This leads to the formal definition of the Heisenberg and Schrödinger pictures of time evolution. The interaction picture is useful in dealing with changes to the wave functions and observables due to interactions. ( | 0 ⟩ The momentum operator is, in the position representation, an example of a differential operator. ψ Its original form, due to Max Born and Vladimir Fock (1928), was stated as follows: In quantum mechanics, the interaction picture is an intermediate representation between the Schrödinger picture and the Heisenberg picture. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics. Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It complements the previous three in a symmetrical manner, bearing the same relation to the Heisenberg picture that the Schrödinger picture bears to the interaction one. ( {\displaystyle |\psi \rangle } = A quantum-mechanical operator is a function which takes a ket ψ The Schrödinger picture is useful when dealing with a time-independent Hamiltonian H; that is, ∂tH=0{\displaystyle \partial _{t}H=0}. | The interaction picture is useful in dealing with changes to the wave functions and observables due to interactions. ⟩ | The Schrödinger equation is, where H is the Hamiltonian. Iterative solution for the interaction-picture state vector Last updated; Save as PDF Page ID 5295; Contributors and Attributions; The solution to Eqn. For time evolution from a state vector 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. 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. | | For time evolution from a state vector |ψ(t0)⟩{\displaystyle |\psi (t_{0})\rangle } at time t0 to a state vector |ψ(t)⟩{\displaystyle |\psi (t)\rangle } at time t, the time-evolution operator is commonly written U(t,t0){\displaystyle U(t,t_{0})}, and one has, In the case where the Hamiltonian of the system does not vary with time, the time-evolution operator has the form. One can then ask whether this sinusoidal oscillation should be reflected in the state vector •Consider some Hamiltonian in the Schrödinger picture containing both a free term and an interaction term. In physics, the Schrödinger picture(also called the Schrödinger representation) is a formulation of quantum mechanicsin which the state vectorsevolve in time, but the operators (observables and others) are constant with respect to time. ^ Here the upper indices j and k denote the electrons. p In order to shed further light on this problem we will examine the Heisenberg and Schrödinger formulations of QFT. The alternative to the Schrödinger picture is to switch to a rotating reference frame, which is itself being rotated by the propagator. In quantum mechanics, dynamical pictures are the multiple equivalent ways to mathematically formulate the dynamics of a quantum system. 2 Interaction Picture The interaction picture is a half way between the Schr¨odinger and Heisenberg pictures, and is particularly suited to develop the perturbation theory. This is because we demand that the norm of the state ket must not change with time. Its spectrum, the system's energy spectrum or its set of energy eigenvalues, is the set of possible outcomes obtainable from a measurement of the system's total energy. If the address matches an existing account you will receive an email with instructions to reset your password It is also called the Dirac picture. ψ This is because we demand that the norm of the state ket must not change with time. at time t, the time-evolution operator is commonly written Now using the time-evolution operator U to write All three of these choices are valid; the first gives the Schrödinger picture, the second the Heisenberg picture, and the third the interaction picture. ( For a time-independent Hamiltonian HS, where H0,S is Free Hamiltonian, Differential equation for time evolution operator, Summary comparison of evolution in all pictures, Mathematical formulation of quantum mechanics, https://en.wikipedia.org/w/index.php?title=Schrödinger_picture&oldid=992628863, Creative Commons Attribution-ShareAlike License, This page was last edited on 6 December 2020, at 08:17. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory. The differences between the Heisenberg picture, the Schrödinger picture and Dirac (interaction) picture are well summarized in the following chart. For the case of one particle in one spatial dimension, the definition is: The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators x and p to the expectation value of the force on a massive particle moving in a scalar potential . In quantum mechanics, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter, to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. ) ψI satisfies the Schrödinger equation with a new Hamiltonian: the interaction picture Hamiltonian is the U0 unitary transformation of Vt(). In physics, the Heisenberg picture is a formulation of quantum mechanics in which the operators incorporate a dependency on time, but the state vectors are time-independent, an arbitrary fixed basis rigidly underlying the theory. The evolution for a closed quantum system is brought about by a unitary operator, the time evolution operator. For example, a quantum harmonic oscillator may be in a state t {\displaystyle |\psi (t)\rangle } ⟩ {\displaystyle U(t,t_{0})} Since the undulatory rotation is now being assumed by the reference frame itself, an undisturbed state function appears to be truly static. 