Bound states in quantum mechanics

This part introduces variational principles relevant to the quantum mechanics of bound stationary states. Chapter 4 covers well-known variational theory that underlies modern computational methodology for electronic states of atoms and molecules. Extension to condensed matter is deferred until Part III, since continuum theory is part of the formal basis of the multiple scattering theory that has been developed for applications in this subfield. Chapter 5 develops the variational theory that underlies independent-electron models, now widely used to transcend the practical limitations of direct variational methods for large systems. This is extended in Chapter 6 to time-dependent variational theory in the context of independent-electron models, including linear-response theory and its relationship to excitation energies. 

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Calculating the Energy of Excited Bound States in Quantum Mechanics. 

Variational principles are widely applied to calculations of bound states in quantum mechanics [18,231]. One usually considers the expectation value of the energy of the system, FJ[ ] = ( ]ff ), and looks for stationary points (i.e., 5E = 0) of this functional with respect to arbitrary variations... 

L. I. SchifF, Approximation methods for bound states, in Quantum Mechanics, McGraw-Hill, New York, 1968, pp. 279-295. 

In simple English, this implies that if you were in a 4-D universe and launched planets toward a sun, the planets would either fly away to infinity or spiral into the sun. (This is in contrast to a (3 + 1) universe that obviously can, for example, have stable orbits of moons around planets.) A similar problem occurs in quantum mechanics, in which a study of the Schrodinger equation shows that the hydrogen atom has no bound states for n > 5. This seems to suggest that it is difficult for higher universes to be stable over time and contain creatures that can make observations about the universe. 

In order to discuss the fundamental problems that are connected with the bound states in kinetic theory, we first restrict ourselves to systems with two-particle bound states only. The states of the two-particle system are determined by Eq. (2.12). Furthermore, we remark that to describe the formation of two-particle bound states by a collision, at least three particles are necessary in order to fulfill energy and momentum conservation. Thus, it is necessary to consider the quantum mechanics of three-particle systems. 

The electron and positron in positronium move with a typical velocity a and have a momentum mot and energy mo . If we are interested in a precision level where relativistic effects become important, the usual quantum mechanical treatment of the bound state has to be merged with the complete field theory description. Then, no matter how weak the coupling is, we deal with the problem of bound states in the field theory, which is rather complicated. One of the possible computational approaches (not always the most convenient) is the Bethe-Salpeter equation. 

The author examines with success the efficiency of the methods by their applications to bound states for the one-dimensional harmonic oscillator, anharmonic oscillators, the Morse potential, and the modified Poschl-Teller potential in quantum mechanics. 

In classical mechanics, positions and momenta are treated on an equal footing in the Hamiltonian picture. In quantum mechanics, they become operators, but it is true that the position r and momentum p of a particle are appropriate conjugate variables that can entirely equivalently describe a state of a system under the commutation relation [r, p] = i (Dirac, 1958). This equivalence is usually demonstrated by the example of the onedimensional harmonic oscillator. The choice of the most appropriate representation depends on convenient description of the phenomenon considered. Generally, the position representation is useful for most bound-state problems such as atomic and molecular electronic structures as well as for many scattering problems. The momentum-space treatment... 

The interparticle distances ry can be easily expressed in terms of Jacobi coordinates, and the reduced masses (J, have been defined in terms of the particle masses m as /u23 = rn2m3/(m2 + m3), /j, 1,23 = mi(m2 + m3)/[mi + (m2 + m3)]. A bound state in a quantum mechanical system is described by a real valued energy and the corresponding real valued eigenfunction. However we above set out to study fragmentation phenomena. We thus need to study complex energies and wavefunctions. 

It is only possible to understand how two electrons can be bound to one proton by considering the electron wave functions. In quantum mechanics, the electrons cannot be modeled as pointlike particles orbiting the nucleus, but must be pictured as fuzzy distributions of probability. In H, the electrons are in close enough proximity that their probability distributions, or wave functions, overlap. This overlap induces a positive correlation that allows the bound state of the ion. This means that the electrons do not have simple individual independent wave functions, but share a different and more complicated wave function. 

Sturmians are not simply an alternative orbital set closely related to hydrogenic functions and to Slater-type orbitals. In the last decade the connection between their use and the ever-increasing exploitation of hyperspherical methods to treat the A -body problem in quantum mechanics was established, for both bound-state and scattering problems in nuclear, atomic and molecular physics and this became an important source of progress, as recorded in a recent book [5]. 

Fig. 2.5. Functions of class Q (i.e. wave functions allowed in quantum mechanics) - examples and counterexamples. A wave function (a) must not be zero everywhere in space (b) has to be continuous (c) cannot tend to infinity even at a single point (d) cannot tend to infinity (e) its first derivative cannot be discontinuous for infinite number of points (f) its first derivative may be discontinuous for a finite number of points (g) has to be defined uniquely in space (for angular variable 0) (h) cannot correspond to multiple values at a point in space (for angular variable 6) (i) for bound states must not be non-zero in infinity (j) for bound states has to vanish in infinity.

As noted in the Introduction, in this presentation, we will limit our formalism and analysis to one dimensional, rational fraction, bound state potentials, for simplicity. Our intention is to motivate what we perceive to be the principal importance of Continuous Wavelet Transform (CWT) theory in quantum mechanics, that of facilitating the multiscale analysis of singular systems, particularly those associated with multiple (complex) turning point interactions. The understanding of these issues rests on a clear appreciation of the significant role Moment Quantization methods bear on the multiscale analysis of quantum operators. 

Draw approximate energy levels and wavefunctions for the bound states (E < 0) of the square well potential. THINKING AHEAD [How is this like other cases in quantum mechanics that you may know from earlier chapters ]... 

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