Haag's Theorem - Wikipedia, the Free Encyclopedia

Haag's theorem - Wikipedia, the free encyclopedia

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Haag's theorem

From Wikipedia, the free encyclopedia Main page Contents Rudolf Haag postulated [1] that the interaction picture does not exist in an interacting, relativistic Featured content quantum field theory (QFT), something now commonly known as Haag's Theorem. Haag's Current events original proof was subsequently generalized by a number of authors, notably Hall and Random article Wightman,[2] who reached the conclusion that a single, universal Hilbert space representation Donate to Wikipedia does not suffice for describing both free and interacting fields. In 1975, Reed and Simon proved Wikimedia Shop [3] that a Haag-like theorem also applies to free neutral scalar fields of different masses, which Interaction implies that the interaction picture cannot exist even under the absence of interactions. Help About Wikipedia Contents [hide] Community portal 1 Formal description of Haag's theorem Recent changes 2 Physical (heuristic) point of view Contact page 3 Workarounds Tools 4 Conflicting reactions of the practitioners of QFT What links here 5 References Related changes 6 Further reading Upload file Special pages Permanent link [edit] Page information Formal description of Haag's theorem Wikidata item In its modern form, the Haag theorem may be stated as follows:[4] Cite this page Consider two representations of the canonical commutation relations (CCR), and Print/export (where denote the respective Hilbert spaces and the collection of Create a book operators in the CCR). Both representations are called unitarily equivalent if and only if there Download as PDF Printable version exists some unitary mapping from Hilbert space to Hilbert space such that for each operator there exists an operator . Unitary Languages Deutsch equivalence is a necessary condition for both representations to deliver the same expectation Português values of the corresponding observables. Haag's theorem states that, contrary to ordinary non- Edit links relativistic quantum mechanics, within the formalism of QFT such a unitary mapping does not exist, or, in other words, the two representations are unitarily inequivalent. This confronts the practitioner of QFT with the so-called choice problem, namely the problem of choosing the 'right' representation among a non-denumerable set of inequivalent representations. To date, the choice problem has not found any solution.

Physical (heuristic) point of view [edit]

As was already noticed by Haag in his original work, it is the vacuum polarization that lies at the

http://en.wikipedia.org/wiki/Haag's_theorem[10/01/2015 19:03:14] Haag's theorem - Wikipedia, the free encyclopedia

core of Haag's theorem. Any interacting quantum field (including non-interacting fields of different masses) is polarizing the vacuum, and as a consequence its vacuum state lies inside a renormalized Hilbert space that differs from the Hilbert space of the free field. Although an isomorphism could always be found that maps one Hilbert space into the other, Haag's theorem implies that no such mapping would deliver unitarily equivalent representations of the corresponding CCR, i.e. unambiguous physical results.

Workarounds [edit]

Among the assumptions that lead to Haag's theorem is translation invariance of the system. Consequently, systems that can be set up inside a box with periodic boundary conditions or that interact with suitable external potentials escape the conclusions of the theorem.[5] Haag[6] and Ruelle[7] have presented the Haag-Ruelle scattering theory that is dealing with asymptotic free states and thereby serving to formalize some of the assumptions needed for the LSZ reduction formula.[8] These techniques, however, cannot be applied to massless particles and have unsolved issues with bound states.

Conflicting reactions of the practitioners of QFT [edit]

While some physicists and philosophers of physics have repeatedly emphasized how seriously Haag's theorem is shaking the foundations of QFT, the majority of QFT practitioners simply dismiss the issue. Most quantum field theory texts geared to practical appreciation of the Standard Model of elementary particle interactions do not even mention it, implicitly assuming that some rigorous set of definitions and procedures may be found to firm up the powerful and well-confirmed heuristic results they report on.

They shrug off asymptotic structure (cf. QCD jets), as they have not stumbled on a specific calculation in agreement with experiment but nevertheless failing by dint of Haag's theorem. As was pointed out by P. Teller: Everyone must agree that as a piece of mathematics Haag's theorem is a valid result that at least appears to call into question the mathematical foundation of interacting quantum field theory, and agree that at the same time the theory has proved astonishingly successful in application to experimental results.[9] T. Lupher has suggested that the wide range of conflicting reactions to Haag's theorem may partly be caused by the fact that the same exists in different formulations, which in turn were proved within different formulations of QFT such as Wightman's axiomatic approach or the LSZ formalism.[10] According to Lupher, The few who mention it tend to regard it as something important that someone (else) should investigate thoroughly.

