Euler's Identity and it's geometrical interpretation

The most beautiful equation in mathematics that combines five of the most important constants of nature: 0, 1, π, e and i, with the three fundamental operations: addition, multiplication and exponentiation.

Euler’s identity is an equality found in mathematics that has been compared to a Shakespearean sonnet and described as "the most beautiful equation." It is a special case of a foundational equation in complex arithmetic called Euler’s Formula, which the late great physicist Richard Feynman called in his lectures "our jewel" and "the most remarkable formula in mathematics."

The five constants are:

1. The 0( zero)

2. The number 1.

3. The number π, an irrational number (with unending digits) that is the ratio of the circumference of a circle to its diameter. It is approximately 3.14159…

4. The number e, also an irrational number. It is the base of natural logarithms that arises naturally through study of compound interest and calculus. The number e pervades math, appearing seemingly from nowhere in a vast number of important equations. It is approximately 2.71828….

5. The number i, defined as the square root of negative one: √(-1). The most fundamental of the imaginary numbers, so called because, in reality, no number can be multiplied by itself to produce a negative number (and, therefore, negative numbers have no real square roots). But in math, there are many situations where one is forced to take the square root of a negative. The letter i is therefore used as a sort of stand-in to mark places where this was done.

Prolific mathematician

Leonhard Euler was an 18th-century Swiss-born mathematician who developed many concepts that are integral to modern mathematics. He spent most of his career in St. Petersburg, Russia. He was one of the most prolific mathematicians of all time, according to the U.S. Naval Academy (USNA), with 886 papers and books published. Much of his output came during the last two decades of his life, when he was totally blind. There was so much work that the St. Petersburg Academy continued publishing his work posthumously for more than 30 years.

Euler's important contributions include Euler's Formula and Euler's Theorem, both of which can mean different things depending on the context. According to the USNA, in mechanics, there are "Euler angles (to specify the orientation of a rigid body), Euler's theorem (that every rotation has an axis), Euler's equations for motion of fluids, and the Euler-Lagrange equation (that comes from calculus of variations)."

Euler's identity and Euler's formula are deeply embedded in the mathematical framework of quantum physics. They are essential tools for expressing, manipulating, and interpreting complex-valued functions and operators that describe quantum systems.

Wave Functions and Schrödinger Equation:

1. Complex Wave Functions: Quantum mechanics often describes the state of a system using wave functions, which are generally complex-valued functions. Euler's formula is crucial in expressing these complex wave functions in terms of their real and imaginary parts.

Schrödinger Equation Solutions: Solutions to the Schrödinger equation, which governs the behavior of quantum systems, often involve complex exponentials.

2. Quantum Operators:

Translation and Rotation Operators: In quantum mechanics, operators representing physical quantities or transformations are often expressed using exponential functions of complex numbers. For example, the time evolution operator, which dictates how a quantum state evolves over time, is H is the Hamiltonian of the system. Euler's identity and formula are integral to understanding these operators.

Spin and Angular Momentum: The mathematics of spin and angular momentum in quantum mechanics relies heavily on complex exponentials.

3. Quantum Field Theory:

Path Integrals: In quantum field theory, the path integral formulation involves summing over all possible paths.

4. Fourier Transform:

Momentum Representation: The Fourier transform, which uses Euler's formula, is essential in quantum mechanics for transforming wave functions between position and momentum representations.

5. Quantum Harmonic Oscillator:

Energy Eigenstates: In the case of the quantum harmonic oscillator, the energy eigenstates can be expressed using Hermite polynomials multiplied by Gaussian functions. These solutions can also be represented in terms of complex exponentials, facilitating their manipulation and interpretation.