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Laddas ned direkt. Introduces number operators with a focus on the relationship between quantum mechanics and social science Mathematics is increasingly applied to classical problems in finance, biology, economics, and elsewhere. Quantum Dynamics for Classical Systems describes how quantum tools the number operator in particular can be used to create dynamical systems in which the variables are operator-valued functions and whose results explain the presented model.
The book presents mathematical results and their applications to concrete systems and discusses the methods used, results obtained, and techniques developed for the proofs of the results.
Then, the motion along one of the F or G directions is determined by the corresponding conjugate variable. These vector fields in general are not orthogonal, nor parallel. If the motion of phase space points is governed by the vector field 15 , F remains constant because. In contrast, when motion occurs in the F direction, by means of Eq.
Hence, motion originated by the conjugate variables F z and G z occurs on the shells of constant F z or of constant G z , respectively. Thus, the motions associated to each of these conjugate variables preserve the phase space area. Then, the magnitudes of the vector fields and the angle between them changes in such a way that the cross product remains constant when the Poisson bracket is equal to one, i. The Jacobian for transformations from phase space coordinates to f , g variables is one for each type of motion:.
We have seen some properties related to the motion of phase space points caused by conjugate variables. The Poisson bracket can also be written in two ways involving a commutator. One form is. These operators generate complementary motion of functions in phase space.
MATH3111 Quantum Mechanics III
Note that now, we also have operators and commutators as in Quantum Mechanics. Conserved motion of phase space functions moving along the f or g directions can be achieved with the above Liouvillian operators as. Indeed, with the help these definitions and of the chain rule, we have that the total derivative of functions vanishes, i. Also, note that for any function u z of a phase space point z , we have that.
The formal solutions to these equations are. As in quantum theory, we have found commutators and there are many properties based on them, taking advantage of the fact that a commutator is a derivation. Since the commutator is a derivation, for conjugate variables F z and G z we have that, for integer n ,. From Eq. But, if we multiply by u - 1 L G from the right, we arrive to. This is a generalized version of a shift of F , and the classical analogue of a generalization of the quantum Weyl relationship.
Mathematical Methods of Quantum Mechanics
A simple form of the above equality, a familiar form, is obtained with the exponential function, i. This is a relationship that indicates how to translate the function F z as an operator. When this equality is acting on the number one, we arrive at the translation property for F as a function. Continuing in a similar way, we can obtain the relationships shown in the following diagram.
This operator is also the propagator for the evolution of functions along the g direction. The variable g is more than just a shift parameter; it actually labels the values that G z takes, the classical analogue of the spectrum of a quantum operator. But L F commutes with F z and then it cannot be used to translate functions of F z , F z is a conserved quantity when motion occurs along the G z direction.
The variable f is more than just a parameter in the shift of s L F , it actually is the value that s L F can take, the classical analogue of the spectra of a quantum operator. These comments involve the left hand side of the above diagram.
There are similar conclusions that can be drawn by considering the right hand side of the diagram. Remember that the above are results valid for classical systems. Below we derive the corresponding results for quantum systems. We now derive the quantum analogues of the relationships found in previous section. The eigenvectors of the position, momentum and energy operators have been used to provide a representation of wave functions and of operators. In an abuse of notation, we have that. We can take advantage of this fact and derive the quantum versions of the equalities found in the classical realm.
A set of equalities is obtained from Eq.
Next, we multiply these equalities by the inverse operator to the right or to the left in order to obtain. The usual shift relationships are obtained when u x is the exponential function, i. Now, as in Classical Mechanics, the commutator between two operators can be seen as two different derivatives introducing quantum dynamical system as.
We can define many of the classical quantities but now in the quantum realm. Liouville type operators are. There are many equalities that can be obtained as in the classical case. The following diagram shows some of them:.
Mathematical formulation of quantum mechanics - Wikipedia
Note that the conclusions mentioned at the end of the previous section for classical systems also hold in the quantum realm. As a brief application of the abovee ideas, we show how to use the energy-time coordinates and eigenfunctions in the reversible evolution of probability densities.
Earlier, there was an interest on the classical and semi classical analysis of energy transfer in molecules. In those earlier calculations, an attempt to use the eigenfunctions of a complex classical Liouville operator was made [ 5 - 8 ].
The results in this chapter show that the eigenfunction of the Liouville operator L H is e g T z and that it do not seems to be a good set of functions in terms of which any other function can be written, as is the case for the eigenfunctions of the Hamiltonian operator in Quantum Mechanics. In this section, we use the time eigenstates instead.