On the anharmonic oscillator in quantum mechanics and in field theory

1967 ◽  
Vol 205 (2) ◽  
pp. 149-158 ◽  
Author(s):  
Harald J. W. M�ller
Keyword(s):  
Field Theory ◽  
Quantum Mechanics ◽  
Anharmonic Oscillator

THE UNCERTAINTY OF THE GAUSSIAN EFFECTIVE POTENTIAL

1988 ◽  
Vol 03 (09) ◽  
pp. 2143-2163 ◽  
Author(s):  
R. MUÑOZ-TAPIA ◽  
J. TARON ◽  
R. TARRACH
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Mechanics ◽  
Effective Potential ◽  
Anharmonic Oscillator ◽  
Liouville Theory ◽  
Quantum Field ◽  
First Case ◽  
Gaussian Effective Potential

An uncertainty is introduced for the Gaussian Effective Potential. The definition is quite straightforward for quantum mechanics but fairly subtle for quantum field theory. The uncertainty provides a good estimation of the error in the first case, but renormalization seems to spoil its usefulness in the second case. The examples considered are the anharmonic oscillator, λϕ4 in 3+1 dimensions and the Liouville theory in 1+1 dimensions.

Quantum mechanics

2018 ◽  
Author(s):  
Michael Kachelriess
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Mechanics ◽  
Relativistic Quantum ◽  
Transition Amplitude ◽  
External Source ◽  
Quantum Field ◽  
Damping Term ◽  
Relativistic Quantum Theory ◽  
Generating Functional

After a brief review of the operator approach to quantum mechanics, Feynmans path integral, which expresses a transition amplitude as a sum over all paths, is derived. Adding a linear coupling to an external source J and a damping term to the Lagrangian, the ground-state persistence amplitude is obtained. This quantity serves as the generating functional Z[J] for n-point Green functions which are the main target when studying quantum field theory. Then the harmonic oscillator as an example for a one-dimensional quantum field theory is discussed and the reason why a relativistic quantum theory should be based on quantum fields is explained.

QLB for Quantum Many-Body and Quantum Field Theory

2018 ◽  
Author(s):  
Sauro Succi
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Mechanics ◽  
Quantum Computing ◽  
Single Particle ◽  
Body Problem ◽  
Three Dimensions ◽  
Quantum Field ◽  
Many Body ◽  
Quantum Particles

Chapter 32 expounded the basic theory of quantum LB for the case of relativistic and non-relativistic wavefunctions, namely single-particle quantum mechanics. This chapter goes on to cover extensions of the quantum LB formalism to the overly challenging arena of quantum many-body problems and quantum field theory, along with an appraisal of prospective quantum computing implementations. Solving the single particle Schrodinger, or Dirac, equation in three dimensions is a computationally demanding task. This task, however, pales in front of the ordeal of solving the Schrodinger equation for the quantum many-body problem, namely a collection of many quantum particles, typically nuclei and electrons in a given atom or molecule.

Quantum mechanics of the anharmonic oscillator

1973 ◽  
Vol 14 (2) ◽  
pp. 219-227 ◽  
Author(s):  
Francis R. Halpern
Keyword(s):  
Quantum Mechanics ◽  
Anharmonic Oscillator

Hyper-asymptotic expansions and instantons

2019 ◽  
pp. 471-492
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Mechanics ◽  
Asymptotic Expansions ◽  
Perturbative Expansion ◽  
Quantum Field ◽  
Wkb Method ◽  
Additional Information ◽  
Spectral Equation ◽  
Double Well Potential

Chapter 24 examines the topic of hyper–asymptotic expansions and instantons. A number of quantum mechanics and quantum field theory (QFT) examples exhibit degenerate classical minima connected by quantum barrier penetration effects. The analysis of the large order behaviour, based on instanton calculus, shows that the perturbative expansion is not Borel summable, and does not define unique functions. An important issue is then what kind of additional information is required to determine the exact expanded functions. While the QFT examples are complicated, and their study is still at the preliminary stage, in quantum mechanics, in the case of some analytic potentials that have degenerate minima (like the quartic double–well potential), the problem has been completely solved. Some examples are described in Chapter 24. There, the perturbative, complete, hyper–asymptotic expansion exhibits the resurgence property. The perturbative expansion can be related to the calculation of the spectral equation via the complex WKB method.

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