Two different analyses on derivation of PYTHAGORAS Theorem (569-479 BC): Discrete continuum states

TL;DR

This paper presents two novel derivations of the Pythagoras Theorem and applies them to analyze both discrete and continuum states in mathematical physics.

Contribution

It introduces two distinct derivations of the Pythagoras Theorem and explores their application to different states, bridging classical geometry and physical state analysis.

Findings

Successful derivation of Pythagoras Theorem using two methods

Application of derivations to discrete and continuum states

Insights into geometric principles in physical systems

Abstract

We propose two different derivations of Pythagoras Theorem and apply the same to study discrete and continuum states.

Tables

Fig.1. Any triangle

Fig.2. Wave function and continuum states

n	Energy	Energy (scattering model)
0	10.4624	1.0080
1	28.3872	1.0210
2	44.3121	1.0494
3	50.2370	1.0840

Fig.2. Wave function and continuum states

n	Energy	Energy (Scattering model)
0	9.8265	50.5113
1	25.2187	50.5121
2	29.8358	50.5454
3	36.3890	50.5486

Equations

$$a^2+b^2=c^2$$

$$\frac{a}{c}=\frac{1}{\cosh (x)}\to a=csech(x)$$

$$\frac{b}{c}=\tanh (x)$$

$$b=a\sinh (x)$$

$$a^2+b^2=a^2(\cosh (x){)}^2=[c^2(sech(x){)}^2][\cosh (x){]}^2=c^2$$

$$a^2+b^2=c^2$$

$$U(x)=\frac{a^2}{c^2}+\lambda \frac{b^2}{c^2}$$

$$U(x)=(sech(x){)}^2+\lambda (\tanh (x){)}^2$$

$$H=p^2+U(x)$$

$$a^2+b^2=c^2$$

$$\frac{b^2}{c^2}=\frac{[\cosh (x){]}^2}{\cosh (2x)}$$

$$\frac{a^2}{c^2}=\frac{[\sinh (x){]}^2}{\cosh (2x)}$$

$$\frac{b^2}{c^2}+\frac{a^2}{c^2}=\frac{{\cosh }^{2}x}{\cosh (2x)}+\frac{{\sinh }^{2}x}{\cosh (2x)}=1$$

$$a^2+b^2=c^2$$

$$V(x)=\frac{a^2}{c^2}+\lambda \frac{b^2}{c^2}$$

$$H=p^2+V(x)$$

$$H=p^2+\frac{[\cosh (x){]}^2}{\cosh (2x)}+\lambda \frac{[\sinh (x){]}^2}{\cosh (2x)}$$

$$H\Psi =E\Psi$$

$$|\Psi >=\sum A_m|m>$$

$$[p^2+x^2]|m>=(2m+1)|m>$$

Full text

Two different analysis on derivation of PYTHAGORAS Theorem (569 -479 B.C): Discrete continuum states

Biswanath Rath

Department of Physics, Maharaja Sriram Chandra Bhanj Deo University, Takatpur, Baripada -757003, Odisha, India. e.mail:biswanathrath10@gmail.com

$\bf{Abstract:}$ We prese3nt two different ways to derive PYTHOGORAS theorem without assuming right angle concept. In each case we generate a model quantum well and notice that energy levels are discrete but continuous in nature. We also present wave function nature in each case.

PACS: 02.30.Hq; 03.65.Db, o3.65.Ge

Key words : Pythagoras theorem, quantum well, discrete levels,continuum nature

Correspondence: biswanathrath10@gmail.com

**1.Introduction **

Nearly 4000 years ago, a famous Greek mathematician cum philosopher named ” Pythogoras” proposed that if sum of squares of two sides of a triangle becomes squares of the third side, then the triangle must be ”right angle triangle”. Mathematically

[TABLE]

where $a,b$ and $c$ are the sides of the triangle.

Derivation of this relation is still interesting. In fact practical application of this theorem to quantum physics is still lacking interest. In thiis paper we propose two different ways to derive the theorem, In each case a quantum well has been proposed to study nature of quantum states as follows

**2A. Model derivation using $\tanh(x)$ and $sech(x)$ **

Let us define the following

[TABLE]

[TABLE]

Hence it is to show that

[TABLE]

Hence using the relation

[TABLE]

[TABLE]

Hence the theorem is proved.

**2,B- Model potential and Spectral nature **

Here we construct a model potential as

[TABLE]

whose explict form is

[TABLE]

The corresponding Hamiltonian for $\lambda=100$ is

[TABLE]

In this case energy levels are discrete and reflect continuum behaviour as seen below.

The first four eigenstates energy are given below

**3A. Second derivation of Pythagoras theorem using $\sinh(x)$ and $\cosh(x)$ **

Here, we consider another model derication of Pythagoras theorem using

[TABLE]

Mathematically consider that

[TABLE]

and

[TABLE]

Now adding we get

[TABLE]

Hence

[TABLE]

**3B. Potential model and Bound states **

Here, we choose the potential as

[TABLE]

Here the corresponding Hamiltonian is written as

[TABLE]

in its explict form

[TABLE]

For $\lambda=100$ we present potential , wave function nature and duiscrete energy levels nature. below

The first four eigenstates energy are given below

**4. Method of calculation **

Here we solve the eigenvalue relation

[TABLE]

where

[TABLE]

Here $m>$ satisfies the eigenvalue relation

[TABLE]

**5.Conclusion ** We have presented two different ways to derive the PYTHAGORAS theorem without assuming the concept of right angle triangle.In thhis derivation, we use hyperbolic functions. Further model Hamiltonian reported here shows discrete energy levels on quantum well and inverted scattering model potential. In fact continuum levels are well demarcated in lower and higher energy levels. However in the case of well continuum energy lavels are seen in higher quantum states.

**Author’s contribution: **

B.Rath: formulation,computation,writing,finalization.

**Conflict of interest **

Author declares there is no conflict of interest.

**DATA AVAILABILITY **

No additional data is required . All the datas included in this paper are sufficient.

**Declaration **

Present paper is a modified version of arxiv paper.