In quantum mechanics the eigenstates of the Hamiltonian form a complete basis. However, physicists conventionally express completeness as a formal sum over the eigenstates, and this sum is typically a divergent series if the Hilbert space is infinite dimensional. Furthermore, while the Hamiltonian can be reconstructed formally as a sum over its eigenvalues and eigenstates, this series is typically even more divergent. For the simple cases of the square-well and the harmonic-oscillator potentials, this paper explains how to use the elementary procedure of Euler summation to sum these divergent series and thereby to make sense of the formal statement of the completeness of the formal sum that represents the reconstruction of the Hamiltonian.
There are a few exactly solvable potentials in quantum mechanics for which the completeness relation of the energy eigenstates can be explicitly verified. In this article, we give an elementary proof that the set of bound (discrete) states together with the scattering (continuum) states of the reflectionless potential form a complete set. We also review a direct and elegant derivation of the energy eigenstates with proper normalization by introducing an analog of the creation and annihilation operators of the harmonic oscillator problem. We further show that, in the case of a single bound state, the corresponding wave function can be found from the knowledge of continuum eigenstates of the system. Finally, completeness is shown by using the even/odd parity eigenstates of the Hamiltonian, which provides another explicit demonstration of a fundamental property of quantum mechanical Hamiltonians.
In quantum mechanics the time operator Θ satisfies the commutation relation [Θ, H] = i, and thus it may be thought of as being formally canonically conjugate to the Hamiltonian H. The time operator associated with a given Hamiltonian H is not unique because one can replace Θ by Θ + Θhom, where Θhom satisfies the homogeneous condition [Θhom, H] = 0. To study this nonuniqueness the matrix elements of Θ for the harmonic-oscillator Hamiltonian are calculated in the eigenstate basis. This calculation requires the summation of divergent series, and the summation is accomplished by using zeta-summation techniques. It is shown that by including appropriate homogeneous contributions, the matrix elements of Θ simplify dramatically. However, it is still not clear whether there is an optimally simple representation of the time operator.
This is the first in a series of papers. It deals with the linked-cluster summation of the perturbation series in quantum mechanics and quantum statistical physics. In particular, a formalism is developed for calculating the matrix element of the operator
exp(+sH), where
H is the quantum-mechanical Hamiltonian operator. It is found that the matrix element may be written as an ordinary
c-number integral of a
c-number function over the phase space. This function is found by doing a linked-cluster summation of the usual perturbation series. It is in the form of an exponential of an effective Hamiltonian,
Heff, which is a function of the parameter
s and the
c-number momentum and position coordinates. We have assumed that the Hamiltonian has no spin-dependent terms. The matrix element is written in such a form that it may be used for calculating the transition matrix elements in time-dependent perturbation theory as well as for calculating the canonical partition function
ZN in quantum statistical mechanics. The former is obtained by substituting $s=\ensuremath{-}\frac{\mathrm{it}}{\ensuremath{\hbar}}$ and the latter by substituting $s=\ensuremath{-}\ensuremath{\beta}$. The series for
Heff depends only on the potential and is independent of the statistics. The effects of the statistics (Bose-Einstein, Fermi-Dirac, or Maxwell-Boltzmann) are all included in a certain factor multiplying
exp(sHeff). Thus we have succeeded in separating the statistical and quantum corrections. These latter arise from the noncommutativity of the position and momentum operators. The effective Hamiltonian is an infinite series. A diagrammatic technique is given for evaluating this series. It is an expansion in powers of the coupling constant and can also be expanded in powers of Planck's constant. In general, it is not possible to give a closed-form expression for
Heff. But for most practical cases it is sufficient to keep the first few powers of Planck's constant $\ensuremath{\hbar}$, because it is very small. We therefore have an entirely new method of approximation. The statistical effect may always be included exactly. No numerical estimates are made in this paper.
It is shown that it is possible to establish sum rules that must be satisfied at the nodes and extrema of the eigenstates of confining potentials which are functions of a single variable. At any boundstate energy the Schrödinger equation has two linearly independent solutions one of which is normalizable while the other is not. In the domain after the last node of a boundstate eigenfunction the unnormalizable linearly independent solution has a simple form which enables the construction of functions analogous to Green's functions that lead to certain sum rules. One set of sum rules gives conditions that must be satisfied at the nodes and extrema of the boundstate eigenfunctions of confining potentials. Another sum rule establishes a relation between an integral involving an eigenfunction in the domain after the last node and a sum involving all the eigenvalues and eigenstates. Such sum rules may be useful in the study of properties of confining potentials. The exactly solvable cases of the particle in a box and the simple harmonic oscillator are used to illustrate the procedure. The relations between one of the sum rules and two-particle densities and a construction based on supersymmetric quantum mechanics are discussed.
We delineate the scope of research on the completeness of eigenstates in quantum mechanics. Based on the limit of the potential function at infinity, the proof of completeness is divided into eight cases, and theoretical proofs or numerical simulations are provided for each case. We present the definition of orthonormalization for general free states and the solution to the normalization coefficients, as well as a general set of initial states, which simplifies and concretizes the proof of completeness. Additionally, we define the spectral function for continuous energy eigenvalues. By taking the spectral function as the original integral variable of the expansion function, the relationship between the measured probability amplitude and the expansion function is endowed with the physical meaning of coordinate-momentum transformation.