Abstract This chapter discusses some applications of path integrals. First, it illustrates how their qualitative use can yield insights. As an example, it applies it to the so-called Bohm–Aharonov effect. Then, it discusses the generalization of the path-integral formalism for the so-called many-body problem. It only treats distinguishable particles (what we call Boltzmannons) in this chapter. The book will discuss indistinguishable particles in a future chapter. This chapter will also discuss the formulation of quantum statistical mechanics of Boltzmannons.
Abstract This chapter discusses some applications of path integrals. First, it illustrates how their qualitative use can yield insights. As an example, it applies it to the so-called Bohm–Aharonov effect. Then, it discusses the generalization of the path-integral formalism for the so-called many-body problem. It only treats distinguishable particles (what we call Boltzmannons) in this chapter. The book will discuss indistinguishable particles in a future chapter. This chapter will also discuss the formulation of quantum statistical mechanics of Boltzmannons.
The aim of this chapter is to introduce the concept of the Feynman path integral. As well as developing the general construction scheme, particular emphasis is placed on establishing the interconnections between the quantum mechanical path integral, classical Hamiltonian mechanics, and classical statistical mechanics. The practice of path integration is discussed in the context of several pedagogical applications. As well as the canonical examples of a quantum particle in a single and a double potential well, we discuss the generalization of the path integral scheme to tunneling of extended objects (quantum fields), dissipative and thermally assisted quantum tunneling, and the quantum mechanical spin. In this chapter we temporarily leave the arena of many-body physics and second quantization and, at least superficially, return to single-particle quantum mechanics. By establishing the path integral approach for ordinary quantum mechanics, we will set the stage for the introduction of field integral methods for many-body theories explored in the next chapter. We will see that the path integral not only represents a gateway to higher-dimensional functional integral methods but, when viewed from an appropriate perspective, already represents a field theoretical approach in its own right. Exploiting this connection, various concepts of field theory, namely stationary phase analysis, the Euclidean formulation of field theory, instanton techniques, and the role of topology in field theory, are introduced in this chapter. The path integral: general formalism Broadly speaking, there are two approaches to the formulation of quantum mechanics: the “operator approach” based on the canonical quantization of physical observables and the associated operator algebra, and the Feynman path integral.
These lectures are intended as an introduction to the technique of path integrals and their applications in physics. The audience is mainly first-year graduate students, and it is assumed that the reader has a good foundation in quantum mechanics. No prior exposure to path integrals is assumed, however. The path integral is a formulation of quantum mechanics equivalent to the standard formulations, offering a new way of looking at the subject which is, arguably, more intuitive than the usual approaches. Applications of path integrals are as vast as those of quantum mechanics itself, including the quantum mechanics of a single particle, statistical mechanics, condensed matter physics and quantum field theory. After an introduction including a very brief historical overview of the subject, we derive a path integral expression for the propagator in quantum mechanics, including the free particle and harmonic oscillator as examples. We then discuss a variety of applications, including path integrals in multiply-connected spaces, Euclidean path integrals and statistical mechanics, perturbation theory in quantum mechanics and in quantum field theory, and instantons via path integrals. For the most part, the emphasis is on explicit calculations in the familiar setting of quantum mechanics, with some discussion (often brief and schematic) of how these ideas can be applied to more complicated situations such as field theory.
The aim of this chapter is to introduce the concept of the Feynman path integral. As well as developing the general construction scheme, particular emphasis is placed on establishing the interconnections between the quantum mechanical path integral, classical Hamiltonian mechanics, and classical statistical mechanics. The practice of path integration is discussed in the context of several pedagogical applications. As well as the canonical examples of a quantum particle in a single and a double potential well, we discuss the generalization of the path integral scheme to tunneling of extended objects (quantum fields), dissipative and thermally assisted quantum tunneling, and the quantum mechanical spin. In this chapter we will temporarily leave the arena of many-body physics and second quantization and, at least superficially, return to single-particle quantum mechanics. By establishing the path integral approach for ordinary quantum mechanics, we will set the stage for the introduction of functional field integral methods for many-body theories explored in the next chapter. We will see that the path integral not only represents a gateway to higher-dimensional functional integral methods but, when viewed from an appropriate perspective, already represents a field theoretical approach in its own right. Exploiting this connection, various techniques and concepts of field theory, namely stationary phase analyses of functional integrals, the Euclidean formulation of field theory, instanton techniques, and the role of topological concepts in field theory, will be motivated and introduced in this chapter.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism. It is a generalization of the classical least action principle. The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state. Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral. We look in to the subject of two kinds of slit-experiments: The square slit, the single- and the double-Gaussian slit. Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics. It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral. Further applications of the path integral are discussed.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism.It is a generalization of the classical least action principle.The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state.Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral.We look in to the subject of two kinds of slit-experiments: The square slit, the single-and the double-Gaussian slit.Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics.It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral.Further applications of the path integral are discussed.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism.It is a generalization of the classical least action principle.The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state.Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral.We look in to the subject of two kinds of slit-experiments: The square slit, the single-and the double-Gaussian slit.Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics.It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral.Further applications of the path integral are discussed.