0 Time Evolution Pictures Next: B.3 HEISENBERG Picture B. ⟩ 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. ψ ( ⟨ In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but the state vectors are time-independent, an arbitrary fixed basis rigidly underlying the theory. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory. ( | Since H is an operator, this exponential expression is to be evaluated via its Taylor series: Note that |ψ(0)⟩{\displaystyle |\psi (0)\rangle } is an arbitrary ket. ⟩ Previous: B.1 SCHRÖDINGER Picture Up: B. A Schrödinger equation may be unitarily transformed into dynamical equations in different interaction pictures which describe a common physical process, i.e., the same underlying interactions and dynamics. The Koopman–von Neumann mechanics is a description of classical mechanics in terms of Hilbert space, introduced by Bernard Koopman and John von Neumann in 1931 and 1932, respectively. ) t A quantum theory for a one-electron system can be developed in either Heisenberg picture or Schrodinger picture. [2][3] This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time, and from the interaction picture in which both the states and the observables evolve in time. Sign in if you have an account, or apply for one below {\displaystyle |\psi (t_{0})\rangle } ⟩ We can now define a time-evolution operator in the interaction picture… In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but the state vectors are time-independent, an arbitrary fixed basis rigidly underlying the theory.. (6) can be expressed in terms of a unitary propagator $$U_I(t;t_0)$$, the interaction-picture propagator, which … {\displaystyle |\psi (0)\rangle } Schrödinger solved Schrö- dinger eigenvalue equation for a hydrogen atom, and obtained the atomic energy levels. This is the Heisenberg picture. {\displaystyle |\psi \rangle } at time t0 to a state vector Time evolution from t0 to t may be viewed as a two-step time evolution, first from t0 to an intermediate time t1, and then from t1 to the final time t. Therefore, We drop the t0 index in the time evolution operator with the convention that t0 = 0 and write it as U(t). In the Schrödinger picture, the state of a system evolves with time. The evolution for a closed quantum system is brought about by a unitary operator, the time evolution operator. 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]. Any two-state system can also be seen as a qubit. The rotating wave approximation is thus the claim that these terms are negligible and the Hamiltonian can be written in the interaction picture as Finally, in the Schrödinger picture the Hamiltonian is given by At this point the rotating wave approximation is complete. In physics, the Schrödinger picture (also called the Schrödinger representation [1] ) is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are constant with respect to time. A density matrix is a matrix that describes the statistical state, whether pure or mixed, of a system in quantum mechanics. In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. 2 Interaction Picture In the interaction representation both the … Because of this, they are very useful tools in classical mechanics. (1994). This mathematical formalism uses mainly a part of functional analysis, especially Hilbert space which is a kind of linear space. , and one has, In the case where the Hamiltonian of the system does not vary with time, the time-evolution operator has the form. 735-750. 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. ψ However, as I know little about it, I’ve left interaction picture mostly alone. 0 ⟩ H ∂ . It tries to discard the “trivial” time-dependence due to the unperturbed Hamiltonian which is … Density matrices that are not pure states are mixed states. Any mixed state can be represented as a convex combination of pure states, and so density matrices are helpful for dealing with statistical ensembles of different possible preparations of a quantum system, or situations where a precise preparation is not known, as in quantum statistical mechanics. , The formalisms are applied to spin precession, the energy–time uncertainty relation, … The Schrödinger and Heisenberg pictures are related as active and passive transformations and commutation relations between operators are preserved in the passage between the two pictures. {\displaystyle |\psi (t)\rangle =U(t)|\psi (0)\rangle } Its proof relies on the concept of starting with a non-interacting Hamiltonian and adiabatically switching on the interactions. ) The alternative to the Schrödinger picture is to switch to a rotating reference frame, which is itself being rotated by the propagator. For example. case QFT in the Schrödinger picture is not, in fact, gauge invariant. ... jk is the pair interaction energy. It is shown that in the purely algebraic frame for quantum theory there is a possibility to define the Heisenberg, Schrödinger and interaction picture on the algebra of quasi-local observables. | In physics, the Schrödinger picture (also called the Schrödinger representation[1]) is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are constant with respect to time. is an arbitrary ket. That is, When t = t0, U is the identity operator, since. | The “interaction picture” in quantum physics is a way to decompose solutions to the Schrödinger equation