Physics
Heisenberg Picture
The Heisenberg picture is a formulation of quantum mechanics in which the operators representing physical observables evolve with time, while the state vectors remain constant. This approach allows for a different perspective on the behavior of quantum systems and is particularly useful for studying the dynamics of systems in time-dependent situations.
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9 Key excerpts on "Heisenberg Picture"
eBook - PDF- F. Constantinescu, E. Magyari, J.A. Spiers(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
CHAPTER V Pictures and Representations 1. The Schrödinger, the Heisenberg and the Interaction Pictures In quantum mechanics, the state of a system at any given time is described by a unit vector in a Hubert space, in which sets of axes can be defined by the eigenvectors of complete sets of observables of the system. Any change with time in the state of the system can be investigated by keeping the axes fixed and allowing the state vector to rotate, or by keeping the state vector fixed and allowing the axes to rotate, or by permitting simultaneous rotation of the state vector and of the axes, using in each case the appropriate equations of motion of the vectors concerned. The three possibilities described above are called the Schrödinger, the Heisenberg and the interaction pictures respectively. Table V.l gives the equations of motion of the state vector ψ) and of any observable A of TABLE V. 1. Schrödinger picture Heisenberg Picture Interaction picture dy>(t)) « -! g ^ = JBTIV(0> (V.l) IV>(0>= tf(i,fo)IV('o)> (V.2) (V.3) Adf) = V+(t, t 0 ) AU(t, t 0 ) (V.4) dA 7SA ih -^ = M„ H 0I ] + «£/«·>♦ °-^j-£/«·> (V.5) IVX0>= t/ (0,+ (Mo)lvW>, A,(t) = U m+ (t, f 0 ) A t/<°»(/„ i 0 ). (V.6) 112 Ch.V Pictures and Representations the system in each of the three pictures (using the subscripts H and I to denote Heisenberg and interaction respectively. Quantities without subscripts refer to the Schrödinger picture.) Relations between corresponding entities in the different pictures are also given. In the interaction picture H T = H 0I +H' l9 i.e., the Hamiltonian is split into two parts: H 0I , the free Hamiltonian, and H' j9 the interaction Hamiltonian. U(t, t 0 ) and U (0 t 9 t 0 ) are unitary operators satisfying the differential equations ίΛ *ψί = Ηυ{ί , ίο) . m I^L = HoU , Kt , to) , (v.7). with the initial conditions U(t 0 , to) = 1, J7 (0) (fo, to) = 1. If H and Ho are time-independent, we obtain as solutions U(t,t 0 ) = e h , £/«»(/,/o)= h .
- Sow-Hsin Chen, Michael Kotlarchyk;;;(Authors)
- 2007(Publication Date)
- WSPC(Publisher)
Therefore, this picture shall be introduced first, followed by a transformation to the Schrödinger picture. A third viewpoint, referred to as the interaction picture , will be introduced in Chapter 6 in the context of time-dependent perturbation the-ory. The elegant Dirac formulation naturally facilitates transformations between the different pictures. The chapter closes with a discussion of angular momentum and quantum mechanics in three-dimensions. 3.1 Basic Dirac Formulation Whenoneisintroducedtoquantumphenomenaforthefirsttime,thetheoryisusually castintheformof wavemechanics . The Dirac formulation is a more fundamental ap-proach that elucidates the very underpinnings of quantum mechanics. The following sections attempt to provide an overview of the basic elements of this formalism. 46 The Transition to Quantum Mechanics 3.1.1 The State Vector: Kets, Bras, and Inner Products The wave-mechanical treatment of quantum mechanics is centered around the con-cept of a complex wavefunction , or probability amplitude , representing the state of a system. For a particle in one dimension, the wavefunction is denoted by ψ ( x,t ) . Although ψ ( x,t ) cannot be measured, the quantity | ψ ( x,t ) | 2 , called the probability density , can be accessed by experiment. | ψ ( x,t ) | 2 dx represents the probability that a measurement of the particle’s position at time t produces an outcome between x and x + dx . Wave mechanics is extremely powerful when one is interested in visualizing spatial probabilities, however, it is limited in that it only offers a partial glimpse of the total quantum-mechanical picture. A more abstract, but general, way to specify the state of a system is through the concept of a state function or state vector , denoted by | ψ angbracketright .
