The one-dimensional harmonic oscillator is one of the few models in quantum mechanics that can be solved analytically, and it is used to exam the theories and methods of quantum mechanics, possessing rich connotations. Using the Fourier transform and the generating function of Hermite polynomials, the transformation from the eigenfunction in the coordinate representation to that in the momentum representation for a one-dimensional harmonic oscillator has been achieved. The equivalence and connection between the two representations for describing a one-dimensional harmonic oscillator are verified. The eigenenergies and eigenfunctions of the one-dimensional harmonic oscillator in the coordinate representation and momentum representation are also obtained by using the algebraic solution method, respectively. The equivalence of the two representations is further discussed. The idea of psolving physical problems in higher dimensional spaceq is emphasized, which deepens students' understanding of quantum mechanics and stimulates their interest in learning.The one-dimensional harmonic oscillator is one of the few models in quantum mechanics that can be solved analytically, and it is used to exam the theories and methods of quantum mechanics, possessing rich connotations. Using the Fourier transform and the generating function of Hermite polynomials, the transformation from the eigenfunction in the coordinate representation to that in the momentum representation for a one-dimensional harmonic oscillator has been achieved. The equivalence and connection between the two representations for describing a one-dimensional harmonic oscillator are verified. The eigenenergies and eigenfunctions of the one-dimensional harmonic oscillator in the coordinate representation and momentum representation are also obtained by using the algebraic solution method, respectively. The equivalence of the two representations is further discussed. The idea of psolving physical problems in higher dimensional spaceq is emphasized, which deepens students' understanding of quantum mechanics and stimulates their interest in learning.

Wuli yu gongcheng.Jun 1, 20250 citations

The Variational Methods of Quantum Systems in Holomorphic Representation.

M. W. AlMasri, Mohamed Ridza Wahiddin

We show that variational method for harmonic oscillator in Bargmann representation breaks down since we always end up with one of the exact eigenvalues of Hamiltonian using any normalized monomials in the complex plane. Using the holomorphic representation of pure spin systems , we also encounter the same result found earlier for harmonic oscillators in Bargmann representation. We conclude that the exact energy eigenvalues of these systems are unavoidable starting from any normalized monomials of the complex phase-space coordinate. Since $\{\frac{z^{n}}{\sqrt{n!}}\}$ form an orthonormal basis, one could in principle expand any complex function $f(z)$ in powers of $z$ and use this expansion as energy eigenstates. Such expansion is valid since the magnitude of $z$ carry no physical information only their non-negative exponents $n$ are physically important. These results may lead to new selection principle regarding the quantized energy spectrum of physical systems and enhance our understanding of quantum world.We show that variational method for harmonic oscillator in Bargmann representation breaks down since we always end up with one of the exact eigenvalues of Hamiltonian using any normalized monomials in the complex plane. Using the holomorphic representation of pure spin systems , we also encounter the same result found earlier for harmonic oscillators in Bargmann representation. We conclude that the exact energy eigenvalues of these systems are unavoidable starting from any normalized monomials of the complex phase-space coordinate. Since $\{\frac{z^{n}}{\sqrt{n!}}\}$ form an orthonormal basis, one could in principle expand any complex function $f(z)$ in powers of $z$ and use this expansion as energy eigenstates. Such expansion is valid since the magnitude of $z$ carry no physical information only their non-negative exponents $n$ are physically important. These results may lead to new selection principle regarding the quantized energy spectrum of physical systems and enhance our understanding of quantum world.

Nov 28, 20210 citations

Quantum Mechanics of the Harmonic Oscillator

WORLD SCIENTIFIC eBooksSep 1, 20121 citation

Factorization of analytic representations in the unit disc and number-phase statistics of a quantum harmonic oscillator

Apostol Vourdas, Constantin Brif, A. Mann

The inner - outer part factorization of analytic representations in the unit disc is used for an effective characterization of the number-phase statistical properties of a quantum harmonic oscillator. It is shown that the factorization is intimately connected with the number-phase Weyl semigroup and its properties. In the Barut - Girardello analytic representation the factorization is implemented as a convolution. Several examples are given which demonstrate the physical significance of the factorization and its role for quantum statistics. In particular, we study the effect of phase-space interference on the factorization properties of a superposition state.The inner - outer part factorization of analytic representations in the unit disc is used for an effective characterization of the number-phase statistical properties of a quantum harmonic oscillator. It is shown that the factorization is intimately connected with the number-phase Weyl semigroup and its properties. In the Barut - Girardello analytic representation the factorization is implemented as a convolution. Several examples are given which demonstrate the physical significance of the factorization and its role for quantum statistics. In particular, we study the effect of phase-space interference on the factorization properties of a superposition state.

arXiv:quant-ph/9606023Journal of Physics A Mathematical and GeneralSep 21, 199618 citations

Quantum Mechanics of the Harmonic Oscillator

WORLD SCIENTIFIC eBooksMar 1, 20061 citation

\{\frac{z^{n}}{\sqrt{n!}}\}

form an orthonormal basis, one could in principle expand any complex function

f(z)

in powers of

z

and use this expansion as energy eigenstates. Such expansion is valid since the magnitude of

z

carry no physical information only their non-negative exponents

n

are physically important. These results may lead to new selection principle regarding the quantized energy spectrum of physical systems and enhance our understanding of quantum world.