**Supersymmetry and quantum mechanics**

In the past 10 years, the ideas of scientific theory are fruitfully applied to several nonrelativistic quantum mechanical issues. particularly, there’s currently a way deeper understanding of why bound potentials ar associatealytically soluble and an array of powerful new approximation strategies for handling potentials that don’t seem to be specifically soluble. during this report, we tend to review the theoretical formulation of supersymmetric quantum physics and discuss several applications. specifically soluble potentials may be understood in terms of a couple of basic ideas that embrace supersymmetric partner potentials, form invariableness and operator transformations. acquainted soluble potentials all have the property of form invariableness. we tend to describe new specifically soluble form invariant potentials that embrace the recently discovered self-similar potentials as a special case. **[1]**

**Complex Extension of Quantum Mechanics**

Requiring that a Hamiltonian be Hermitian is to a fault restrictive. a regular physical theory of quantum physics will be engineered on a posh Hamiltonian that’s not Hermitian however satisfies the less restrictive and additional vigor of coordinate system reflection symmetry (PT symmetry). One may expect a non-Hermitian Hamiltonian to guide to a violation of unitarity. However, if noble metal symmetry isn’t impromptu broken, it’s doable to construct a antecedently unperceived symmetry C of the Hamiltonian. Using C, associate degree dot product whose associated norm is positive definite will be created. The procedure is general and works for any PT-symmetric Hamiltonian. Observables exhibit CPT symmetry, and also the dynamics is ruled by unitary time evolution. This work isn’t in conflict with standard quantum physics however is quite a posh generalization of it. **[2]**

**Quantum mechanics as a statistical theory**

An attempt is formed to interpret quantum physics as a applied mathematics theory, or additional precisely as a variety of non-deterministic applied mathematics dynamics. The paper falls into 3 elements. within the 1st, the distribution functions of the entire set of driving variables specifying a system (phase-space distributions), that area unit elementary in any variety of applied mathematics dynamics, area unit expressed in terms of the wave vectors of scientific theory. this can be shown to be comparable to specifying a theory of functions of non-commuting operators, and should thus be thought of as AN interpretation of quantum mechanics. within the second half, the laws governing the transformation with time of those phase-space distributions area unit derived from the equations of motion of quantum dynamics and located to be of the desired type for a driving framework.** [3]**

**A Relational Formulation of Quantum Mechanics**

Non-relativistic quantum physics is reformulated here supported the concept that relative properties among quantum systems, rather than the freelance properties of a quantum system, are the foremost elementary parts to construct quantum physics. This idea, combining with the stress that measuring of a quantum system could be a biface interaction method, ends up in a brand new framework to calculate the chance of AN outcome once activity a quantum system. during this framework, the foremost basic variable is that the relative chance amplitude. chance is calculated as summation of weights from the choice measuring configurations. **[4]**

**Discrete Phase Space, String-Like Phase Cells, and Relativistic Quantum Mechanics**

The distinct space illustration of quantum physics involving a characteristic length is investigated. the continual (1 + 1)-dimensional space is initial mentioned for the sake of simplicity. it’s discretized into numerable infinite variety of concentric circles. These circles, blessed with “unit area”, ar degenerate section cells resembling closed strings.

Next, Schrödinger differential equation for one particle within the 3 dimensional area underneath the influence of a static potential is studied within the distinct space illustration of quantum mechanics. The Schrödinger equation within the arena of distinct space may be a partial distinction equation. The energy eigenvalue downside for a 3 dimensional generator is precisely resolved. **[5]**

**Reference**

**[1]** Cooper, F., Khare, A. and Sukhatme, U., 1995. Supersymmetry and quantum mechanics. Physics Reports, 251(5-6), (Web Link)

**[2]** Bender, C.M., Brody, D.C. and Jones, H.F., 2002. Complex extension of quantum mechanics. Physical Review Letters, 89(27), (Web Link)

**[3]** Moyal, J.E., 1949, January. Quantum mechanics as a statistical theory. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 45, No. 1, pp. 99-124). Cambridge University Press. (Web Link)

**[4]** A Relational Formulation of Quantum Mechanics

Jianhao M. Yang

Scientific Reports volume 8, Article number: 13305 (2018) (Web Link)

**[5]** Das, A. and Haldar, S. (2018) “Discrete Phase Space, String-Like Phase Cells, and Relativistic Quantum Mechanics”, Physical Science International Journal, 17(2), (Web Link)