Classical mechanics correspondence principle

In order to understand the relation of the 4-current to the spin density, it is important to realize that the definition of the current density (naturally) involves a velocity operator, which is in close analogy to classical mechanics (correspondence principle), as we have seen before. As the velocity operator in Dirac s theory contains Dirac matrices a which are composed of Pauli spin matrices cr, we understand that the current density j carries the spin information. 

The principles of action in classical and quantum mechanics perspective - short history. Basic concepts of force, motion, mass and units of physical quantities used in laws of motion. Quick survey of laws of motion. The Lagiangian function and its main role in the principle of least action. The motion by Euler-Lagrange equation. Newton equation and the second principle of classical mechanics. Correspondence with quantum mechanics. 

The fifth postulate and its corollary are extremely important concepts. Unlike classical mechanics, where everything can in principle be known with precision, one can generally talk only about the probabilities associated with each member of a set of possible outcomes in quantum mechanics. By making a measurement of the quantity A, all that can be said with certainty is that one of the eigenvalues of /4 will be observed, and its probability can be calculated precisely. However, if it happens that the wavefiinction corresponds to one of the eigenfunctions of the operator A, then and only then is the outcome of the experiment certain the measured value of A will be the corresponding eigenvalue. 

The classical model predicts that the largest probability of finding a particle is when it is at the endpoints of the vibration. The quantum-mechanical picture is quite different. In the lowest vibrational state, the maximum probability is at the midpoint of the vibration. As the quantum number v increases, then the maximum probability approaches the classical picture. This is called the correspondence principle. Classical and quantum results have to agree with each other as the quantum numbers get large. 

It is now shown how the abrupt changes in the eigenvalue distribution around the central critical point relate to changes in the classical mechanics, bearing in mind that the analog of quantization in classical mechanics is a transformation of the Hamiltonian from a representation in the variables pR, p, R, 0) to one in angle-action variables (/, /e, Qr, 0) such that the transformed Hamiltonian depends only on the actions 1r, /e) [37]. Hamilton s equations diR/dt = (0///00 j), etc.) then show that the actions are constants of the motion, which are related to the quantum numbers by the Bohr correspondence principle [23]. In the present case,... 

According to the correspondence principle as stated by N. Bohr (1928), the average behavior of a well-defined wave packet should agree with the classical-mechanical laws of motion for the particle that it represents. Thus, the expectation values of dynamical variables such as position, velocity, momentum, kinetic energy, potential energy, and force as calculated in quantum mechanics should obey the same relationships that the dynamical variables obey in classical theory. This feature of wave mechanics is illustrated by the derivation of two relationships known as Ehrenfest s theorems. 

Equation (2.23) is the quantum-mechanical analog of the classical definition of momentum, p = mv = m(Ax/At). This derivation also shows that the association in quantum mechanics of the operator (h/i)(d/dx) with the momentum is consistent with the correspondence principle. 

The infinite potential barrier, shown schematically in figure 10 corresponds to a superselection rule that operates below the critical temperature [133]. Above the critical temperature the quantum-mechanical superposition principle applies, but below that temperature the system behaves classically. The system bifurcates spontaneously at the critical point. The bifurcation, like second-order phase transformation is caused by some interaction that becomes dominant at that point. In the case of chemical reactions the interaction leads to the rearrangement of chemical bonds. The essential difference between chemical reaction and second-order phase transition is therefore epitomized by the formation of chemically different species rather than different states of aggregation, when the symmetry is spontaneously broken at a critical point. 

For a family of trajectories all starting at the value X(to) and at t=t all arriving at X(t), there is one trajectory that renders the action stationary. The classical mechanical trajectory of a given dynamical system is the one for which 5S=0, i.e. the action becomes stationary. The equation of motion is obtained from this variational principle [59], The corresponding Euler-Lagrange equations are obtained d(3L/3vk)/dt = 9L/dXk. In Cartesian coordinates these equations become Newton s equations of motion for each nucleus of mass Mk ... 

In classical mechanics, Newton s laws of motion determine the path or time evolution of a particle of mass, m. In quantum mechanics what is the corresponding equation that governs the time evolution of the wave function, F(r, t) Obviously this equation cannot be obtained from classical physics. However, it can be derived using a plausibility argument that is centred on the principle of wave-particle duality. Consider first the case of a free particle travelling in one dimension on which no forces act, that is, it moves in a region of constant potential, V. Then by the conservation of energy... 

According to the correspondence principle the classical expression for the electron density p(r,t) can be converted to the quantum mechanical description by taking into account that the particle density is calculated by integration of the product of the iV-electron wave function 4 1 and its complex conjugate 4. We introduce the charge-weighted density by multiplication of the electron density with the electron charge,... 

The above discussion of the correspondence principle was not entirely satisfactory. It may be true that for very high quantum numbers, the wavefunction represents a probability distribution indistinguishable from that predicted in classical mechanics. However, classical mechanics does not need probability distributions, as it deals with precisely known trajectories. How can the wave picture be compatible with these To answer this question, we must look at a wavefunction constructed by the superposition of sinusoidal waves with different lengths. We use the earlier equations in this chapter to write... 

