects the anticommutator with its bilinear form.

Examples

The anticommutator of a pair of different imaginary units in quaternions is equal to zero.
Using the anti-commutator , Dirac gamma matrices are determined.

Literature

Blokhintsev D.I. Fundamentals of quantum mechanics. - 5th ed. - M .: Nauka, 1976 .-- 664 p.
Boome A. Quantum Mechanics: Fundamentals and Applications. - M.: Mir, 1990 .-- 720c.
Dirac P. Principles of quantum mechanics. - 2nd ed. - M .: Nauka, 1979.- 480 s.
Landau, L.D. , Lifshits, E.M. Quantum mechanics (nonrelativistic theory). - 4th edition. - M .: Nauka , 1989 .-- 768 p. - (“ Theoretical Physics ”, Volume III). - ISBN 5-02-014421-5 .

Notes

↑ 3.7. Simultaneous measurement of different physical quantities

[1] 3.7. Simultaneous measurement of different physical quantities