arXiv:quant-ph/0702260

Nodes of Wavefunctions

Marco Moriconi

Published Feb 27, 2007• 0 citations

Abstract

We give a simple argument to show that the $n$th wavefunction for the one-dimensional Schrödinger equation has $n-1$ nodes. We also show that if $n_1 < n_2$, then between two consecutive zeros of $ψ_{n_1}$, there is a zero of $ψ_{n_2}$.

DOI: 10.48550/arxiv.quant-ph/0702260

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Nodes of wavefunctions

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arXiv:quant-ph/0702260

Nodes of Wavefunctions

Marco Moriconi

Published Feb 27, 2007• 0 citations

arXiv HTML PDF

Abstract

We give a simple argument to show that the

n

th wavefunction for the one-dimensional Schrödinger equation has

n-1

nodes. We also show that if

n_1 < n_2

, then between two consecutive zeros of

ψ_{n_1}

, there is a zero of

ψ_{n_2}

DOI: 10.48550/arxiv.quant-ph/0702260

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Nodes of wavefunctions

Marco Moriconi

American Journal of PhysicsFeb 12, 200714 citations

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Vladimir I. Pupyshev, E. A. Pazyuk, A. V. Stolyarov +2 more

Conventional one-dimensional oscillation theorem is found to be violated for multi-component Schrödinger equations in a general case while for two-component eigenstates coupled by the sign-constant potential operator the following statements are valid: (1) the ground state ( $v = 0$ ) is not degenerate; and (2) the arithmetic mean of nodes $n_1$ , $n_2$ for the two-component wavefunction never exceeds the ordering number $v$ of eigenstate: $(n_1 + n_2)/2\leq v$ .Conventional one-dimensional oscillation theorem is found to be violated for multi-component Schrödinger equations in a general case while for two-component eigenstates coupled by the sign-constant potential operator the following statements are valid: (1) the ground state ($v = 0$) is not degenerate; and (2) the arithmetic mean of nodes $n_1$, $n_2$ for the two-component wavefunction never exceeds the ordering number $v$ of eigenstate: $(n_1 + n_2)/2\leq v$.

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We analyze two simple models derived from a quantum-mechanical particle on an elliptical path. The first Hamiltonian operator is non-Hermitian but equivalent to an Hermitian operator. It exhibits the same two-fold degeneracy as the particle on a circular path. More precisely, the spectrum is $E_{n}=n^{2}E_{1},\ n=0,\pm 1,\pm 2,\ldots $, $E_{1}>0$ . The second Hamiltonian operator is Hermitian and does not exhibit such degeneracy. In this case the nth excited energy level splits at the nth order of perturbation theory. Both models can be described in terms of symmetry point groups with one-dimensional irreducible representations but the non-Hermitian example exhibits an additional symmetry that explains the two-fold degeneracy. The Schrödinger equation for the first example is exactly solvable.We analyze two simple models derived from a quantum-mechanical particle on an elliptical path. The first Hamiltonian operator is non-Hermitian but equivalent to an Hermitian operator. It exhibits the same two-fold degeneracy as the particle on a circular path. More precisely, the spectrum is $E_{n}=n^{2}E_{1},\ n=0,\pm 1,\pm 2,\ldots $, $E_{1}>0$. The second Hamiltonian operator is Hermitian and does not exhibit such degeneracy. In this case the nth excited energy level splits at the nth order of perturbation theory. Both models can be described in terms of symmetry point groups with one-dimensional irreducible representations but the non-Hermitian example exhibits an additional symmetry that explains the two-fold degeneracy. The Schrödinger equation for the first example is exactly solvable.

quant-pharXiv:2512.09905Dec 11, 20250 citations

We give a simple argument to show that the

n

th wavefunction for the one-dimensional Schrödinger equation has

n-1

nodes. We also show that if

n_1 < n_2

, then between two consecutive zeros of

ψ_{n_1}

, there is a zero of

ψ_{n_2}

v = 0

) is not degenerate; and (2) the arithmetic mean of nodes

n_1

n_2

for the two-component wavefunction never exceeds the ordering number

v

of eigenstate:

(n_1 + n_2)/2\leq v

E_{1}>0

. The second Hamiltonian operator is Hermitian and does not exhibit such degeneracy. In this case the nth excited energy level splits at the nth order of perturbation theory. Both models can be described in terms of symmetry point groups with one-dimensional irreducible representations but the non-Hermitian example exhibits an additional symmetry that explains the two-fold degeneracy. The Schrödinger equation for the first example is exactly solvable.