Paolo Amore, Francisco M. Fernández
In this paper we develop the Hellmann-Feynman theorem in statistical mechanics without resorting to the eigenvalues and eigenvectors of the Hamiltonian operator. Present approach does not require the quantum-mechanical version of the theorem at $T=0$ and bypasses any discussion about degenerate states.
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The Hellmann-Feynman theorem and the spectrum of some Hamiltonian operators
本文利用Hellmann-Feynman定理证明了某些非相对论哈密顿算符具有无穷多个束缚态。
In this paper we discuss the validity of the Hellmann-Feynman theorem (HFT) for degenerate states. We derive it in a general way and apply it to simple illustrative examples. We also analyze a recent paper that shows results that apparently suggest that the HFT does not apply to degenerate states.
The well known Hellmann-Feynman theorem of Quantum Mechanics connected with the derivative of the eigenvalues with respect to a parameter upon which the Hamiltonian depends, is generalized to include cases in which the domain of definition of the Hamiltonian of the system also depends on that parameter.
By virtue of the generalized Hellmann-Feynman theorem for ensemble average, we obtain internal energy and average energy consumed by the resistance R in a quantized RLC electric circuit. We also calculate entropy-variation with respect to R. The relation between entropy and R is also derived. By depicting figures we indeed see that the entropy increases with the increment of R.
In this work, we used a tool of conventional Nikiforov-Uvarov method to determine bound state solution of Schrodinger equation with quantum interaction potential called Hulthen-Yukawa inversely quadratic potential (HYIQP). We obtained the energy eigen values and the total wave function . We employed Hellmann-Feynmann Theorem (HFT) to compute expectation values for four different diatomic molecules: Hydrogen molecule (H2), Lithium hydride molecule (LiH), Hydrogen Chloride molecule (HCl) and Carbon(II)Oxide molecule. The resulting energy equation reduces to three well known potentials which are:…
Modified Hellmann Feynman Theorem
本文将 Hellmann Feynman 定理扩展至非厄米量子力学领域，推导出适用于 PT 破缺相、未破缺相及例外点的修正形式 MHFT，并基于 G 内积证明其在系统是否满足幺正时间演化时均有效。
Using well-known methods we generalize (hyper)virial theorems to case of singular potential. Discussion is performed for most general second order differential equation, which involves all physically interesting cases, as Schrodinger and Klein-Gordon equations with singular potentials. Some physical consequences are discussed. The connection with Feynman-Hellmann like theorems are also considered and some relevant differences are underlined.
We present a simple derivation of the Hellmann-Feynman theorem at finite temperature. We illustrate its validity by considering three relevant examples which can be used in quantum mechanics lectures: the one-dimensional harmonic oscillator, the one-dimensional Ising model and the Lipkin model. We show that the Hellmann-Feynman theorem allows one to calculate expectation values of operators that appear in the Hamiltonian. This is particularly useful when the total free-energy is available, but there is not direct access to the thermal average of the operators themselves.
In a computer experiment the choice of suitable estimators to measure a physical quantity plays an important role. We propose a new direct route to determine estimators for observables which do not commute with the Hamiltonian. Our new route makes use of the Hellmann and Feynman theorem and in a diffusion Monte Carlo simulation it introduces a new bias to the measure due to the choice of the auxiliary function. This bias is independent from the usual one due to the choice of the trial wave function. We used our route to measure the radial distribution function of a spin one half Fermion fluid.