eBook - PDF- P I Pronin, Gennadi A Sardanashvily(Authors)
- 1996(Publication Date)
- World Scientific(Publisher)
Heisenberg Picture FOR QUANTIZED FIELDS INTERACTING WITH NONSTATIONARY ELECTROMAGNETIC OR GRAVITATIONAL BACKGROUND A.A.LOBASHOV D.I.Mendeleev Research Institute of Metrology, Moskowsky pr. 19, 198005, St.Petersburg, Russia V.M.MOSTEPANENKO St.Petersburg State Technological Institute, Department of Mathematics, Moskowsky pr. 26, 198013, St.Petersburg, Russia Abstract The Heisenberg formalism for the creation and annihilation operators of quantized scalar and spinor fields in nonstationary external electromagnetic or gravitational fields is developed. An operator connected with the Hamiltonian is constructed that depends on time as on parameter and whose eigenfunc-tions can be used to expand the field variables in the Heisenberg represen-tation. Heisenberg equations of motion are obtained both for fields and for the creation-annihilation operators. The additional terms which arise in these equations take into account the effect of particle creation from vacuum by the external field. 1 Introduction Heisenberg representation is the well-known and commonly used formalism of a stan-dard quantum field theory (see, e.g., [1, 2]). In this formalism the time dependence of the quantized fields is transferred to the particle creation and annihilation oper-ators whereas the wave functions remain constant. Such situation, however, holds only in the theory of free fields. For the case of quantized field interacting with some nonstationary external field the time dependence of the Schrodinger wave functions cannot be completely transferred to the creation and annihilation operators. This circumstance was established for the special case of a spatially homogeneous non-stationary electric fields [3, 4] (see also [5, 6]). As it was shown in [6-8], the same situation takes place for the quantized fields interacting with nonstationary space homogeneous gravitational fields.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
Within the Copenhagen interpretation of quantum mechanics the uncertainty principle was taken to mean that on ________________________ WORLD TECHNOLOGIES ________________________ an elementary level, the physical universe does not exist in a deterministic form, but rather as a collection of probabilities, or possible outcomes. For example, the pattern (probability distribution) produced by millions of photons passing through a diffraction slit can be calculated using quantum mechanics, but the exact path of each photon cannot be predicted by any known method. The Copenhagen interpretation holds that it cannot be predicted by any method, not even with theoretically infinitely precise measurements. Historical introduction Werner Heisenberg formulated the uncertainty principle at Niels Bohr's institute in Copenhagen, while working on the mathematical foundations of quantum mechanics. In 1925, following pioneering work with Hendrik Kramers, Heisenberg developed matrix mechanics, which replaced the ad-hoc old quantum theory with modern quantum mechanics. The central assumption was that the classical motion was not precise at the quantum level, and electrons in an atom did not travel on sharply defined orbits. Rather, the motion was smeared out in a strange way: the Fourier transform of time only involving those frequencies that could be seen in quantum jumps. Heisenberg's paper did not admit any unobservable quantities like the exact position of the electron in an orbit at any time; he only allowed the theorist to talk about the Fourier components of the motion. Since the Fourier components were not defined at the classical frequencies, they could not be used to construct an exact trajectory, so that the formalism could not answer certain overly precise questions about where the electron was or how fast it was going. The most striking property of Heisenberg's infinite matrices for the position and momentum is that they do not commute.
