Experiment has shown that scattered x-rays contain both a coherent and an incoherent portion. Classical electron theory provides a formula for scattering closely similar to that based on wave mechanics, but includes both coherent and incoherent components only if the atom consists of discrete electrons. From a comparison of the classical and wave-mechanics theories of scattering it is concluded that the most accurate classical analog of Schr\"odinger's $\ensuremath{\psi}{\ensuremath{\psi}}^{*}$ is the probability of occurrence of discrete electrons, and that a particular electron is associated with each particular function ${\ensuremath{\psi}}_{n}{{\ensuremath{\psi}}_{n}}^{*}$.
Experiment has shown that scattered x-rays contain both a coherent and an incoherent portion. Classical electron theory provides a formula for scattering closely similar to that based on wave mechanics, but includes both coherent and incoherent components only if the atom consists of discrete electrons. From a comparison of the classical and wave-mechanics theories of scattering it is concluded that the most accurate classical analog of Schr"odinger's \ensuremathψ\ensuremathψ∗ is the probability of occurrence of discrete electrons, and that a particular electron is associated with each particular function \ensuremathψn\ensuremathψn∗.
Quantum mechanics is introduced here as a fundamental theory that must treat particles equally often as waves. Here, classical waves are introduced in some detail, with discussions of the important concepts of waves: medium, amplitude, frequency, wavelength, period, harmonics, interference, standing waves, etc. Waves can be oscillatory waves with a repeating pattern, or pulse waves, which are localized in space. All of the elements of waves are important in understanding how the Schrödinger equation describes particles. Classical standing waves must have wave lengths that fit between boundaries giving them discrete allowed frequencies, a harbinger of quantized energies in quantum mechanics. Complicated waves can be analyzed in terms of their component frequencies, the fundamental idea of Fourier analysis. This discussion shows that a classical wave has a limit in how accurately one can specify simultaneously the wave’s position and frequency, in analogy with Heisenberg uncertainty principle of quantum mechanics.
Introducing a dimensional unit \hbar into classical mechanics, we make classical wave functions √ρ1exp (iS1/\hbar)-√ρ2exp (iS2/\hbar) for a one-particle system in the configuration space where √ρ1exp (iS1/\hbar) corresponds to the incident path (or one half) of an orbit and √ρ2exp (iS2/\hbar) to the outgoing (or another half), and ρ's and S's are densities and actions respectively. These classical wave functions are compared with the corresponding wave functions of solutions of the Schrödinger equation for three one-particle systems, the scattering by a Coulomb potential, the linear harmonic oscillator and an electron bound by a Coulomb potential. From the essential correspondence between them it is concluded that the Schrödinger wave function describes the steady, causal but a little imaginary motion of the one-particle system as √ρexp (iS/\hbar) in the configuration space where ρ is a density and S an action, and that the states which the wave equation describes exist continuously (not discrete) and the (discrete) stationary states are the ones where √ρexp (iS/\hbar) is uniquely determined throughout an orbit. The Dirac equation shows that an electron of optics (or the electromagnetic wave equation) can be considered as the Schrödinger (or Dirac) equation for the motion of a photon, a particle, in a potential. A quantum-electrodynamical wave equation which is composed of the Dirac equation with a Coulomb potential and the electromagnetic wave equation is constructed for the steady motion of a one-electron system without recource to the so-called second-quantization method. It gives finite energy shifts. A line shift is calculated: ΔE(2S1/2 →2P1/2) \cong709 Mc/s.
At the atomic scale, the distinction between particles and waves breaks down. The realization, early in the twentieth century, that light consists of discrete packages, or “photons,” prompted the converse suggestion that particles could behave like waves. Confirmation of this idea came in 1927, from an experiment involving the scattering of electrons from a crystal. Today, “wave-particle duality” is central to quantum mechanics, and the wave properties of subatomic particles are widely used in microscopy and crystallography. Electron diffraction techniques are widely used to characterize atomic arrangements on solid surfaces and inside thin films, and to investigate the structures of molecules in gases. The dual nature of waves and particles is part of the “spookiness” of quantum mechanics, which cannot be explained in any classical way. It used to be that one could only imagine an experiment that would simultaneously demonstrate the particle and wave nature of electrons.
Assuming a background in basic classical physics, multivariable calculus, and differential equations, A Concise Introduction to Quantum Mechanics provides a self-contained presentation of the mathematics and physics of quantum mechanics. The relevant aspects of classical mechanics and electrodynamics are reviewed, and the basic concepts of wave-particle duality are developed as a logical outgrowth of experiments involving blackbody radiation, the photoelectric effect, and electron diffraction. The Copenhagen interpretation of the wave function and its relation to the particle probability density is presented in conjunction with Fourier analysis and its generalization to function spaces. These concepts are combined to analyze the system consisting of a particle confined to a box, developing the probabilistic interpretation of observations and their associated expectation values. The Schrödinger equation is then derived by using these results and demanding both Galilean invariance of the probability density and Newtonian energy-momentum relations. The general properties of the Schrödinger equation and its solutions are analyzed, and the theory of observables is developed along with the associated Heisenberg uncertainty principle. Basic applications of wave mechanics are made to free wave packet spreading, barrier penetration, the simple harmonic oscillator, the Hydrogen atom, and an electric charge in a uniform magnetic field. In addition, Dirac notation, elements of Hilbert space theory, operator techniques, and matrix algebra are presented and used to analyze coherent states, the linear potential, two state oscillations, and electron diffraction. Applications are made to photon and electron spin and the addition of angular momentum, and direct product multiparticle states are used to formulate both the Pauli exclusion principle and quantum decoherence. The book concludes with an introduction to the rotation group and the general properties of angular momentum.