Grüneisen parameter as an entanglement compass and the breakdown of the Hellmann-Feynman theorem
提出Grüneisen比的量子类比以通过纠缠探测量子相变，证明QPT仅发生在基态能量非线性依赖参数的系统，并展示Hellmann-Feynman定理在热力学极限下QCP处的失效。
Hellmann Feynman Theorem in Non-Hermitian system
本研究在PT不变非厄米量子框架下推导了修正版Hellmann-Feynman定理，证明其在多类相及例外点均成立，还基于此推出非厄米系统广义位力定理，为相关实验验证提供了理论基础。
An extension of the Hellmann-Feynman theorem to one employing dynamical parameters that vary with time according to quantum dynamics is rigorously derived, avoiding any linear response or other approximations. The resulting theorem for the dynamics of observables, valid to all orders in external fields, is found to contain generalized Berry curvature type of quantities that incorporate the dynamics through explicit and nontrivial time-dependence; these Berry quantities resemble the so called anomalous terms (in semiclassical equations of motion in solids) but are both of magnetic and electric …
We point out that two of the most important theorems of Quantum Mechanics, the Ehrenfest theorem and the Hellmann-Feynman theorem, lack in their standard form important information: there are cases where non-Hermitian boundary contributions emerge. These contributions actually appear naturally, in order for the above theorems to be valid and applicable (i.e. in multiply-connected spaces), and this occurs for physical quantities that are not represented by well-defined self-adjoint operators (such as the position operator in a periodic potential, or in general Aharonov-Bohm configurations, eith…
Using well-known methods we generalize (hyper)virial theorems to case of singular potential. Discussion is carried on for most general second order differential equation, which involves all physically interesting cases, such as Schrödinger and two-body Klein-Gordon equations with singular potentials. Some physical consequences are discussed. The connection with Feynman-Hellmann like theorems are also considered and some relevant differences are underlined.
The Hellmann-Feynman, virial and comparison theorems are three fundamental theorems of quantum mechanics. For the first two, counterparts exist for classical mechanics with relativistic or nonrelativistic kinetic energy. It is shown here that these three theorems are valid for classical mechanics with a nonstandard kinetic energy. This brings some information about the connections between the quantum and classical worlds. Constraints about the functional form of the kinetic energy are also discussed.
The Feynman integral is given a stochastic interpretation in the framework of Nelson's stochastic mechanics employing a time-symmetric variant of Nelson's kinematics recently developed by the author.
Recent developments in elementary quantum mechanics have seen a number of extraordinary claims regarding quantum behaviour, and even questioning internal consistency of the theory. These are, we argue, different disguises of what Feynman described as quantum theory's "only mystery".
In this paper we will demonstrate the use of Feynman Diagrams for one dimensional scattering in quantum mechanics. We will evaluate the S-Matrix explicitly for the Dirac delta and finite wall potentials by summing the full series of Feynman diagrams, illustrating the spirit of perturbation theory. This technique may be useful in introductory quantum mechanics courses, and provides the student with intuition about conservation laws in the context of scattering problems by connecting Feynman diagrams, free propagation, and conservation of the corresponding observable. It also provides a toy mode…
The five laws of relativistic quantum mechanics, according to Feynman's path integral formulation, are concisely stated and applied to experiments. Reflection diffraction grating experiments for both photons and electrons are analysed, in particular the Davisson-Germer experiment in which the wave-like property of electrons was first established. It is shown how classical, purely spatial, effective wave theories for both photons and electrons are predicted by the path integral formulation of quantum mechanics. The standard Copenhagen interpretation of wave mechanics is critically discussed in …
Interpretation of Feynman formalism of quantum mechanics in terms of probabilities of paths
本文重新审视费曼路径积分，提出将跃迁概率视为路径概率之和的重新表述，并由此推导出散射的玻恩近似，从而无需玻恩假设即可实现基于粒子的量子力学诠释。
Both Bohmian mechanics, a version of quantum mechanics with trajectories, and Feynman's path integral formalism have something to do with particle paths in space and time. The question thus arises how the two ideas relate to each other. In short, the answer is, path integrals provide a re-formulation of Schroedinger's equation, which is half of the defining equations of Bohmian mechanics. I try to give a clear and concise description of the various aspects of the situation.
