It is known that classical waves such as waves on a string, sound and light (from Maxwell’s em equations) take the argument -Et+px where E is associated with a frequency and p with 1/wavelength. We argue that these classical nonrelativistic treatments already bring in features of special relativity (as does Schrodinger equation quantum mechanics for a free particle) as -Et+px is a Lorentz invariant. In fact we argue that Lagrangian formalism: E=pv-L and p=dL/dv is compatible with special relativity. In a previous note (1) we argued that d/dx partial A(x,t) = p and d/dt partial A(x,t) = -E for a free particle with A=tL(v). We also noted that A= -Et+px is a solution, but that one may also write E(v)= E(x/t) and p(x/t) and take the derivatives d/dx partial A and d/dt partial A and obtain the same results. In this note we show how this follows from regular Lagrange theory and note that one already associates d/dx partial with p and d/dt partial with -E i.e. the quantum theory for a free particle is present.It is known that classical waves such as waves on a string, sound and light (from Maxwell’s em equations) take the argument -Et+px where E is associated with a frequency and p with 1/wavelength. We argue that these classical nonrelativistic treatments already bring in features of special relativity (as does Schrodinger equation quantum mechanics for a free particle) as -Et+px is a Lorentz invariant. In fact we argue that Lagrangian formalism: E=pv-L and p=dL/dv is compatible with special relativity. In a previous note (1) we argued that d/dx partial A(x,t) = p and d/dt partial A(x,t) = -E for a free particle with A=tL(v). We also noted that A= -Et+px is a solution, but that one may also write E(v)= E(x/t) and p(x/t) and take the derivatives d/dx partial A and d/dt partial A and obtain the same results. In this note we show how this follows from regular Lagrange theory and note that one already associates d/dx partial with p and d/dt partial with -E i.e. the quantum theory for a free particle is present.

Jan 15, 20220 citations

Some Remarks on Hamiltonian Systems and Quantum Mechanics

Paul Robert Chernoff, Jerrold Eldon Marsden

Springer eBooksJan 1, 197611 citations

Geometric momentum: The proper momentum for a free particle on a two-dimensional sphere

Quanhui Liu, Liang-Hui Tang, D. M. Xun

In Dirac's canonical quantization theory on systems with second-class constraints, the commutators between the position, momentum, and Hamiltonian form a set of algebraic relations that are fundamental in construction of both the quantum momentum and the Hamiltonian. For a free particle on a two-dimensional sphere or a spherical top, results show that the well-known canonical momentum ${p}_{\ensuremath{\theta}}$ breaks one of the relations, while three components of the momentum expressed in the three-dimensional Cartesian system of axes as ${p}_{i}$ $(i=1,2,3)$ are satisfactory all around. This momentum is not only geometrically invariant but also self-adjoint, and we call it geometric momentum. The nontrivial commutators between ${p}_{i}$ generate three components of the orbital angular momentum; thus the geometric momentum is fundamental to the angular one. We note that there are five different forms of the geometric momentum proposed in the current literature, but only one of them turns out to be meaningful.In Dirac's canonical quantization theory on systems with second-class constraints, the commutators between the position, momentum, and Hamiltonian form a set of algebraic relations that are fundamental in construction of both the quantum momentum and the Hamiltonian. For a free particle on a two-dimensional sphere or a spherical top, results show that the well-known canonical momentum ${p}_{\ensuremath{\theta}}$ breaks one of the relations, while three components of the momentum expressed in the three-dimensional Cartesian system of axes as ${p}_{i}$ $(i=1,2,3)$ are satisfactory all around. This momentum is not only geometrically invariant but also self-adjoint, and we call it geometric momentum. The nontrivial commutators between ${p}_{i}$ generate three components of the orbital angular momentum; thus the geometric momentum is fundamental to the angular one. We note that there are five different forms of the geometric momentum proposed in the current literature, but only one of them turns out to be meaningful.

arXiv:1109.5223Physical Review AOct 3, 201156 citations

Building confidence in the Dirac δ -function

Asim Gangopadhyaya, Constantin Rasinariu

In this note we present an example from undergraduate quantum mechanics designed to highlight the versatility of the Dirac $δ$ -function. Namely, we compute the expectation value of the Hamiltonian of a free-particle in a state described by a triangular wave function $ψ(x)$ . Since the first derivative of $ψ(x)$ is piecewise constant, and because this Hamiltonian is proportional to the second order spatial derivative, students often end up finding the expectation value to be zero --an unphysical answer. This problem provides a pedagogical application of the Dirac $δ$ -function. By arriving at the same result via alternate pathways, this exercise reinforces students' confidence in the Dirac $δ$ -function and highlights its efficiency and elegance.In this note we present an example from undergraduate quantum mechanics designed to highlight the versatility of the Dirac $δ$-function. Namely, we compute the expectation value of the Hamiltonian of a free-particle in a state described by a triangular wave function $ψ(x)$. Since the first derivative of $ψ(x)$ is piecewise constant, and because this Hamiltonian is proportional to the second order spatial derivative, students often end up finding the expectation value to be zero --an unphysical answer. This problem provides a pedagogical application of the Dirac $δ$-function. By arriving at the same result via alternate pathways, this exercise reinforces students' confidence in the Dirac $δ$-function and highlights its efficiency and elegance.

arXiv:2007.13478European Journal of PhysicsAug 30, 20182 citations

Hamiltonian aspects of quantum theory

Valerij V. Kozlov, Oleg Georgievich Smolyanov

Doklady MathematicsMay 1, 201210 citations

{p}_{i}

(i=1,2,3)

are satisfactory all around. This momentum is not only geometrically invariant but also self-adjoint, and we call it geometric momentum. The nontrivial commutators between

{p}_{i}

generate three components of the orbital angular momentum; thus the geometric momentum is fundamental to the angular one. We note that there are five different forms of the geometric momentum proposed in the current literature, but only one of them turns out to be meaningful.

In this note we present an example from undergraduate quantum mechanics designed to highlight the versatility of the Dirac

δ

-function. Namely, we compute the expectation value of the Hamiltonian of a free-particle in a state described by a triangular wave function

ψ(x)

. Since the first derivative of

ψ(x)

is piecewise constant, and because this Hamiltonian is proportional to the second order spatial derivative, students often end up finding the expectation value to be zero --an unphysical answer. This problem provides a pedagogical application of the Dirac

δ

-function. By arriving at the same result via alternate pathways, this exercise reinforces students' confidence in the Dirac

δ

-function and highlights its efficiency and elegance.