eBook - PDF- Michael V. Sadovskii(Author)
- 2019(Publication Date)
- De Gruyter(Publisher)
6 Invariant perturbation theory 6.1 Schroedinger and Heisenberg representations Let us proceed to a systematic presentation of mathematical apparatus of perturba-tion theory over interactions in quantum field theory. It is well known that there exist two main formulations for equations of motion in quantum theory. In Schroedinger representation the quantum state at a given moment of time t is represented by the state vector Ψ S ( t ) , containing the complete set of all possible results of measurements, applied to the system at this moment of time. The further evolution of the system is described by the time dependence of this state vector (wave function), described by the Schroedinger equation i ℏ 𝜕 Ψ S 𝜕 t = H S Ψ S ( t ) . (6.1) In this representation, the operators of physical variables F S do not depend on time; for all t , they are the same: dF S / dt = 0. At the same time, the average value of the operator ⟨ F S ⟩ = ⟨ Ψ S ( t ) ? ? ? ? F S ? ? ? ? Ψ S ( t )⟩, (6.2) in the general case, will depend on time as i ℏ d dt ⟨ F S ⟩ = ⟨ Ψ S ( t ) ? ? ? ? [ F S , H ] ? ? ? ? Ψ S ( t )⟩ . (6.3) Let us make the following time-dependent unitary transformation of vector Ψ S ( t ) : Φ ( t ) = V ( t ) Ψ S ( t ) , (6.4) where V ( t ) V + ( t ) = V + ( t ) V ( t ) = 1 , V + ( t ) = V − 1 ( t ) . (6.5) Then, the new state vector Φ ( t ) satisfies the equation 1 i ℏ 𝜕 Φ ( t ) 𝜕 t = ( i ℏ 𝜕 V 𝜕 t V − 1 + VH S V − 1 ) Φ ( t ) . (6.6) Let us choose V ( t ) satisfying the equation − i ℏ 𝜕 V 𝜕 t = ( VH S V − 1 ) V = VH S . (6.7) 1 We have i ℏ 𝜕 Φ ( t ) 𝜕 t = i ℏ 𝜕 V 𝜕 t Ψ S ( t )+ i ℏ V 𝜕 Ψ S 𝜕 t = i ℏ 𝜕 V 𝜕 t V − 1 Φ ( t )+ VH S Ψ S = i ℏ 𝜕 V 𝜕 t V − 1 Φ ( t )+ VH S V − 1 Φ ( t ) , which coincides with (6.6). https://doi.org/10.1515/9783110648522-006 136 | 6 Invariant perturbation theory Then, the transformed state vector will not depend on time, which is directly seen from (6.6).
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
Within the Copenhagen interpretation of quantum mechanics the uncertainty principle was taken to mean that on ________________________ WORLD TECHNOLOGIES ________________________ an elementary level, the physical universe does not exist in a deterministic form, but rather as a collection of probabilities, or possible outcomes. For example, the pattern (probability distribution) produced by millions of photons passing through a diffraction slit can be calculated using quantum mechanics, but the exact path of each photon cannot be predicted by any known method. The Copenhagen interpretation holds that it cannot be predicted by any method, not even with theoretically infinitely precise measurements. Historical introduction Werner Heisenberg formulated the uncertainty principle at Niels Bohr's institute in Copenhagen, while working on the mathematical foundations of quantum mechanics. In 1925, following pioneering work with Hendrik Kramers, Heisenberg developed matrix mechanics, which replaced the ad-hoc old quantum theory with modern quantum mec-hanics. The central assumption was that the classical motion was not precise at the quantum level, and electrons in an atom did not travel on sharply defined orbits. Rather, the motion was smeared out in a strange way: the Fourier transform of time only involving those frequencies that could be seen in quantum jumps. Heisenberg's paper did not admit any unobservable quantities like the exact position of the electron in an orbit at any time; he only allowed the theorist to talk about the Fourier components of the motion. Since the Fourier components were not defined at the classical frequencies, they could not be used to construct an exact trajectory, so that the formalism could not answer certain overly precise questions about where the electron was or how fast it was going. The most striking property of Heisenberg's infinite matrices for the position and momentum is that they do not commute.