Sklar[11] further points out:There may be a presence within a theory of conceptual problems that appear to be the result of mathematical artifacts. These seem to the theoretician to be not fundamental problems rooted in some deep physical mistake in the theory, but, rather, the consequence of some misfortune in the way in which the theory has been expressed. Haag’s Theorem is, perhaps, a difficulty of this kind.

References [edit] 1. ^ Haag, R: On quantum field theories , Matematisk-fysiske Meddelelser, 29, 12 (1955).

http://en.wikipedia.org/wiki/Haag's_theorem[10/01/2015 19:03:14] Haag's theorem - Wikipedia, the free encyclopedia

2. ^ Hall, D. and Wightman, A.S.: A theorem on invariant analytic functions with applications to relativistic quantum field theory, Matematisk-fysiske Meddelelser, 31, 1 (1957) 3. ^ Reed, M. and Simon, B.: Methods of modern mathematical physics, Vol. II, 1975, Fourier analysis, self-adjointness, Academic Press, New York 4. ^ John Earman, Doreen Fraser, Haag's Theorem and Its Implications for the Foundations of Quantum Field Theory, Erkenntnis 64, 305(2006) online at philsci-archive 5. ^ Reed, M.; Simon, B. (1979). Scattering theory. Methods of modern mathematical physics III. New York: Academic Press. 6. ^ Haag, R. (1958). "Quantum field theories with composite particles and asymptotic conditions". Phys. Rev. 112 (2): 669–673. Bibcode:1958PhRv..112..669H . doi:10.1103/PhysRev.112.669 . 7. ^ Ruelle, D. (1962). "On the asymptotic condition in quantum field theory". Helvetica Physica Acta 35: 147–163. 8. ^ Fredenhagen, Klaus (2009). Quantum field theory . Lecture Notes, Universität Hamburg. 9. ^ Teller, Paul (1997). An interpretive introduction to quantum field theory. Princeton University Press. p. 115. 10. ^ Lupher, T. (2005). "Who proved Haag's theorem?". International Journal of Theoretical Physics 44: 1993–2003. 11. ^ Sklar, Lawrence (2000), Theory and Truth: Philosophical Critique within Foundational Science. Oxford University Press.

Further reading [edit] Fraser, Doreen (2006). Haag’s Theorem and the Interpretation of Quantum Field Theories with Interactions . Ph.D. thesis. U. of Pittsburgh. Arageorgis, A. (1995). Fields, Particles, and Curvature: Foundations and Philosophical Aspects of Quantum Field Theory in Curved Spacetime. Ph.D. thesis. Univ. of Pittsburgh. Bain, J. (2000). "Against Particle/field duality: Asymptotic particle states and interpolating fields in interacting QFT (or: Who's afraid of Haag's theorem?)". Erkenntnis 53: 375–406.

Categories: Quantum field theory Theorems in quantum physics Theorems in mathematical physics

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Canonical commutation relation

From Wikipedia, the free encyclopedia Main page (Redirected from Canonical commutation relations) Contents Featured content In quantum mechanics (physics), the canonical commutation relation is the fundamental Current events relation between canonical conjugate quantities (quantities which are related by definition such Random article that one is the Fourier transform of another). For example, Donate to Wikipedia Wikimedia Shop between the position and momentum in the direction of a point particle in one dimension, Interaction x px x Help where [x , px] = x px − px x is the commutator of x and px , i is the imaginary unit, and is the About Wikipedia h reduced Planck's constant 2π . In general, position and momentum are vectors and their Community portal commutation relation between different components of position and momentum can be Recent changes expressed as Contact page . Tools What links here This relation is attributed to Max Born (1925),[1] who called it a "quantum condition" serving as a Related changes postulate of the theory; it was noted by E. Kennard (1927)[2] to imply the Heisenberg Upload file uncertainty principle. Special pages Permanent link Contents [hide] Page information 1 Relation to classical mechanics Wikidata item 2 Representations Cite this page 3 Generalizations Print/export 4 Gauge invariance Create a book 5 Angular momentum operators Download as PDF 6 See also Printable version 7 References Languages Deutsch Հայերեն Relation to classical mechanics [edit] Қазақша By contrast, in classical physics, all observables commute and the commutator would be zero. 日本語 Русский However, an analogous relation exists, which is obtained by replacing the commutator with the 中文 Poisson bracket multiplied by i : Edit links

