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CRC Press eBooks

THE DENSITY MATRIX

Murray, Iii Sargent

Published Mar 5, 2018• 0 citations

Abstract

In Chap. 2 we solved for the probability amplitudes Ca and C.b explicitly. The results of our discussion, such as the probability of a transition or the value of the induced dipole moment, were invariably expressed in terms of bilinear combinations of the amplitudes, such as CaCa* and CaCi>*. In fact, the expectation value of any observable involves bilinear combinations. Hence an alternative formulation of quantum mechanics consists of dealing with the bilinear quantities directly. The method involves organizing the quantities in a matrix form called the density matrix. There are two principal advantages in doing so. First, the resulting mathematics is often simpler. Sec ond, when the wave function for an ensemble of systems is not known but the probabilities for having various different wave functions are known, the en semble can be described by a weighted sum of individual density matrices. In particular, the amplitudes cannot themselves describe simply certain com mon statistical phenomena, such as the effects of elastic collisions on the in duced dipole moments of atoms.

DOI: 10.1201/9780429493515-14

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In Chap. 2 we solved for the probability amplitudes Ca and C.b explicitly. The results of our discussion, such as the probability of a transition or the value of the induced dipole moment, were invariably expressed in terms of bilinear combinations of the amplitudes, such as CaCa* and CaCi>*. In fact, the expectation value of any observable involves bilinear combinations. Hence an alternative formulation of quantum mechanics consists of dealing with the bilinear quantities directly. The method involves organizing the quantities in a matrix form called the density matrix. There are two principal advantages in doing so. First, the resulting mathematics is often simpler. Sec ond, when the wave function for an ensemble of systems is not known but the probabilities for having various different wave functions are known, the en semble can be described by a weighted sum of individual density matrices. In particular, the amplitudes cannot themselves describe simply certain com mon statistical phenomena, such as the effects of elastic collisions on the in duced dipole moments of atoms.

Non-relativistic quantum mechanics is reformulated here based on the idea that relational properties among quantum systems, instead of the independent properties of a quantum system, are the most fundamental elements to construct quantum mechanics. This idea, combining with the emphasis that measurement of a quantum system is a bidirectional interaction process, leads to a new framework to calculate the probability of an outcome when measuring a quantum system. In this framework, the most basic variable is the relational probability amplitude. Probability is calculated as summation of weights from the alternative measurement configurations. The properties of quantum systems, such as superposition and entanglement, are manifested through the rules of counting the alternatives. Wave function and reduced density matrix are derived from the relational probability amplitude matrix. They are found to be secondary mathematical tools that equivalently describe a quantum system without explicitly calling out the reference system. Schrödinger Equation is obtained when there is no entanglement in the relational probability amplitude matrix. Feynman Path Integral is used to calculate the relational probability amplitude, and is further generalized to formulate the reduced density matrix. In essence, quantum mechanics is reformulated as a theory that describes physical systems in terms of relational properties.

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In this study, expectation value dynamics of the quantum harmonic oscillator under the influence of the external dipol effects is discussed in some details. The dipole function of the system has been taken as an odd cubic spatial polynomial and the external field amplitude has been assumed to be known. Instead of the wave function of the system, expectation values of the position and momentum operators are considered as unknowns and certain matrix ordinary differential equations are obtained by using the fluctuation free matrix representation concept. The resulting equations are in the category of Quantum Mechanical Matrix Ordinary Differential Equations (QMMODE) and can be solved by a recently developed scheme called Characteristic Evolutions Method (CEM). We mention also how to implement this approach. The purpose of this work can be considered as a preparation to quantum optimal control problem solutions via expectation value dynamics.