Feynman's laws of quantum dynamics are concisely stated, discussed in comparison with other formulations of quantum mechanics and applied to selected problems in the physical optics of photons and massive particles as well as flavour oscillations. The classical wave theory of light is derived from these laws for the case in which temporal variation of path amplitudes may be neglected, whereas specific experiments, sensitive to the temporal properties of path amplitudes, are suggested. The reflection coefficient of light from the surface of a transparent medium is found to be markedly different…
We propose a formulation of an absorbing boundary for a quantum particle. The formulation is based on a Feynman-type integral over trajectories that are confined by the absorbing boundary. Trajectories that reach the absorbing wall are instantaneously terminated and their probability is discounted from the population of the surviving trajectories. This gives rise to a unidirectional absorption current at the boundary. We calculate the survival probability as a function of time. Several modes of absorption are derived from our formalism: total absorption, absorption that depends on energy level…
Our previous work on quantum mechanics in Hilbert spaces of finite dimensions N is applied to elucidate the deep meaning of Feynman's path integral pointed out by G. Svetlichny. He speculated that the secret of the Feynman path integral may lie in the property of mutual unbiasedness of temporally proximal bases. We confirm the corresponding property of the short-time propagator by using a specially devised N x N -approximation of quantum mechanics in L^2(R) applied to our finite-dimensional analogue of a free quantum particle.
Quantum optical coherence theory based on Feynman's path integral
该研究介绍并综述了基于费曼路径积分的量子光学相干理论，指出其能更好关联数学计算与物理解释，预测热光电场经典模型或不准确，统一解释了双光子聚束与Hong-Ou-Mandel凹陷的物理本质，助力理解光的相干性及光子本质问题。
Power-Duality in Path Integral Formulation of Quantum Mechanics
本文研究费曼路径积分表述下的幂对偶性，发现修改后的promotor在经典角动量下形式不变，角动量量子化后对偶性破坏，进而提出准幂对偶性概念并给出由已知格林函数确定对偶系统格林函数的公式。
The Einstein-Podolsky-Rosen (EPR) paradox that argues for the incompleteness of quantum mechanics as a description of physical reality has been put to rest by John Bell's famous theorem, which inspired numerous experimental tests and brought about further affirmations of quantum reality. Nevertheless, in his writings and public presentations, Richard Feynman never acknowledged the significance of Bell's contribution to the resolution of the EPR paradox. In this paper, we discuss several variants of the Bell inequalities (including one that was specifically espoused by Feynman), and explore the…
The extended Wigner's friend problem deals with two Observers each measuring a sealed laboratory in which a friend is making a quantum measurement. We investigate this problem by relying on the basic rules of quantum mechanics as exposed by Feynman in the well-known "Feynman Lectures on Physics". Although recent discussions have suggested that the extended Wigner's friend problem cannot consistently be described by quantum theory, we show here that a straightforward application of these standard rules results in a non-ambiguous and consistent account of the measurement outcomes for all agents …
Similarities between the non-deterministic nature of quantum theory and the unpredictable patterns of human cognition and decision making have been observed and commented on many times since the invention of Quantum Mechanics in the first part of the 20th century. Niels Bohr himself took note of the parallels.[3] In fact, an entire field of study, Quantum Cognition, has been borne from the study of this analogy. [3] However, many of the attempts to model human behavior with quantum mechanics conflate the identity of a particle with its own wavefunction, which is incorrect and invalidates the a…
In this paper we show how Feynman diagrams, which are used as a tool to implement perturbation theory in quantum field theory, can be very useful also in classical mechanics, provided we introduce also at the classical level concepts like path integrals and generating functionals.
Gaussian Reformulation of the Feynman Path Integral for Quantum Statistical Mechanics with Results for the Second Virial Coefficient of $^4$He
研究针对量子统计力学的费曼路径积分进行高斯重整化，通过晶格上环聚合物统计和维格纳 - 柯克伍德函数高温展开获方差均值，避免传统问题，给出氦第二维里系数的解析与模拟结果并与实验比较。
A history of Feynman's sum over histories is presented in brief. A focus is placed on the progress of path-integration techniques for exactly path-integrable problems in quantum mechanics.