This observation led Dirac to propose that the quantum counterparts f̂, g ̂ of classical observables f, g satisfy

http://en.wikipedia.org/wiki/Canonical_commutation_relation[10/01/2015 19:13:25] Canonical commutation relation - Wikipedia, the free encyclopedia

In 1946, Hip Groenewold demonstrated that a general systematic correspondence between quantum commutators and Poisson brackets could not hold consistently.[3] However, he did appreciate that such a systematic correspondence does, in fact, exist between the quantum commutator and a deformation of the Poisson bracket, the Moyal bracket, and, in general, quantum operators and classical observables and distributions in phase space. He thus finally elucidated the correspondence mechanism, Weyl quantization, that underlies an alternate equivalent mathematical approach to quantization known as deformation quantization.[3]

Representations [edit]

The group H3( ) generated by exponentiation of the Lie algebra specified by these commutation relations, [x, p] = i , is called the Heisenberg group. According to the standard mathematical formulation of quantum mechanics, quantum observables such as x and p should be represented as self-adjoint operators on some Hilbert space. It is relatively easy to see that two operators satisfying the above canonical commutation relations cannot both be bounded—try taking the Trace of both sides of the relations and use the relation Trace(A B ) = Trace(B A ); one gets a finite number on the right and zero on the left.[4]

These canonical commutation relations can be rendered somewhat "tamer" by writing them in terms of the (bounded) unitary operators exp(i t x) and exp(i s p), which do admit finite- dimensional representations. The resulting braiding relations for these are the so-called Weyl relations exp(i t x) exp(i s p) = exp(−i s t) exp(i s p) exp(i t x). The corresponding group commutator is then exp(i t x) exp(i s p) exp(−i t x) exp(−i s p) = exp(−i s t). The uniqueness of the canonical commutation relations between position and momentum is then guaranteed by the Stone–von Neumann theorem.

Generalizations [edit]

The simple formula

valid for the quantization of the simplest classical system, can be generalized to the case of an arbitrary Lagrangian .[5] We identify canonical coordinates (such as x in the example above, or a field Φ(x) in the case of quantum field theory) and canonical momenta πx (in the example above it is p, or more generally, some functions involving the derivatives of the canonical coordinates with respect to time):

This definition of the canonical momentum ensures that one of the Euler–Lagrange equations has the form

http://en.wikipedia.org/wiki/Canonical_commutation_relation[10/01/2015 19:13:25] Canonical commutation relation - Wikipedia, the free encyclopedia

The canonical commutation relations then amount to

where δij is the Kronecker delta. Further, it can be easily shown that

Gauge invariance [edit]

Canonical quantization is applied, by definition, on canonical coordinates. However, in the presence of an electromagnetic field, the canonical momentum p is not gauge invariant. The correct gauge-invariant momentum (or "kinetic momentum") is

(SI units) (cgs units),

where q is the particle's electric charge, A is the vector potential, and c is the speed of light. Although the quantity pkin is the "physical momentum", in that it is the quantity to be identified with momentum in laboratory experiments, it does not satisfy the canonical commutation relations; only the canonical momentum does that. This can be seen as follows.

The non-relativistic Hamiltonian for a quantized charged particle of mass m in a classical electromagnetic field is (in cgs units)

where A is the three-vector potential and φ is the scalar potential. This form of the Hamiltonian, as well as the Schrödinger equation Hψ = iħ∂ψ/∂t, the Maxwell equations and the Lorentz force law are invariant under the gauge transformation

where

and Λ=Λ(x,t) is the gauge function.

The angular momentum operator is

and obeys the canonical quantization relations http://en.wikipedia.org/wiki/Canonical_commutation_relation[10/01/2015 19:13:25] Canonical commutation relation - Wikipedia, the free encyclopedia

defining the Lie algebra for so(3), where is the Levi-Civita symbol. Under gauge transformations, the angular momentum transforms as

The gauge-invariant angular momentum (or "kinetic angular momentum") is given by

which has the commutation relations

where

is the magnetic field. The inequivalence of these two formulations shows up in the Zeeman effect and the Aharonov–Bohm effect.