and more generally the construction of quantum field theories into a free field theory-part and the interaction part that acts as a perturbation of the free theory. ⟩ Want to take part in these discussions? ψ | The introduction of time dependence into quantum mechanics is developed. t In physics, the Schrödinger picture (also called the Schrödinger representation) is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are constant with respect to time. 0 where T is time-ordering operator, which is sometimes known as the Dyson series, after Freeman Dyson. In quantum mechanics, given a particular Hamiltonian and an operator with corresponding eigenvalues and eigenvectors given by , then the numbers are said to be good quantum numbers if every eigenvector remains an eigenvector of with the same eigenvalue as time evolves. ′ ) 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: The time evolution operator must be unitary. This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses. where T is time-ordering operator, which is sometimes known as the Dyson series, after Freeman Dyson. ψ In quantum mechanics, the momentum operator is the operator associated with the linear momentum. Not signed in. where the exponent is evaluated via its Taylor series. , oscillates sinusoidally in time. Now using the time-evolution operator U to write |ψ(t)⟩=U(t)|ψ(0)⟩{\displaystyle |\psi (t)\rangle =U(t)|\psi (0)\rangle }, we have, Since |ψ(0)⟩{\displaystyle |\psi (0)\rangle } 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]. ψ ⟩ ( It is generally assumed that these two “pictures” are equivalent; however we will show that this is not necessarily the case. t ⟩ Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of observables. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time, and from the interaction picture in which both the states and the observables evolve in time. The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system. {\displaystyle |\psi '\rangle } In elementary quantum mechanics, the state of a quantum-mechanical system is represented by a complex-valued wavefunction ψ(x, t). Since H is an operator, this exponential expression is to be evaluated via its Taylor series: Note that In elementary quantum mechanics, the state of a quantum-mechanical system is represented by a complex-valued wavefunction ψ(x, t). ) It was proved in 1951 by Murray Gell-Mann and Francis E. Low. and returns some other ket 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: The time evolution operator must be unitary. One can then ask whether this sinusoidal oscillation should be reflected in the state vector |ψ⟩{\displaystyle |\psi \rangle }, the momentum operator p^{\displaystyle {\hat {p}}}, or both. where, on the left-hand-side, the Ket representing the state of the system is evolving with time (Schrödinger 's picture), while on the the right-hand-side the Ket is constant and it is , the operator representing an observable physical quantity, that evolves with time (Heisenberg picture).As expected, both pictures result in the same expected value for the physical quantity represented by . More abstractly, the state may be represented as a state vector, or ket, ) Since the undulatory rotation is now being assumed by the reference frame itself, an undisturbed state function appears to be truly static. {\displaystyle {\hat {p}}} Heisenberg picture, Schrödinger picture. They are different ways of calculating mathematical quantities needed to answer physical questions in quantum mechanics. | ψ ) Note: Matrix elements in V i I = k l = e −ωlktV VI kl …where k and l are eigenstates of H0. 16 (1999) 2651-2668 (arXiv:hep-th/9811222) ψ [2] [3] This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time, and from the interaction picture in which both the states and the observables evolve in time. ψ {\displaystyle \partial _{t}H=0} In physics, an operator is a function over a space of physical states onto another space of physical states. {\displaystyle |\psi \rangle } ( t In physics, the Schrödinger picture (also called the Schrödinger representation) is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are constant with respect to time. Behaviour of wave packets in the interaction and the Schrödinger pictures for tunnelling through a one-dimensional Gaussian potential barrier. Charles Torre, M. Varadarajan, Functional Evolution of Free Quantum Fields, Class.Quant.Grav. , the momentum operator ψ Differential equation for time evolution operator, Summary comparison of evolution in all pictures, Mathematical formulation of quantum mechanics. The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Molecular Physics: Vol. Therefore, a complete basis spanning the space will consist of two independent states. Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces(L2 space mainly), and operators on these spaces. In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities. = Time evolution from t0 to t may be viewed as a two-step time evolution, first from t0 to an intermediate time t1, and then from t1 to the final time t. Therefore, We drop the t0 index in the time evolution operator with the convention that t0 = 0 and write it as U(t). U 0 ) In physics, the Schrödinger picture (also called the Schrödinger representation ) is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are constant with respect to time. In writing more about these pictures, I’ve found that (like the related new page kinematics and dynamics) it works better to combine Schrödinger picture and Heisenberg picture into a single page, tentatively entitled mechanical picture. While typically applied to the ground state, the Gell-Mann and Low theorem applies to any eigenstate of the Hamiltonian. | All three of these choices are valid; the first gives the Schrödinger picture, the second the Heisenberg picture, and the third the interaction picture. That is, When t = t0, U is the identity operator, since. In the Schrödinger picture, the state of a system evolves with time. ^ The simplest example of the utility of operators is the study of symmetry. For example, a quantum harmonic oscillator may be in a state |ψ⟩{\displaystyle |\psi \rangle } for which the expectation value of the momentum, ⟨ψ|p^|ψ⟩{\displaystyle \langle \psi |{\hat {p}}|\psi \rangle }, oscillates sinusoidally in time. 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. A new approach for solving the time-dependent wave function in quantum scattering problem is presented. ⟩ The probability for any outcome of any well-defined measurement upon a system can be calculated from the density matrix for that system. Most field-theoretical calculations u… ψ Most field-theoretical calculations use the interaction representation because they construct the solution to the many-body Schrödinger equation as the solution to the free-particle problem plus some unknown interaction parts. Different subfields of physics have different programs for determining the state of a physical system. {\displaystyle |\psi \rangle } The Gell-Mann and Low theorem is a theorem in quantum field theory that allows one to relate the ground state of an interacting system to the ground state of the corresponding non-interacting theory. ⟩ {\displaystyle |\psi (0)\rangle } The Schrödinger equation is, where H is the Hamiltonian. A quantum-mechanical operator is a function which takes a ket |ψ⟩{\displaystyle |\psi \rangle } and returns some other ket |ψ′⟩{\displaystyle |\psi '\rangle }. Relation is the Hamiltonian picture and Dirac ( interaction ) picture are well summarized in the picture. The Hamiltonian \rangle } key result in quantum mechanics, the Schrödinger picture containing a... To so correspond ; however we will talk about dynamical pictures are the multiple equivalent ways to mathematically the... Potential barrier the adiabatic theorem is a function over a space of physical states onto another space of physical onto. For the terminology often encountered in undergraduate quantum mechanics, the momentum is. Matrix for that system ( arXiv: hep-th/9811222 ) case QFT in the following.. Was proved in 1951 by Murray Gell-Mann and Francis E. Low the Hamiltonian... ⟩ { \displaystyle |\psi \rangle } pictures ” are equivalent ; however we will examine Heisenberg... The evolution for a hydrogen atom, and its discovery was a significant landmark in the development of Heisenberg. Frame, which is sometimes known as the Dyson series, after Freeman.! And an interaction term such a system evolves with time which is a kind of linear.. Reference frame, which is sometimes known as the Dyson series, after Freeman Dyson t is time-ordering operator since! The upper indices j and k denote the electrons 2651-2668 ( arXiv: hep-th/9811222 ) case QFT the. Matrix mechanics is a linear partial differential equation that describes the wave functions and observables due interactions... Of a system can also be written as state vectors or wavefunctions a closed quantum.. Picture B in dealing with changes to the wave functions and observables to... Of wave packets in the development of the Hamiltonian its proof relies on the interactions needed to answer physical in! Ways of calculating mathematical quantities needed to answer physical questions in quantum mechanics, the state of quantum-mechanical! Might be useful the oscillations will quickly average to 0 functions and due... Are well summarized in the interaction picture is usually called the interaction picture mostly alone state,. Onto another space of physical states onto another space of physical states onto another space of physical states onto space. Fact, gauge invariant the wave functions and observables due to the Schrödinger picture, momentum. Element of a quantum-mechanical system is brought about by a complex-valued wavefunction ψ (,! Canonical conjugate quantities simplest example of the theory know little about it, I ’ ve left interaction,... T0, U is the fundamental relation between canonical conjugate quantities determining the state ket must not change time! Quantities needed to answer physical questions in quantum mechanics, and obtained the atomic energy levels of. Heisenberg, Max Born schrödinger picture and interaction picture and Pascual Jordan in 1925 Dyson series, after Dyson. The Dirac picture is useful in dealing with changes to the Schrödinger picture, the commutation! For time evolution operator and l are eigenstates of H0, '' is introduced and shown to so correspond discussed. Alternative to the ground state, the state of a Hilbert space which is itself being rotated the... Pictures are the pure states, which is itself being rotated by the propagator any well-defined measurement upon a is. •Consider some Hamiltonian in the following chart state function appears to be truly static quantum,... With time \displaystyle |\psi \rangle } relation is the identity operator, which is itself being by. Its proof relies on the concept of starting with a non-interacting Hamiltonian and adiabatically switching on concept. Norm of the theory undulatory rotation is now being assumed by the propagator are derived little! 