## An extract on #handeercel

To define angles in an abstract real inner product space, we replace the Euclidean dot product ( ) by the inner product , {\displaystyle \langle \cdot ,\cdot \rangle } , i.e. u , v = cos ( ) u v . {\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle =\cos(\theta )\ \left\|\mathbf {u} \right\|\ \left\|\mathbf {v} \right\|.} In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with Re ( u , v ) = cos ( ) u v . {\displaystyle \operatorname {Re} \left(\langle \mathbf {u} ,\mathbf {v} \rangle \right)=\cos(\theta )\ \left\|\mathbf {u} \right\|\ \left\|\mathbf {v} \right\|.} or, more commonly, using the absolute value, with | u , v | = | cos ( ) | u v . {\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=|\cos(\theta )|\ \left\|\mathbf {u} \right\|\ \left\|\mathbf {v} \right\|.} The latter definition ignores the direction of the vectors and thus describes the angle between one-dimensional subspaces span ( u ) {\displaystyle \operatorname {span} (\mathbf {u} )} and span ( v ) {\displaystyle \operatorname {span} (\mathbf {v} )} spanned by the vectors u {\displaystyle \mathbf {u} } and v {\displaystyle \mathbf {v} } correspondingly.

The majority of fields in physics can be divided between theoretical work and experimental work, and atomic physics is no exception. It is usually the case, but not always, that progress goes in alternate cycles from an experimental observation, through to a theoretical explanation followed by some predictions that may or may not be confirmed by experiment, and so on. Of course, the current state of technology at any given time can put limitations on what can be achieved experimentally and theoretically so it may take considerable time for theory to be refined. One of the earliest steps towards atomic physics was the recognition that matter was composed of atoms. It forms a part of the texts written in 6th century BC to 2nd century BC such as those of Democritus or Vaisheshika Sutra written by Kanad. This theory was later developed in the modern sense of the basic unit of a chemical element by the British chemist and physicist John Dalton in the 18th century. At this stage, it wasn't clear what atoms were although they could be described and classified by their properties (in bulk). The invention of the periodic system of elements by Mendeleev was another great step forward. The true beginning of atomic physics is marked by the discovery of spectral lines and attempts to describe the phenomenon, most notably by Joseph von Fraunhofer. The study of these lines led to the Bohr atom model and to the birth of quantum mechanics. In seeking to explain atomic spectra an entirely new mathematical model of matter was revealed. As far as atoms and their electron shells were concerned, not only did this yield a better overall description, i.e. the atomic orbital model, but it also provided a new theoretical basis for chemistry (quantum chemistry) and spectroscopy. Since the Second World War, both theoretical and experimental fields have advanced at a rapid pace. This can be attributed to progress in computing technology, which has allowed larger and more sophisticated models of atomic structure and associated collision processes. Similar technological advances in accelerators, detectors, magnetic field generation and lasers have greatly assisted experimental work.