Angular momentum operators [edit]

From Lx = y pz − z py, etc., it follows directly from the above that

where is the Levi-Civita symbol and simply reverses the sign of the answer under pairwise interchange of the indices. An analogous relation holds for the spin operators.

All such nontrivial commutation relations for pairs of operators lead to corresponding uncertainty relations,[6] involving positive semi-definite expectation contributions by their respective commutators and anticommutators. In general, for two Hermitian operators A and B, consider expectation values in a system in the state ψ, the variances around the corresponding expectation values being (ΔA)2 ≡ (A − A )2 , etc.

Then

where [A, B] ≡ A B − B A is the commutator of A and B, and {A, B} ≡ A B + B A is the anticommutator.

This follows through use of the Cauchy–Schwarz inequality, since | A2 | | B2 | ≥ | A B |2, and A B = ([A, B] + {A, B})/2 ; and similarly for the shifted operators A − A and B − B . (cf. Uncertainty principle derivations.)

Judicious choices for A and B yield Heisenberg's familiar uncertainty relation for x and p, as usual.

[6] Here, for Lx and Ly , in angular momentum multiplets ψ = |ℓ,m , one has

http://en.wikipedia.org/wiki/Canonical_commutation_relation[10/01/2015 19:13:25] Canonical commutation relation - Wikipedia, the free encyclopedia

2 2 2 2 Lx = Ly = (ℓ (ℓ + 1) − m ) /2 , so the above inequality yields useful constraints such as a lower bound on the Casimir invariant ℓ (ℓ + 1) ≥ m (m + 1), and hence ℓ ≥ m, among others.

See also [edit] Canonical quantization CCR algebra Lie derivative Moyal bracket

References [edit] 1. ^ Born, M.; Jordan, P. (1925). "Zur Quantenmechanik". Zeitschrift für Physik 34: 858. doi:10.1007/BF01328531 . 2. ^ Kennard, E. H. (1927). "Zur Quantenmechanik einfacher Bewegungstypen". Zeitschrift für Physik 44 (4–5): 326–352. doi:10.1007/BF01391200 . 3. ^ a b Groenewold, H. J. (1946). "On the principles of elementary quantum mechanics". Physica 12 (7): 405–460. doi:10.1016/S0031-8914(46)80059-4 . n n − 1 n n − 1 4. ^ More directly, note [x , p] = i n x , hence 2 ‖ p ‖ ‖ x ‖ ≥ n ‖ x ‖ , so that, n: 2 ‖ p ‖ ‖ x ‖ ≥ n . However, n can be arbitrarily large. Utilizing the Weyl relations, below, it can actually be shown that both operators are unbounded. 5. ^ Townsend, J. S. (2000). A Modern Approach to Quantum Mechanics. Sausalito, CA: University Science Books. ISBN 1-891389-13-0. 6. ^ a b Robertson, H. P. (1929). "The Uncertainty Principle". Physical Review 34 (1): 163–164. Bibcode:1929PhRv...34..163R . doi:10.1103/PhysRev.34.163 .

Categories: Quantum mechanics Mathematical physics

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Interaction picture

From Wikipedia, the free encyclopedia Main page Contents In quantum mechanics , the interaction picture (also known as the Dirac picture) is an intermediate Quantum mechanics Featured content representation between the Schrödinger picture and the Heisenberg picture. Whereas in the other two Current events pictures either the state vector or the operators carry time dependence, in the interaction picture both carry Random article [1] Donate to Wikipedia part of the time dependence of observables . The interaction picture is useful in dealing with the changes Uncertainty principle to the wave functions and observable due to the interactions. Most field theoretical calculations[2][3] use the Wikimedia Shop Introduction · Glossary · History interaction representation because they construct the solution to the many body Schrödinger equation as the Interaction Background [show] solution to the free particle problem plus some unknown interaction part. Help Fundamentals [show] About Wikipedia Equations that include operators acting at different times, which hold in the interaction picture, don't Experiments [show] Community portal necessarily hold in the Schrödinger or the Heisenberg picture. This is because time-dependent unitary Recent changes Formulations [hide] transformations relate operators in one picture to the analogous operators in the others. Contact page Overview Heisenberg · Interaction · Matrix · Tools Contents [hide] Phase-space · Schrödinger · What links here 1 Definition Sum-over-histories (path integral) Related changes 1.1 State vectors Equations [show] Upload file 1.2 Operators Special pages 1.2.1 Hamiltonian operator Interpretations [show] Permanent link 1.2.2 Density matrix Advanced topics [show] Page information 2 Time-evolution equations in the interaction picture Scientists [show] Wikidata item 2.1 Time-evolution of states Cite this page V · T · E 2.2 Time-evolution of operators Print/export 2.3 Time-evolution of the density matrix Create a book 3 Use of interaction picture Download as PDF 4 References Printable version 5 See also Languages Deutsch Español Definition [edit] Français Operators and state vectors in the interaction picture are related by a change of basis (unitary transformation ) to those same operators and state vectors in 한국어 Italiano the Schrödinger picture. 日本語 To switch into the interaction picture, we divide the Schrödinger picture Hamiltonian into two parts, Português Русский Suomi Татарча/tatarça Українська Any possible choice of parts will yield a valid interaction picture; but in order for the interaction picture to be useful in simplifying the analysis of a 中文 problem, the parts will typically be chosen so that H0,S is well understood and exactly solvable, while H1,S contains some harder-to-analyze Edit links perturbation to this system.