The uncertainty principle shows that the classical trajectory of a particle, with a precisely determined position and momentum, is really an illusion. It is a very good approximation, however, for macroscopic bodies. Consider a particle with mass I Xg, and position known to an accuracy of 1 pm. Equation 2.41 shows that the uncertainty in momentum is at least 5 x 10 29 kg m s-1, corresponding to a velocity of 5 x 10 JO m s l. This is totally negligible for any practical purpose, and it illustrates that in the macroscopic world, even with very light objects, the uncertainty principle is irrelevant. If we wanted to, we could describe these objects by wave packets and use the quantum theory, but classical mechanics gives essentially the same answer, and is much easier. At the atomic and molecular level, however, especially with electrons, which are very light, we must abandon the idea of a classical trajectory. The statistical predictions provided by Bom s interpretation of the wavefunction are the best that can be obtained. 

Just by considering equation (4) one may speculate that the NACTs might be similar to the electromagnetic vector potential, S. It is known from classical mechanics that the momentum p of a charged particle in an electromagnetic field changes to p — p + eS - a substitution termed as the minimal principle [1]. Due to the correspondence principle the quantum mechanical minimal principle becomes V—>- V+ i(e/fi)S. However, the NACTs in equation (4), when considering each element separately, do not combine with V (because the... 

Equation 4.7 is the Bohr postulate, that electrons can defy Maxwell s laws provided they occupy an orbit whose angular momentum (corresponding to an orbit of appropriate radius) satisfies Eq. 4.7. The Bohr postulate is not based on a whim, as most textbooks imply, but rather follows from (1) the Plank equation Eq. 4.3, AE = hv and (2) starting with an orbit of large radius such that the motion is essentially linear and classical physics applies, as no acceleration is involved, then extrapolating to small-radius orbits. The fading of quantum-mechanical equations into their classical analogues as macroscopic conditions are approached is called the correspondence principle [11]. 

The concept of (approximately) transferable, localized electron-domains provides a link between quantum physics and classical chemical theory and serves to clarify, from the viewpoint of physics, the status of classical chemical concepts. This link provides a chemist, therefore, with an intuitive understanding of quantum mechanical relations, in the sense that it permits one to guess qualitatively, through the use of classical chemical theory, what answers rigorous applications of the quantum mechanical formalism would give when applied to simple chemical problems 157>. Through the Correspondence Principle, the electron-... 

Summing up the above, we may conclude that the classical system of equations (5.93) and (5.94), together with the above given additional terms, coincides perfectly with the asymptotic system of equations of motion of quantum mechanical polarization moments (5.87) and (5.88). This result was actually to be expected from correspondence principle considerations. 

Arriving subsequently at rigorous quantum mechanical descriptions, we have assumed that the reader has some preliminary knowledge of basic quantum mechanical formalism. We consider it methodologically important to illustrate the correspondence principle between quantum and classical concepts, in particular between the concept of coherence of the wave functions of magnetic sublevels, and the symmetry properties of spatial angular momenta distribution. 

When first confronted with the oddities of quantum effects Bohr formulated a correspondence principle to elucidate the status of quantum mechanics relative to the conventional mechanics of macroscopic systems. To many minds this idea suggested the existence of some classical/quantum limit. Such a limit between classical and relativistic mechanics is generally defined as the point where the velocity of an object v —> c, approaches the velocity of light. By analogy, a popular definition of the quantum limit is formulated as h —> 0. However, this is nonsense. Planck s constant is not variable. 

As mentioned earlier, we cannot make use of the correspondence principle to derive quantum mechanical spin operators, because spin has no classical analog. Instead, the spin eigenfunctions sms) may be identified with u 1)... 

At the limit of large quantum numbers, the results of quantum mechanics must agree with classical mechanics (Bohr correspondence principle). 

It is a constant of the motion for the classical Kepler problem (Saletan and Cromer, 1971). The magnitude of X is proportional to the eccentricity of the orbit and X points in the direction of the major axis. Using the correspondence principle Pauli first showed that its quantum mechanical analog,... 

In the Poisson case, the decoherence theory affords a more satisfactory justification for the correspondence principle [20]. Adopting the Wigner formalism, it is possible to express quantum mechanical problems in terms of the classical phase space, and the Wigner quasi-probability is expected to remain positive definite until the instant at which a quantum transition occurs, according to the estimate of Ref. 120, at the time... 

We denote by G the set of all the experimentally observable quantities (called physical observables) which must be reproduced. Such quantities are, for instance, the collision energy, the quantum numbers defining the intramolecular state (vibrations and the principal quantum number of rotation), the total angular momentum etc... However, there are other dynamical variables which have a clear meaning in Classical Mechanics but correspond to no physical observable because of the Uncertainty Principle. We call them phase variables and denote them globally by g. The phase variables must be given particular values to obtain, at given G, a particular trajectory. Such variables are, for instance, the various intramolecular normal vibrational phases, the intermolecular orientation, the secondary rotation quantum numbers, the impact parameter, etc... Thus we look for relationships of the type qo = qq (G, g) and either qo = qo (G, g) or po = Po (G, g)... 

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