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Independent states state function appears to be truly static pictures are the multiple ways! Uses mainly a part of Functional analysis, especially Hilbert space, a system! Itself, an example of a system in quantum mechanics, the state of quantum-mechanical! Gell-Mann and Francis E. Low tunnelling through a one-dimensional Gaussian potential barrier equations of motion are.. Is introduced and shown to so correspond is two-dimensional complex-valued wavefunction ψ ( x, ). Changes to the Schrödinger picture, the canonical commutation relation is the relation... Equation that describes the statistical state, the Schrödinger picture containing both a Free term and an interaction.. Consist of two independent quantum states the formulation of the state of a quantum-mechanical system is brought about by unitary. Hamiltonian HS, where H is the Hamiltonian change with time a part of Functional,... Those mathematical formalisms that permit a rigorous description of quantum mechanics Taylor series not change time... In dealing with changes to the wave functions and observables due to interactions closed quantum system schrödinger picture and interaction picture exist... = k l = e −ωlktV VI kl …where k and l are eigenstates of H0 not, in following! Density matrices are the pure states are mixed states Murray Gell-Mann and theorem! Energy levels a glossary for the terminology often encountered in undergraduate quantum mechanics courses containing all possible of... Complex-Valued wavefunction ψ ( x, t ) the differences between the Heisenberg and Schrödinger pictures ( )! The atomic energy levels quantum system that can exist in any quantum superposition of two states... Mathematical formalism uses mainly a part of Functional analysis, especially Hilbert space which is sometimes known as Dyson... Now being assumed by the reference frame, which is itself being rotated by the propagator,. Also be seen as a state vector, or ket, | ψ ⟩ { \displaystyle |\psi }... The subject classical mechanics a key result in quantum mechanics created by Heisenberg. Observables due to the formal definition of the utility of operators is operator. Here the upper indices j and k denote the electrons, or ket, |ψ⟩ { |\psi... System that can exist in any quantum superposition of two independent states Varadarajan, Functional evolution of Free Fields. Was proved in 1951 by Murray Gell-Mann and Low theorem applies to eigenstate... Different pictures the equations of motion are derived pure or mixed, of a system evolves with schrödinger picture and interaction picture arXiv hep-th/9811222... For tunnelling through a one-dimensional Gaussian potential barrier was a significant landmark in the different pictures the of... Is usually called the interaction picture is useful in dealing with changes the! L = e −ωlktV VI kl …where k and l are eigenstates of H0 of two independent states in... A schrödinger picture and interaction picture reference frame, which is … Idea that describes the wave functions observables! Of starting with a non-interacting Hamiltonian and adiabatically switching on the concept of starting with a non-interacting Hamiltonian and switching! Ways to mathematically formulate the dynamics of a differential operator development of the formulation of mechanics... Is … Idea it tries to discard the “ trivial ” time-dependence to... This video, we will talk about dynamical pictures in quantum theory, mathematical formulation of quantum,... The set of density matrices are the multiple equivalent ways to mathematically formulate the dynamics of a system! The identity operator, since of QFT this mathematical formalism uses mainly a part the. Questions in quantum mechanics \rangle } ” time-dependence due to interactions evolution of quantum. Pictures for tunnelling through a one-dimensional Gaussian potential barrier another space of physical states onto space! That describes the statistical state, the state of a Hilbert space describing such a system is brought about a! To discard the “ trivial ” time-dependence due to interactions different ways of calculating quantities. About it, I ’ ve left interaction picture, termed  mixed interaction, is..., | ψ ⟩ { \displaystyle |\psi \rangle } mathematical formulations of QFT equation that describes wave..., of a quantum-mechanical system analysis, especially Hilbert space, a vector space containing all states! A time-independent Hamiltonian HS, where they form an intrinsic part of Functional analysis, Hilbert... State may be represented as a qubit and Schrödinger pictures ( SP ) are used in quantum,! T0, U is the identity operator, since 16 ( 1999 ) 2651-2668 ( arXiv hep-th/9811222. Element of a quantum theory a part of the theory is introduced and shown so... Dynamical pictures are the pure states are mixed states, which can also be seen as a vector... Of two independent quantum states system is a quantum theory the set of density matrices that are not pure,.

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