If the Hamiltonian has explicit time-dependence (for example, if the quantum system interacts with an applied external electric field that varies in

time), it will usually be advantageous to include the explicitly time-dependent terms with H1,S, leaving H0,S time-independent. We proceed assuming that this is the case. If there is a context in which it makes sense to have H0,S be time-dependent, then one can proceed by replacing by the corresponding time-evolution operator in the definitions below.

State vectors [edit] A state vector in the interaction picture is defined as [4]

where | ψS(t)〉is the state vector in the Schrödinger picture.

Operators [edit] An operator in the interaction picture is defined as

http://en.wikipedia.org/wiki/Interaction_picture[10/01/2015 19:10:00] Interaction picture - Wikipedia, the free encyclopedia

Note that AS(t) will typically not depend on t, and can be rewritten as just AS. It only depends on t if the operator has "explicit time dependence", for example due to its dependence on an applied, external, time-varying electric field.

Hamiltonian operator [edit]

For the operator H0 itself, the interaction picture and Schrödinger picture coincide,

This is easily seen through the fact that operators commute with differentiable functions of themselves. This particular operator then can be called H0 without ambiguity.

For the perturbation Hamiltonian H1,I, however,

where the interaction picture perturbation Hamiltonian becomes a time-dependent Hamiltonian—unless [H1,S, H0,S] = 0 .

It is possible to obtain the interaction picture for a time-dependent Hamiltonian H0,S(t) as well, but the exponentials need to be replaced by the unitary propagator for the evolution generated by H0,S(t), or more explicitly with a time-ordered exponential integral.

Density matrix [edit]

The density matrix can be shown to transform to the interaction picture in the same way as any other operator. In particular, let ρI and ρS be the density matrix in the interaction picture and the Schrödinger picture, respectively. If there is probability pn to be in the physical state |ψn〉, then

Evolution Picture

of: Heisenberg Interaction Schrödinger

Ket state constant

Observable constant

Density constant matrix

Time-evolution equations in the interaction picture [edit]

Time-evolution of states [edit] Transforming the Schrödinger equation into the interaction picture gives:

This equation is referred to as the Schwinger–Tomonaga equation.

Time-evolution of operators [edit]

If the operator AS is time independent (i.e., does not have "explicit time dependence"; see above), then the corresponding time evolution for AI(t) is given by

In the interaction picture the operators evolve in time like the operators in the Heisenberg picture with the Hamiltonian H' =H0.

Time-evolution of the density matrix [edit] Transforming the Schwinger–Tomonaga equation into the language of the density matrix (or equivalently, transforming the von Neumann equation into the interaction picture) gives:

Use of interaction picture [edit]

The purpose of the interaction picture is to shunt all the time dependence due to H0 onto the operators, thus allowing them to evolve freely, and leaving only H1,I to control the time-evolution of the state vectors.

The interaction picture is convenient when considering the effect of a small interaction term, H1,S, being added to the Hamiltonian of a solved system,

http://en.wikipedia.org/wiki/Interaction_picture[10/01/2015 19:10:00] Interaction picture - Wikipedia, the free encyclopedia

H0,S. By utilizing the interaction picture, one can use time-dependent perturbation theory to find the effect of H1,I, e.g., in the derivation of Fermi's golden rule, or the Dyson series , in quantum field theory : In 1947, Tomonaga and Schwinger appreciated that covariant perturbation theory could be formulated elegantly in the interaction picture, since field operators can evolve in time as free fields, even in the presence of interactions, now treated perturbatively in such a Dyson series.

