Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

American Journal of Physics

Simplified quantum oscillator solution

Thomas F. Jordan

Published Oct 1, 2001• 1 citation
Publisher

Abstract

To find the eigenvalues and eigenvectors of the quantum oscillator Hamiltonian, it is enough to construct eigenvectors with the raising operator and see that these eigenvectors are complete. The subspace they span must be the entire space of square-integrable functions because it is invariant for the raising and lowering operators and therefore invariant for the position and momentum operators. This shows that the Hamiltonian has no continuous spectrum.

DOI
10.1119/1.1387044

Related papers

The raising and lowering operator in Quantum mechanics

Dai Qi

The raising and lowering operator is studied in Quantum Mechanics.By using raising and lowering operaters,one can solve the eigenvalue question of coordinate,momentum and angular momentum,one can also find the energy eigenvalue and eigenfunction of the linear harmonic oscillator and hydrogen atom.

Journal of Xinyang Normal University•Jan 1, 2002•0 citations

A completely algebraic solution of the simple harmonic oscillator

M. Rushka, J. K. Freericks

We present a full algebraic derivation of the wavefunctions of a simple harmonic oscillator. This derivation illustrates the abstract approach to the simple harmonic oscillator by completing the derivation of the coordinate-space or momentum-space wavefunctions from the energy eigenvectors. It is simple to incorporate into the undergraduate and graduate curricula. We provide a summary of the history of operator-based methods as they are applied to the simple harmonic oscillator. We present the derivation of the energy eigenvectors along the lines of the standard approach that was first presented by Dirac in 1947 (and is modified slightly here in the spirit of the Schrödinger factorization method). We supplement it by employing the appropriate translation operator to determine the coordinate-space and momentum-space wavefunctions algebraically, without any derivatives.

arXiv:1912.08355•American Journal of Physics•Oct 21, 2020•22 citations

Solution of quantum eigenvalue problems by means of algebraic consistency conditions

Luis de la Peña, Ana Marı́a Cetto, Andrea Valdés-Hernández

Abstract We present a simple algebraic procedure that can be applied to solve a range of quantum eigenvalue problems without the need to know the solution of the Schrödinger equation. The procedure, presented with a pedagogical purpose, is based on algebraic consistency conditions that must be satisfied by the eigenvalues of a couple of operators proper of the problem. These operators can be either bilinear forms of the raising and lowering operators appropriate to the problem, or else auxiliary operators constructed by resorting to the factorization of the Hamiltonian. Different examples of important quantum-mechanical textbook problems are worked out to exhibit the clarity and simplicity of the algebraic procedure for determining the spectrum of eigenvalues without knowing the eigenfunctions. For this reason the material presented may be particularly useful for undergraduate students or young physicists.

arXiv:2110.03829•European Journal of Physics•Oct 11, 2021•0 citations

Canonical Operators for the Simple Harmonic Oscillator

Boris Leaf

Angle and action operators (w, j) for the simple harmonic oscillator are treated as resulting from a canonical transformation of coordinate and momentum operators (q, k) generated by a one-sided unitary operator U such that U†U = 1 and UU† commutes with k but not with q. From the discrete spectrum of the number operator n, eigenvectors |η〉 are constructed for every real value of η; the set {|η〉} is complete and orthogonal. Another complete set {|W〉} is obtained, consisting of the Fourier transforms of the kets in the set {|η〉}. The angle operator is w=U†qU=∫dW |W〉 W 〈W|. The set {|W〉} is not orthogonal; |W〉 is not an eigenvector of w. If v is defined as ∫dW|W〉exp (−2πiW) 〈W|, then the creation and destruction operators are given by a = vn½, a† = n½v†. v is a one-sided unitary operator such that vv† = 1, but v†v = 1 − |0〉 〈0|, where |0〉 is the ground state of the oscillator; v and v† are similar to the operators E- and E+ of Carruthers and Nieto. The Weyl transforms of w and j = 2πℏ(n + ½1) are the classical angle and action variables of the oscillator. The Weyl transform is formulated in terms of the coherent states of the oscillator. A time operator canonical to the Hamiltonian is defined as t = 2πw/ω (ω/2π = frequency). The observables for the oscillator are also given in the Heisenberg picture and their classical limits are considered.

Journal of Mathematical Physics•Nov 1, 1969•26 citations

q-oscillator from the q-Hermite polynomial

Satoru Odake, Ryu Sasaki B

By factorization of the Hamiltonian describing the quantum mechanics of the continuous q-Hermite polynomial, the creation and annihilation operators of the q-oscillator are obtained. They satisfy a q-oscillator algebra as a consequence of the shape-invariance of the Hamiltonian. A second set of q-oscillator is derived from the exact Heisenberg operator solution. Now the q-oscillator stands on the equal footing to the ordinary harmonic oscillator.

