A Comprehensive Guide to The Mathematics of Quantum Mechanics

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The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which are a kind of linear space.

We will explore the fundamental mathematical concepts and properties needed to understand Quantum Mechanics. These concepts will be imperative in understanding the inner working of Quantum Finance.

Table of Contents

1. Vector Space
   - 1.1 Vector Addition
   - 1.2 Scalar Multiplication
2. Complex Numbers
   - 2.1 Definition of an Imaginary Number
   - 2.2 Definition of a Complex Number
   - 2.3 Euler’s Formula for Complex Numbers
3. Complex Conjugate
   - 3.1 Complex Conjugate Addition
   - 3.2 Complex Conjugate Multiplication
4. Hilbert Space
   - 4.1 Properties of Hilbert Spaces
   - 4.2 Types of Hilbert Spaces
5. Dirac Notation
   - 5.1 First Postulate of Quantum Mechanics
   - 5.2 Notation
   - 5.3 Properties of Dirac Notation
   - 5.4 Braket Properties
6. Operators
   - 6.1 Vector Space with Finite Dimensions
   - 6.2 Function Space
   - 6.3 Properties of Operators
7. Types of Operators
   - 7.1 Inverse Operators
   - 7.2…