References [edit] 1. ^ Albert Messiah (1966). Quantum Mechanics, North Holland, John Wiley & Sons. ISBN 0486409244 ; J. J. Sakurai (1994). Modern Quantum Mechanics (Addison-Wesley) ISBN 9780201539295 . 2. ^ J. W. Negele, H. Orland (1988), Quantum Many-particle Systems, 3. ^ Piers Coleman, The evolving monogram on Many Body Physics 4. ^ The Interaction Picture , lecture notes from New York University Townsend, John S. (2000). A Modern Approach to Quantum Mechanics, 2nd ed. Sausalito, California: University Science Books. ISBN 1- 891389-13-0.

See also [edit] Bra–ket notation Schrödinger equation Haag's theorem

Categories: Quantum mechanics

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Schrödinger picture

From Wikipedia, the free encyclopedia Main page Contents In physics , the Schrödinger picture (also called the Quantum mechanics Featured content Schrödinger representation[1]) is a formulation of Current events quantum mechanics in which the state vectors evolve in time, Random article but the operators (observables and others) are constant with Donate to Wikipedia Uncertainty principle respect to time.[2][3] This differs from the Heisenberg picture Wikimedia Shop Introduction · Glossary · History which keeps the states constant while the observables evolve Interaction Background [show] in time, and from the interaction picture in which both the Help Fundamentals [show] states and the observables evolve in time. The Schrödinger About Wikipedia Experiments [show] Community portal and Heisenberg pictures are related as active and passive Formulations [hide] Recent changes transformations and have the same measurement statistics. Overview Contact page In the Schrödinger picture, the state of a system evolves with Heisenberg · Interaction · Matrix · Tools time. The evolution for a closed quantum system is brought Phase-space · Schrödinger · What links here about by a unitary operator, the time evolution operator . For Sum-over-histories (path integral) Related changes time evolution from a state vector at time to a Equations [show] Upload file state vector at time , the time-evolution operator is Special pages Interpretations [show] Permanent link commonly written , and one has Advanced topics [show] Page information Scientists [show] Wikidata item V · T · E Cite this page In the case where the Hamiltonian of the system does not vary with time, the time-evolution operator has the form Print/export Create a book Download as PDF where the exponent is evaluated via its Taylor series. Printable version

Languages The Schrödinger picture is useful when dealing with a time-independent Hamiltonian H; that is, . العربية Deutsch [hide] Français Contents 한국어 1 Background Italiano 2 The time evolution operator 日本語 2.1 Definition Português 2.2 Properties Русский 2.2.1 Unitarity Suomi 2.2.2 Identity Татарча/tatarça 2.2.3 Closure Türkçe 2.3 Differential equation for time evolution operator Українська 3 See also 中文 4 Notes http://en.wikipedia.org/wiki/Schrödinger_picture[10/01/2015 19:19:32] Schrödinger picture - Wikipedia, the free encyclopedia

Edit links 5 Further reading

Background [edit] In elementary quantum mechanics, the state of a quantum-mechanical system is represented by a complex-valued wavefunction ψ(x, t). More abstractly, the state may be represented as a state vector, or ket, . This ket is an element of a Hilbert space, a vector space containing all possible states of the system. A quantum-mechanical operator is a function which takes a ket and returns some other ket .

The differences between the Schrödinger and Heisenberg pictures of quantum mechanics revolve around how to deal with systems that evolve in time: the time-dependent nature of the system must be carried by some combination of the state vectors and the operators. For example, a quantum harmonic oscillator may be in a state for which the expectation value of the momentum, , oscillates sinusoidally in time. One can then ask whether this sinusoidal oscillation should be reflected in the state vector , the momentum operator , or both. All three of these choices are valid; the first gives the Schrödinger picture, the second the Heisenberg picture, and the third the interaction picture.