Abstract The new wave mechanics has been eminently successful in correlating and accounting for the large amount of experimental data on the periodic properties of the atom. The applications of the new theory to aperiodic phenomena, though not nearly so numerous, have been attended with no less success; indeed, it is in these experiments on free electrons, in which diffraction patterns similar to those produced by beams of X-rays and light are obtained, that the wave nature of electrons is so clearly and objectively demonstrated. Such diffraction effects have been obtained by a number of investigators by scattering beams of homogeneous electrons in crystals, thin films and by a ruled grating. Similar effects are obtained when electrons are scattered by complex molecules, owing to the symmetrically situated nuclei in the molecular structure, and when α-particles are scattered by helium owing to interference between the waves of the scattered incident particles and recoiling helium nuclei.
This thesis concerns quantum states that are superpositions of discrete or of continuum energy levels. It develops mathematical methods to characterize the physical properties of these states and finds important consequences. Regarding rational extensions of the harmonic oscillator, no other authors had constructed the coherent states associated with the ladder operators of Marquette and Quesne. It was our aim to proceed with this construction, then to calculate physical properties of these coherent states, looking for classical or quantum behaviour. The method of partial wave analysis was known to be not applicable to the Coulomb potential for the scattering of plane waves. It was my aim to repeat the calculation for wavepacket scattering, to investigate whether the method became applicable. Other authors have constructed relativistic probability amplitudes and compared them to the scalar Klein-Gordon amplitude. In this context, it was my aim to question the conclusion of a minimum width for a massive particle. In the same context, I investigated the possibility of a four-vector probability current with the Newton-Wigner position probability density as the zero component. In Chapter 12, my aim was to construct a measure of localization for the photon, knowing from the work of other authors that a point-localized photon state is not possible.In Chapters 3, 4 and 5, our measures of classicality for the Barut-Girardello coherent states included plotting energy expectations as functions of the magnitude of the coherent state para- meter, z, position probability densities for the coherent states and the even and odd cat states, the Wigner functions for particular choices of z, the two-photon number probability distribu- tion and the linear entropy for a coherent state on a beamsplitter, position spreads to look for squeezing and the Mandel Q parameter. In Chapter 6, it was necessary to approximate an integral containing a particular Wigner rotation matrix. An approximation of general Wigner rotation matrices, at low angle, was derived and its precision evaluated numerically in Chapter 7. The exact values for the Coulomb phase shifts were used to construct the incoming wavefunction. Time reversal and a rotation were used to construct the outgoing wavefunction. The sums over l were evaluated numerically. In Chapter 10, four arguments were used against the minimum width. Comparison of the scalar amplitude and the relativistic probability amplitude for a Newton-Wigner localized state showed the root of the problem. A Lorentz transformation was applied to the scalar function to see if the “minimum” width became even smaller. A simple construction produced scalar functions with arbitrarily small widths. A relevant argument from Almeida and Jabs was included here. In Chapter 11, the algebra involving the four components of a four-vector current (at the origin) and the boost generators was derived. All of these commutators must be satisfied by a four-vector, providing a test for the claimed four-vector probability currents of other authors. In Chapter 12, field expectations were taken in a coherent state with a mean photon number of unity, built from identical single-photon states.In Chapters 3, 4 and 5, we found a mix of classical and quantum behaviour for the coherent states. In none of the systems studied did we see the high degree of classicality of the unmodified harmonic oscillator. The investigation of cat states was relevant to the fundamental problem of measurement in quantum mechanics. In Chapter 6, the use of wavepackets introduced a convergence factor into the sum over l, which diverges in the plane wave treatment. The elimination of a divergence is the most significant result of this thesis. Scattering probabilities remained finite at all angles, in disagreement with the diverging Rutherford formula, but only on small angular regions around forward scattering. Future work will investigate the measurability of the phenomena predicted here. In Chapter 10, all four arguments pointed to no minimum width for a massive particle. In Chapter 11, it was found that there can be no four-vector current density with the Newton-Wigner position probability density as the zero component, for any spin. Other authors were found to be in error when they claimed to have produced such a four-vector current density. This result strengthens the position of the Dirac current as the appropriate current for an electron. In Chapter 12, the coherent state field expectations were found to provide a meaningful measure of localization. In summary, wavepackets give a realism to the description of a scattering experiment that is absent with the use of plane waves. Phenomena not captured by the plane wave scattering amplitude, such as time delays, are calculable in the wavepacket context. Relativistic probability amplitudes provide a complete description of free particles, consistent with quantum mechanics and with special relativity. They predict agreement with the Heisenberg uncertainty principle and relativistic effects such as Lorentz contraction. The absence of relativistic probability amplitudes from standard textbooks on relativistic quantum mechanics is a serious omission.