arXiv:0710.2209•Physics Letters B•Apr 2, 2008•27 citations

Rigged Hilbert Space Treatment of Continuous Spectrum

Rafael de la Madrid, Arno Böhm, Manuel Gadella

The ability of the Rigged Hilbert Space formalism to deal with continuous spectrum is demonstrated within the example of the square barrier potential. The non-square integrable solutions of the time-independent Schrödinger equation are used to define Dirac kets, which are (generalized) eigenvectors of the Hamiltonian. These Dirac kets are antilinear functionals over the space of physical wave functions. They are also basis vectors that expand any physical wave function in a Dirac basis vector expansion. It is shown that an acceptable physical wave function must fulfill stronger conditions than just square integrability — the space of physical wave function is not the whole Hilbert space but rather a dense subspace of the Hilbert space. We construct the position and energy representations of the Rigged Hilbert Space generated by the square barrier potential Hamiltonian. We shall also construct the unitary operator that transforms from the position into the energy representation. We shall see that in the energy representation the Dirac kets act as the antilinear Schwartz delta functional. In constructing the Rigged Hilbert Space of the square barrier potential, we will find a systematic procedure to construct the Rigged Hilbert Space of a large class of spherically symmetric potentials. The example of the square barrier potential will also make apparent that the natural framework for the solutions of a Schrödinger operator with continuous spectrum is the Rigged Hilbert Space rather than just the Hilbert space.

arXiv:quant-ph/0109154•Fortschritte der Physik•Mar 1, 2002•44 citations
To find the eigenvalues and eigenvectors of the quantum oscillator Hamiltonian, it is enough to construct eigenvectors with the raising operator and see that these eigenvectors are complete. The subspace they span must be the entire space of square-integrable functions because it is invariant for the raising and lowering operators and therefore invariant for the position and momentum operators. This shows that the Hamiltonian has no continuous spectrum.
We present a full algebraic derivation of the wavefunctions of a simple harmonic oscillator. This derivation illustrates the abstract approach to the simple harmonic oscillator by completing the derivation of the coordinate-space or momentum-space wavefunctions from the energy eigenvectors. It is simple to incorporate into the undergraduate and graduate curricula. We provide a summary of the history of operator-based methods as they are applied to the simple harmonic oscillator. We present the derivation of the energy eigenvectors along the lines of the standard approach that was first presented by Dirac in 1947 (and is modified slightly here in the spirit of the Schrödinger factorization method). We supplement it by employing the appropriate translation operator to determine the coordinate-space and momentum-space wavefunctions algebraically, without any derivatives.
Abstract We present a simple algebraic procedure that can be applied to solve a range of quantum eigenvalue problems without the need to know the solution of the Schrödinger equation. The procedure, presented with a pedagogical purpose, is based on algebraic consistency conditions that must be satisfied by the eigenvalues of a couple of operators proper of the problem. These operators can be either bilinear forms of the raising and lowering operators appropriate to the problem, or else auxiliary operators constructed by resorting to the factorization of the Hamiltonian. Different examples of important quantum-mechanical textbook problems are worked out to exhibit the clarity and simplicity of the algebraic procedure for determining the spectrum of eigenvalues without knowing the eigenfunctions. For this reason the material presented may be particularly useful for undergraduate students or young physicists.
Angle and action operators (w, j) for the simple harmonic oscillator are treated as resulting from a canonical transformation of coordinate and momentum operators (q, k) generated by a one-sided unitary operator U such that U†U = 1 and UU† commutes with k but not with q. From the discrete spectrum of the number operator n, eigenvectors |η〉 are constructed for every real value of η; the set {|η〉} is complete and orthogonal. Another complete set {|W〉} is obtained, consisting of the Fourier transforms of the kets in the set {|η〉}. The angle operator is w=U†qU=∫dW |W〉 W 〈W|. The set {|W〉} is not orthogonal; |W〉 is not an eigenvector of w. If v is defined as ∫dW|W〉exp (−2πiW) 〈W|, then the creation and destruction operators are given by a = vn½, a† = n½v†. v is a one-sided unitary operator such that vv† = 1, but v†v = 1 − |0〉 〈0|, where |0〉 is the ground state of the oscillator; v and v† are similar to the operators E- and E+ of Carruthers and Nieto. The Weyl transforms of w and j = 2πℏ(n + ½1) are the classical angle and action variables of the oscillator. The Weyl transform is formulated in terms of the coherent states of the oscillator. A time operator canonical to the Hamiltonian is defined as t = 2πw/ω (ω/2π = frequency). The observables for the oscillator are also given in the Heisenberg picture and their classical limits are considered.
The ability of the Rigged Hilbert Space formalism to deal with continuous spectrum is demonstrated within the example of the square barrier potential. The non-square integrable solutions of the time-independent Schrödinger equation are used to define Dirac kets, which are (generalized) eigenvectors of the Hamiltonian. These Dirac kets are antilinear functionals over the space of physical wave functions. They are also basis vectors that expand any physical wave function in a Dirac basis vector expansion. It is shown that an acceptable physical wave function must fulfill stronger conditions than just square integrability — the space of physical wave function is not the whole Hilbert space but rather a dense subspace of the Hilbert space. We construct the position and energy representations of the Rigged Hilbert Space generated by the square barrier potential Hamiltonian. We shall also construct the unitary operator that transforms from the position into the energy representation. We shall see that in the energy representation the Dirac kets act as the antilinear Schwartz delta functional. In constructing the Rigged Hilbert Space of the square barrier potential, we will find a systematic procedure to construct the Rigged Hilbert Space of a large class of spherically symmetric potentials. The example of the square barrier potential will also make apparent that the natural framework for the solutions of a Schrödinger operator with continuous spectrum is the Rigged Hilbert Space rather than just the Hilbert space.