The broad debate on foundational issues in quantum mechanics, which took place at the famous 1957 Chapel Hill conference on \textit{The Role of Gravitation in Physics}, is here critically analyzed with an emphasis on Richard Feynman's contributions. One of the most debated questions at Chapel Hill was whether the gravitational field had to be quantized and its possible role in wave function collapse. Feynman's arguments in favor of the quantization of the gravitational field, based essentially on a series of gedanken experiments, are here discussed. Then the related problem of the wave functio…
We describe the "Feynman diagram" approach to nonrelativistic quantum mechanics on R^n, with magnetic and potential terms. In particular, for each classical path γconnecting points q_0 and q_1 in time t, we define a formal power series V_γ(t,q_0,q_1) in \hbar, given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V_γ) satisfies Schrödinger's equation, and explain in what sense the t\to 0 limit approaches the δdistribution. As such, our construction gives explicitly the full \hbar\to 0 asymptotics of the fundamental solution to…
In this article we present an analytic solution of the famous problem of diffraction and interference of electrons through one and two slits (for simplicity, only the one-dimensional case is considered). In addition to exact formulas, we exhibit various approximations of the electron distribution which facilitate the interpretation of the results. Our derivation is based on the Feynman path integral formula and this work could therefore also serve as an interesting pedagogical introduction to Feynman's formulation of quantum mechanics for university students dealing with the foundations of qua…
By investigating the Feynman Path Integral we prove that elementary quantum particle dynamics are directly associated to single compact (cyclic) world-line parameters, playing the role of the particles' internal clock, implicit in ordinary undulatory mechanics and indirectly observed for instance in Time Crystals. This allows us to formulate a novel purely four-dimensional stringy description of elementary particles as possible physics beyond quantum mechanics. The novelty of this approach is that quantum mechanics originates from a non-trivial compact nature of the minkowskian space-time. Our…
New applications of Feynman disentangling method in quantum mechanics are studied and the time-dependent singular oscillator problem is solved in this approach. The important role of representation group theory is discussed in this context.
Introduction to the "second quantization" formalism for non-relativistic quantum mechanics: A possible substitution for Sections 6.7 and 6.8 of Feynman's "Statistical Mechanics"
本文面向仅掌握波函数形式量子力学的读者，系统讲解多粒子非相对论量子力学中的产生与湮灭算符（二次量子化）形式，推导其（反）对易关系，并覆盖费曼《统计力学》第6.7、6.8节的主要内容。
Quantum mechanics can be formulated in three ways, as Heisenberg, Schrödinger and Feynman did respectively. For the last way, an unknown (i.e. forgotten) forerunner exists, that we have found in a paper by Gregor Wentzel, published before the famous works by Heisenberg and Schrödinger, and contemporary with the fundamental works of L. de Broglie. In that paper, one can find the basic formulae and their interpretation as they were adopted by Feynman twenty years later. We believe that Wentzel's work was forgotten for several reasons: (I) Schrödinger's equation was much simpler to deal with (Wen…
On the Path Integral Formulation of Wigner-Dunkl Quantum Mechanics
本文将 Feynman 路径积分推广至 Wigner-Dunkl 量子力学，分析高斯波包、谐振子及带跳跃的 Dunkl 过程，并显式计算 Feynman-Kac 路径积分。
Given an arbitrary Lagrangian function on \RR^d and a choice of classical path, one can try to define Feynman's path integral supported near the classical path as a formal power series parameterized by "Feynman diagrams," although these diagrams may diverge. We compute this expansion and show that it is (formally, if there are ultraviolet divergences) invariant under volume-preserving changes of coordinates. We prove that if the ultraviolet divergences cancel at each order, then our formal path integral satisfies a "Fubini theorem" expressing the standard composition law for the time evolution…
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