The time evolution operator [edit]

Definition [edit]

The time-evolution operator U(t, t0) is defined as the operator which acts on the ket at time t0 to produce the ket at some other time t:

For bras , we instead have

Properties [edit]

Unitarity [edit]

The time evolution operator must be unitary . This is because we demand that the norm of the state ket must not change with time. That is,

Therefore,

Identity [edit]

When t = t0, U is the identity operator , since

Closure [edit]

Time evolution from t0 to t may be viewed as a two-step time evolution, first from t0 to an intermediate

http://en.wikipedia.org/wiki/Schrödinger_picture[10/01/2015 19:19:32] Schrödinger picture - Wikipedia, the free encyclopedia

time t1, and then from t1 to the final time t. Therefore,

Differential equation for time evolution operator [edit]

We drop the t0 index in the time evolution operator with the convention that t0 = 0 and write it as U(t). The Schrödinger equation is

where H is the Hamiltonian . Now using the time-evolution operator U to write , we have

Since is a constant ket (the state ket at t = 0 ), and since the above equation is true for any constant ket in the Hilbert space, the time evolution operator must obey the equation

If the Hamiltonian is independent of time, the solution to the above equation is[note 1]

Since H is an operator, this exponential expression is to be evaluated via its Taylor series :

Therefore,

Note that is an arbitrary ket. However, if the initial ket is an eigenstate of the Hamiltonian, with eigenvalue E, we get:

Thus we see that the eigenstates of the Hamiltonian are stationary states: they only pick up an overall phase factor as they evolve with time.

If the Hamiltonian is dependent on time, but the Hamiltonians at different times commute, then the time evolution operator can be written as

If the Hamiltonian is dependent on time, but the Hamiltonians at different times do not commute, then the time evolution operator can be written as

where T is time-ordering operator, which is sometimes known as the Dyson series, after F.J.Dyson.

http://en.wikipedia.org/wiki/Schrödinger_picture[10/01/2015 19:19:32] Schrödinger picture - Wikipedia, the free encyclopedia

The alternative to the Schrödinger picture is to switch to a rotating reference frame, which is itself being rotated by the propagator. Since the undulatory rotation is now being assumed by the reference frame itself, an undisturbed state function appears to be truly static. This is the Heisenberg picture .

See also [edit] Hamilton–Jacobi equation Interaction picture Heisenberg picture

Notes [edit] 1. ^ Here we use the fact that at t = 0, U(t) must reduce to the identity operator. 1. ^ "Schrödinger representation" . Encyclopedia of Mathematics. Retrieved 3 September 2013. 2. ^ Parker, C.B. (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. pp. 786, 1261. ISBN 0-07-051400-3. 3. ^ Y. Peleg, R. Pnini, E. Zaarur, E. Hecht (2010). Quantum mechanics. Schuam's outline series (2nd ed.). McGraw Hill. p. 70. ISBN 9-780071-623582.

Further reading [edit] Principles of Quantum Mechanics by R. Shankar, Plenum Press. Modern Quantum mechanics by J.J. Sakurai.

Categories: Foundational quantum physics Quantum mechanics Erwin Schrödinger

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Heisenberg picture

From Wikipedia, the free encyclopedia Main page Contents [1] In physics , the Heisenberg picture (also called the Heisenberg representation ) is a formulation Quantum mechanics Featured content (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables Current events and others) incorporate a dependency on time, but the state vectors are time-independent, an arbitrary fixed Random article Donate to Wikipedia basis rigidly underlying the theory. Uncertainty principle

Wikimedia Shop It stands in contrast to the Schrödinger picture in which the operators are constant, instead, and the states Introduction · Glossary · History

Interaction evolve in time. The two pictures only differ by a basis change with respect to time-dependency, which Background [show] corresponds to the difference between active and passive transformations. The Heisenberg picture is the Help Fundamentals [show] About Wikipedia formulation of matrix mechanics in an arbitrary basis, in which the Hamiltonian is not necessarily diagonal. Experiments [show] Community portal It further serves to define a third, hybrid, picture, the Interaction picture. Recent changes Formulations [hide] Contact page Overview Contents [hide] Heisenberg · Interaction · Matrix · Tools 1 Mathematical details Phase-space · Schrödinger · What links here 2 Derivation of Heisenberg's equation Sum-over-histories (path integral) Related changes 3 Commutator relations Upload file Equations [show] 4 Summary comparison of evolution in all pictures Special pages Interpretations [show] 5 See also Permanent link Advanced topics [show] Page information 6 References Scientists [show] Wikidata item 7 External links Cite this page V · T · E

Print/export Mathematical details [edit] Create a book Download as PDF In the Heisenberg picture of quantum mechanics the state vectors, |ψ(t)〉, do not change with time, while observables A satisfy Printable version

