Abstract A wave function which is other than an exact eigenfunction, if it obeys appropriate analytical conditions, can be considered to represent the initial configuration of a nonstationary state. In the course of its subsequent time development the quantum system exhibits implicitly its entire eigenvalue spectrum. A method based, in principle, on Fourier analysis of the evolving quantum system is applied to the direct calculation of the energy eigenvalue spectrum. The spectral function is expressed as a moment expansion, in terms of expectation values of powers of the Hamiltonian. When the expansion is truncated, as it must be in any practical application, the corresponding spectral function represents a smeared‐out eigenvalue spectrum. An alternative approximation leads to the quantum‐mechanical method of moments. As the number of terms is increased, the computed spectrum becomes sharper and more accurate. In certain cases the moment expansion can be circumvented, if the action of the evolution operator can be evaluated in closed form. This is equivalent to finding some solution of the time‐dependent Schrödinger equation. The various methods of eigenvalue spectroscopy are applied to the harmonic oscillator.
Abstract A wave function which is other than an exact eigenfunction, if it obeys appropriate analytical conditions, can be considered to represent the initial configuration of a nonstationary state. In the course of its subsequent time development the quantum system exhibits implicitly its entire eigenvalue spectrum. A method based, in principle, on Fourier analysis of the evolving quantum system is applied to the direct calculation of the energy eigenvalue spectrum. The spectral function is expressed as a moment expansion, in terms of expectation values of powers of the Hamiltonian. When the expansion is truncated, as it must be in any practical application, the corresponding spectral function represents a smeared‐out eigenvalue spectrum. An alternative approximation leads to the quantum‐mechanical method of moments. As the number of terms is increased, the computed spectrum becomes sharper and more accurate. In certain cases the moment expansion can be circumvented, if the action of the evolution operator can be evaluated in closed form. This is equivalent to finding some solution of the time‐dependent Schrödinger equation. The various methods of eigenvalue spectroscopy are applied to the harmonic oscillator.
An arbitrary function in the eigenfunction space of some quantum-mechanical Hamiltonian may be thought to represent the initial configuration ψ(q,0) of a nonstationary state. The system develops in time according to ψ(q,t)=exp(−itH)ψ(q,0)= ∑ k=0∞(−it)kk!Hkψ(q,0).Defining F(t)≡〈ψ(q,0), ψ(q,t) 〉, and hk≡〈ψ(q,0), Hkψ(q,0) 〉, and taking the Fourier transformG(ω)= ∫ −∞∞dtexp(−iωt)F(t),we obtain G(ω)=2π ∑ k=0∞hkk!(ω).In terms of the formal expansion in the energy eingenfunctions ψ(q,0)= ∑ n=0∞cnφn(q)+ ∫ 0∞dωc(ω)φ(ω,q),the Fourier transform represents G(ω)=2π ∑ n=0∞|cn|2δ(ω+ωn)+2π|c(−ω)|2,which exhibits, in principle, the entire eigenvalue spectrum. In this paper, a direct method of calculating eigenvalue spectra, based on the foregoing principle, is proposed. Two modifications are required for computational practicability: (i) use of a finite representation for the delta function and truncation of the summation (a); (ii) replacement of the integrals hk by hk(q′)≡Hkψ(q,0)]q=q′. The modified spectral function is taken to be GN(τ,ω,q′)=2π ∑ k=0Nhk(q′)k!χτ(k)(ω),with χτ(ω)≡sinτω/πω. The sequence GN(τ,ω,q′) is shown to converge as N→∞ if in the Expansion (b) the coefficients cn and c(ω) decrease with ω as exp(—ω/λ) or faster. Assuming convergence, the spectral function represents a broadened eigenvalue spectrum.
The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.
Abstract The energy eigenvalue spectrum for a conservative dynamical system is contained implicitly in its Green's function. It becomes explicit in the Fourier transform of either the Green's function or its trace. The trace exists only when the spectrum is entirely discrete. Applications are made to the free particle, the linear harmonic oscillator, and the hydrogen atom. In the latter two cases determination of the Green's function can be considerably simplified by similarity transformations on the Hamiltonian operator.
Abstract The Green function method is a powerful technique for solving the initial value problem in quantum mechanics. Once the Green function is solved the whole wavefunction evolution is represented in a concise form and can be evaluated conveniently by numerical integration. We present a method for constructing the Green functions systematically which is different from the conventional methods of eigenfunction expansion or path integration. By using variable changing, function substitution, and Fourier transforms, the time dependent Schrödinger equations can be simplified and the solutions for the simplified equations can be easily derived. We then obtain the Green functions for the original equations by the reverse transforms. The method is demonstrated for the linear potential, the harmonic oscillator, the centrifugal potential, and the centripetal barrier oscillator, where the Green function for the centripetal barrier oscillator has not been solved previously by conventional methods. The method and examples illustrated in this paper can be utilised to strengthen undergraduate courses on quantum mechanics and/or partial differential equation.
In the previous chapter, the postulates helped us to use the wave function to find the expected results of experimental measurements by letting a dynamical quantity F (x, p, t) → F (xˆ, pˆ, t) and integrating over space. In the examples, however, the wave functions were typically given and not derived from first principles. We now endeavor to find the appropriate wave functions for a variety of physical systems in one dimension by solving the Schro¨dinger equation where for time-independent cases, the total energy is an eigenvalue. In subsequent chapters, we will extend the method to three dimensions and develop the appropriate operators, eigenvalues, and eigenfunctions for angular momentum. We will also develop methods for examining time-dependent systems. The latter chapters address special systems of interest that may be referred to as applications of quantum mechanics.
A basic quantity in quantum mechanics is the wave function ψ. Given a quantum mechanical system we write the Schrödinger equation for it. Its solution is the wave function. When introducing the wave function we stated that the maximum information about the state of a system can be obtained only from ψ. How do we extract observable properties of the system from the wave function? In quantum mechanics for every classical dynamical variable there exists an operator. For example, the operator of momentum p is −i~∇ while the operator of energy variable H = p2/(2m)+V (X) is −(~2/2m)∇2+V (X). Operation of an operator on ψ gives another function. If the new function is λψ, then λ is the eigenvalue of the operator. For example, the energy of a particle in a particular state represented by ψ can be obtained by operating H on ψ. That is, we have Hψ = Eψ where E is the energy eigenvalue. The expectation value of A(X,p) can be obtained from the formula 〈A〉 = ∫ ∞ −∞ ψ∗Aψ dτ , (3.1) where ψ is the solution of the equation Hψ = Eψ. Operators in quantum mechanics which represent observables are of a spe- cial type: They are linear and Hermitian. Such operators not only form computation tools, but also form the most effective language in terms of which the theory can be formulated. The fascinating point is that using linear operators we can understand the quantum mechanics. Therefore, it is important to study the basic theory of linear operators. In this chapter, we present important properties of linear operators and their eigenfunctions. We discuss the features of various operators which are necessary for studying quantum and ematical structure of quantum mechanics is.