eBook - ePub- Mark Julian Everitt, Kieran Niels Bjergstrom, Stephen Neil Alexander Duffus(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
For the specific, standard, example of position and momentum, this yields. Note that we have made no discussion of measurement, which is very deliberate. The connection between the Heisenberg uncertainty relation, the nature of quantum states, and thus the implication for measurement is often overstated. We will delay a detailed discussion of this topic until it can be properly formulated, following the introduction of the phase space description of quantum mechanics due to Wigner and others in the next chapter. 3.2.2 The Main Proposition To extract the main finding from the preceding discussion means that we have the key equations for quantum mechanics, the (tim e-dependent) Schrödinger Equation : (3.8) where we now interpret as a quantum analogue of the classical Hamiltonian, which gives the dynamics of the state vector. Let us make this more clear by way of the example of the simple harmonic oscillator. The classical Hamiltonian is: with. The quantum Hamiltonian should therefore be, by our rule, with. From this, we can now use Ehrenfest's theorem (3.9) to see if our theory so far is reasonable. Exercise 3.2 Apply Ehrenfest's theorem to find and for the simple harmonic oscillator – how do these equations compare to those for the classical coordinate and momenta? We will discuss what you find here and its significance in detail later. What would happen for a more general potential ? What would Newton think of this? Before proceeding to the next section, let us just review what we have ‘determined’ so far: we represent states by vectors in vector spaces. As a corollary, the dynamics are given by the Schrödinger equation, from which we derive Ehrenfest's theorem. By comparing this with classical mechanics, we proposed to replace Poisson brackets by commutators according to the rule given, and classical quantities by their operator counterpart, the only additional restrictions being that the Hamiltonian is Hermitian and that operators have no implicit time dependence
eBook - ePub- Steven A. Adelman(Author)
- 2021(Publication Date)
- CRC Press(Publisher)
3 The Schrodinger Equation and the Particle-in-a-BoxIn Chapters 4 –6 , we lay out the full formal basis of quantum mechanics, and in the succeeding chapters, we describe increasingly advanced applications of this basis. However, in order to convey some feeling for quantum mechanics before developing its full formal machinery in this chapter, we introduce the most useful quantum equation, the time-independent Schrodinger equation, and then apply it to one of the simplest quantum systems, the instructive one-dimensional particle-in-a-box system.The time-independent Schrodinger equation, as we will show in Section 3.2 , may be derived from the more fundamental time-dependent Schrodinger equation, already touched on in Section 1.5 . So we begin with the time-dependent Schrodinger equation. This equation cannot be derived from anything more fundamental. Rather, like Newton’s equation of motion, it is best viewed as a postulate that is accepted because it successfully predicts a vast range of phenomena.While the time-dependent Schrodinger equation cannot be derived, several non-rigorous plausibility arguments for its form exist. We next give one of these.3.1 A Heuristic “Derivation” of the Time-Dependent Schrodinger Equation
In our discussion of the photoelectric effect in Section 1.1 , we noted that Einstein discovered that light exhibits a wave–particle duality, namely that a light wave of frequency υ or wavelengthλ =where c is the speed of light, could also be viewed as a stream of particles called photons each with an energy E and momentumc υ,p .Einstein postulated that the particle properties of light E and p were related to its wave properties υ and λ as follows:E = h υ a n d p =(3.1)h λ.We further noted in Section 1.4 that de Broglie later hypothesized that ordinary particles also exhibit a wave–particle duality. Namely, de Broglie hypothesized that associated with a particle is a wave with de Broglie wavelengthλ .In analogy to Einstein’s photon relationp =, de Broglie postulated that the wavelength of the matter wave associated with a particle of momentum ph λ
- Volodymyr Krasnoholovets(Author)
- 2017(Publication Date)
- Apple Academic Press(Publisher)
p ) . An experimenter cannot know both the position and the momentum at the same moment in time. Heisenberg noted that a selection from an abundance of possibilities takes place in quantum systems, which also puts a limitation on future outcomes. Heisenberg worked with a broadened wave packet with calculable probability-a typical approach to the description of quantum systems.Later on Kennard [128 ] derived a different formulation of the uncertainty principle, which was later generalized by Robertson [129 ]: σ (q )σ (p ) ≥ h /(4π ), i.e., one cannot suppress quantum fuctuations of both position σ (q ) and momentum σ (q ) lower than a certain limit simultaneously. The fuctuations exist within themselves without respect to whether the position q and the momentum p of the quantum system are measured or not. This inequality cannot foresee any behavior of the parameters q and p at the time of measurement. Nowadays it seems that Kennard’s formulation of the uncertainty principle is more oft en used.Other authors deriving the uncertainty relation(3.5)Δ x Δ p ≥1 2ħbased their consideration on the behavior of a wave packet of finite length (see, e.g., Fermi [130 ] who referred to the proof conducted by E. Persico — see Engl. translation: Persico, E. (1950), Fundamentals of Quantum Me chanics , Prentice-Hall, New York) or the representation of the wave ψ -function as a superposition of plane waves corresponding to the discrete spectrum (Born [90 ], p. 383), and so on.In such a manner the uncertainty principle is considered as a corollary of the wave-particle duality of nature when a canonical particle is called a wave-particle and then all the characteristics of classical waves are automatically attributed to the particle. However, a wave packet is not stable and dissipates over time. Hence the description of a particle by using a wave packet is an approximate description. De Broglie [111 ] in his book devoted to the uncertainty principle analyzed this topic as well as the principle of spectral expansion in detail. He showed that the uncertainty relation (3.5 ) is due to: (i) the attribution of a wave function ψ
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