Evolving Quantum Model

1So, by the end of 1924, many of the basic concepts that would come to underpin the next phase of the development of quantum theory were in place. In subsequent sections of this on-going discussion, we will review the development of Heisenberg’s matrix mechanics and Schrodinger’s wave mechanics. However, before moving onto this level of detail, let us take a moment to reflect on what had already started to emerge from the quantum model. Planck’s initial concept of quantized energy, in 1900, is often assumed be the start of the quantum age; although it is not apparent that any real ‘quantum shift' in physics took place at this very initial stage. Again, while Einstein’s work on the photoelectric effect, in 1905, is now recognized as another milestone in the development of quantum theory, the idea of quantized light, i.e. the photon, would not be generally accepted for another 20 years. In this context, many may therefore cite Bohr’s atomic model, in 1913, as a more profound shift towards the quantum nature of the sub-atomic world of the atom. However, there is another argument that quantum physics only really began to take shape once the full nature of the wave-particle duality was recognized through the work of Compton, in respect to the photon, and deBroglie, in respect to matter waves. Initially, in 1913, the concept of Bohr’s atomic model rested on the idea of quantized angular momentum, but did not really provide any physical rationalization for this effect. Only later, in 1924, would the deBroglie wavelength put forward some sort of physical explanation in terms of the restriction being placed on the electron ‘orbits’ due to integral wavelengths.

At this point, it is possibly worth reflecting on whether an underlying wave structure, restricted by the superposition of integral waveforms, provides the more fundamental, and most rational, explanation of the quantization perceived in so many sub-atomic processes.

As has been pointed out, Bohr’s initial atomic model, assumed circular orbits, which could not explain the spectral lines linked to the Zeeman Effect caused by a magnetic field. So, as early as 1916, Sommerfeld had to introduce the idea of elliptical orbits, in addition to circular orbits, in order to explain the spectral doublets of hydrogen. However, as the science of spectroscopy advanced, the atomic spectral lines were seen to split into evermore finer detail, which required even more esoteric concepts to be introduced, e.g. electron spin. These additional concepts are described as ‘esoteric’ in the sense that they still appear to allude to particle-like concepts, but which are often impossible to quantify in terms of the particle model. Therefore, in order to bring the wave model into some sort of initial focus, this specific discussion will depart from the chronology of developments so as to highlight the pending impact of a wave model on the overall development of the atomic model. While, at this point, we have only outlined some of the basic  concepts and issues surrounding matter waves, we might still make the following statement.

Today, as an accepted generalisation, quantum behaviour is based on the assumption that all particles are waves, whose wavelength is inversely proportional to the particle’s momentum [λ=h/p]. Therefore, in the quantum world, the behaviour of a particle is described in terms of quantum wave mechanics, which although still linked to the conservation of energy, replaces Newton’s laws of particle motion.

While we have not yet discussed the full implications of quantum waves, we might initially assume that they still obey certain basic rules of wave mechanics in that they are associated with energy and momentum. However, at this stage, what is possibly of more interest is the description of a quantum wave in terms of some constructive and destructive superposition process, which might explain why there are regions in space, which become quantized into discrete energy bands. It was in this context that deBroglie initially postulated that an electron, orbiting the nucleus of a hydrogen atom, could only occupy discrete orbits due to the wave nature of the electron. However, with the benefit of hindsight, it may be possible to construct an extended wave model of an atom in which the concept of an electron, as a particle orbiting the nucleus, completely disappears. As such, what emerges is a model of allowed wave states that reflect the idea of superposition waves surrounding the 3-dimensional space centred on the nucleus. In this context, a hydrogen atom in its ground state represents the simplest standing wave configuration, which can be extended to more complex 3-dimensional patterns comprising of shorter integral wavelengths, i.e. higher frequencies and therefore higher energy. Unfortunately, even today, the description of the atomic model still appears to be overloaded with particle visualisation, e.g. atomic orbital and electron spin. However, in order to cross-reference standard text describing the molecular model, it is necessary to define the following description of what are called the electron quantum numbers:

  • Principal number N:
    Is often described in terms of the size of the orbital, which can be cross-referenced to what is also called the atomic shell, where n=1 stands for K shell, n=2 for L shell etc.

  • Angular number L:
    Can be thought to describe the shape of the orbital, which can have values of [0 to N-1], where [N] is the principle number above. So, with N=3, the L number can take the values [0..2]. This number can also be aligned to the description of atomic orbitals of the type [s,p,d or f], e.g. [s, L=0], [p, L=1] etc.

  • Magnetic number M:
    Describes the orientation of an orbital in space, which can take the values from -L to +L. So for L=0,1,2 you get M=-2,-1,0,1,2.

  • Spin number MS:
    While often said to describe the direction of spin, this physical visualisation is misleading in the context of a wave model in which the electron does not exist as a particle. As such, it might be better thought of as the phase relationship of the wave in superposition, i.e. [+] or [-].

As indicated, within the context of a wave model, it might be better to consider these numbers in terms of a superposition of harmonic waves that can create different standing wave patterns in the 3-dimensional space surrounding the nucleus of an atom. These superposition patterns require integral wavelengths, which possibly make it easier to see why the energy levels within the atom becomes quantised. Of course, higher order integral wavelength patterns are only possible, if higher energy is available, as associated with a higher frequency. As such, the complexity of what was originally considered to be electrons orbiting the nucleus converts into the complexity of 3-dimensional wave patterns. While this complexity still needs to be understood, it is possibly easier to understand as there appears to now be some physical rationale supporting the process under investigation. In the following diagram, the wave pattern complexity is subdivided into separate rows, aligning to the quantum numbers defined above, i.e. N, L and M.


In this form, we might describe each row, in the diagram above as a cross-sectional plane defined by spherical coordinates [r,φ,θ] of 3-dimensional space, where [N=r], [L=φ] and [M=θ]. In the case of the Bohr model for hydrogen [1,0,0], the wave pattern assumes spherical symmetry around the nucleus. However, as the values of [N,L,M] increase, the collective patterns become increasingly complex, as illustrated in the next diagram, which tries to extend the previous diagram towards a 3D visualisation:


In the first row of the diagram above, the 3-dimensional shell structure is also shown in cross-section, so that the nested shell structure can be seen. These cross-sections align to the first row in the previous diagram. In practise, this wave model of the atomic structure of the heavier elements, having dozens of electrons ‘orbiting’ a nucleus with both protons and neutrons, becomes an increasingly unwieldy description, which possibly explains why it collapses back into the preferred particle description of physics and chemistry. However, it seems clear that the concept of a particle is only a ‘matter’ of practical convenience and not a true reflection of what appears to be the more fundamental wave structure of matter in the quantum universe.