The book is aimed at several levels. First, and foremost, it takes the approach of showing what quantum mechanics can achieve in understanding, by detailed worked examples, from a range of observations such as magnetic susceptibility, emission, optical, infrared, Raman, NMR, EPR and Mssbauer spectroscopy. The emphasis throughout is on mathematical rigor, an essential component of the book. Without this rigor quantum mechanics becomes superficial and, if presented as such, does t lead to an ability to be used in any meaningful way. The approach is based on oral tradition involving a one to one interaction where each step is clarified in a way that is fully understood prior to extending the processes gradually to more advanced levels. Each chapter is presented with the key components highlighted for those wishing to gain on overview without the depth such as usually presented at an advanced under-graduated level. For those wishing for greater depth, detail is presented in handling n-trivial examples often presented in tables appended to each chapter. These tables may be used as reference material or may be used by a reader or lecturer as a means of grasping or presenting the necessary skills to handle quantum mechanics by example.