Classical path from quantum motion for a particle in a transparent box

Abstract

We consider the problem of a free particle inside a one-dimensional box with transparent walls (or equivalently, along a circle with a constant speed) and discuss the classical and quantum descriptions of the problem. After calculating the mean value of the position operator in a time-dependent normalized complex general state and the Fourier series of the function position, we explicitly prove that these two quantities are in accordance by (essentially) imposing the approximation of high principal quantum numbers on the mean value. The presentation is accessible to advanced undergraduate students with a knowledge of the basic ideas of quantum mechanics. Keywords: correspondence principle, classical limit, Ehrenfest theorem.Consideramos o problema de uma partícula livre no interior de uma caixa unidimensional com paredes transparentes (ou equivalentemente, ao longo de um círculo com uma velocidade constante) e discutimos as descrições clássica e quântica do problema. Depois de calcular o valor médio do operador da posição num estado geral complexo normalizado dependente do tempo e a série de Fourier da função de posição, provamos explicitamente que estas duas quantidades estão em correspondência se (essencialmente) impusermos sobre o valor médio a aproximação dos números quânticos principais elevados. A apresentaçãoé acessível a alunos de graduação avançados com conhecimento das idéias básicas da mecânica quântica. Palavras-chave: princípio da correspondência, limite clássico, teorema de Ehrenfest.

Considerations about the incompleteness of the Ehrenfest’s theorem in quantum mechanics

“…Equation (36) is the basis to prove the reality of the eigenvalues and the orthogonality of the eigenfunctions when s = r. If degenerate eigenfunctions turn out to exist then we can make recourse to the Gram-Schmidt recipe [3,8] to orthogonalise them but, for the sake of simplicity, we proceed with the assumption of degeneracy absence. In any case, we are free to choose normalised eigenfunctions…”

“…However, his explanation for such an occurrence, provided at page 738 of his article, is only of mathematical nature. Also, Reif proposed at page 356 of his textbook [22] an interesting problem regarding the motion of a particle on a circumference, again involving periodic (p) boundary conditions therefore, that offers an appropriate scenario to scrutinise the physical meaning of the boundary terms in equation (43)(p); De Vincenzo [36] has discussed classical and quantum descriptions of this problem in details and we refer the interested reader to his well-written paper.…”