Languages العربية Deutsch Español where H is the Hamiltonian and [•,•] denotes the commutator of two operators (in this case H and A). Taking expectation values automatically yields Français the Ehrenfest theorem , featured in the correspondence principle . 한국어 Italiano By the Stone–von Neumann theorem , the Heisenberg picture and the Schrödinger picture are unitarily equivalent, just a basis change in Hilbert space . 日本語 In some sense, the Heisenberg picture is more natural and convenient than the equivalent Schrödinger picture, especially for relativistic theories. Português Lorentz invariance is manifest in the Heisenberg picture, since the state vectors do not single out the time or space. Русский Suomi This approach also has a more direct similarity to classical physics : by simply replacing the commutator above by the Poisson bracket , the Heisenberg Татарча/tatarça equation reduces to an equation in Hamiltonian mechanics. Українська Tiếng Việt Derivation of Heisenberg's equation [edit] 中文 Edit links For pedagogical reasons, the Heisenberg picture is introduced here from the subsequent, but more familiar, Schrödinger picture . The expectation value of an observable A, which is a Hermitian linear operator, for a given Schrödinger state | ψ(t)〉, is given by

In the Schrödinger picture, the state | ψ(t)〉at time t is related to the state |ψ(0)〉at time 0 by a unitary time-evolution operator , U(t),

If the Hamiltonian does not vary with time, then the time-evolution operator can be written as

where H is the Hamiltonian and ħ is the reduced Planck constant . Therefore,

Peg all state vectors to a rigid basis of | ψ(0)〉then, and define

It now follows that

http://en.wikipedia.org/wiki/Heisenberg_picture[10/01/2015 19:26:30] Heisenberg picture - Wikipedia, the free encyclopedia

Differentiation was according to the product rule , while ∂A/∂t is the time derivative of the initial A, not the A(t) operator defined. The last equation holds since exp(−iHt/ħ) commutes with H.

Thus

and hence emerges the above Heisenberg equation of motion, since the convective functional dependence on x(0) and p(0) converts to the same dependence on x(t), p(t), so that the last term converts to ∂A(t)/∂t . [X, Y] is the commutator of two operators and is defined as [X, Y] := XY − YX .

The equation is solved by the A(t) defined above, as evident by use of the standard operator identity ,

which implies

This relation also holds for classical mechanics , the classical limit of the above, given the correspondence between Poisson brackets and commutators ,

In classical mechanics, for an A with no explicit time dependence,

so, again, the expression for A(t) is the Taylor expansion around t = 0.

In effect, the arbitrary rigid Hilbert space basis | ψ(0)〉has receded from view, and is only considered at the very last step of taking specific expectation values or matrix elements of observables.

Commutator relations [edit] Commutator relations may look different than in the Schrödinger picture, because of the time dependence of operators. For example, consider the operators x(t1), x(t2), p(t1) and p(t2). The time evolution of those operators depends on the Hamiltonian of the system. Considering the one- dimensional harmonic oscillator,

the evolution of the position and momentum operators is given by:

Differentiating both equations once more and solving for them with proper initial conditions,

leads to

Direct computation yields the more general commutator relations,

, .

http://en.wikipedia.org/wiki/Heisenberg_picture[10/01/2015 19:26:30] Heisenberg picture - Wikipedia, the free encyclopedia

For , one simply recovers the standard canonical commutation relations valid in all pictures.

Summary comparison of evolution in all pictures [edit]

Evolution Picture

of: Heisenberg Interaction Schrödinger

Ket state constant

Observable constant

Density constant matrix

See also [edit] Interaction picture Bra-ket notation Schrödinger picture

References [edit] 1. ^ "Heisenberg representation" . Encyclopedia of Mathematics. Retrieved 3 September 2013. Cohen-Tannoudji, Claude ; Bernard Diu; Frank Laloe (1977). Quantum Mechanics (Volume One). Paris: Wiley. pp. 312–314. ISBN 0-471- 16433-X. Albert Messiah , 1966. Quantum Mechanics (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons.

External links [edit] Pedagogic Aides to Quantum Field Theory Click on the link for Chap. 2 to find an extensive, simplified introduction to the Heisenberg picture.

Categories: Quantum mechanics

This page was last modified on 27 December 2014 at 13:02.

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