00 Complex Numbers
Complex numbers, and more generally higher-dimensional numbers like quaternions and octonions, are fascinating mathematical constructs that help us describe and understand multidimensional spaces and phenomena.
01 Observables
In quantum mechanics, an observable is a physical quantity that can be measured—such as position, momentum, or energy. Mathematically, an observable is represented by a Hermitian (self-adjoint) operator acting on...
02 Eigenstate
An eigenstate is a state that satisfies the equation
03 Hilbert Space
A Hilbert space is a complete vector space equipped with an inner product. It provides the mathematical setting for quantum mechanics, where states, observables, and dynamics are all formulated. Let’s...
04 Born Rule
The Born rule is a foundational principle in quantum mechanics that connects the abstract mathematical description of a quantum state with the probabilities of observable outcomes. Here’s a detailed explanation:...
05 Superposition
In quantum mechanics, a qubit is not simply a bit that is either 0 or 1. Instead, it exists in a superposition of both states. Its state is typically written...
06 Euler Formula
1. The Role of Euler’s Formula in Representing Complex Numbers
07 Angles and Phases
Let’s break it down:
08 Bloch Sphere
The state of a qubit is much more than a simple bit—it’s a quantum state that can exist as a superposition of two basis states. Geometrically, we represent a pure...
09 Unitary
In quantum computing, unitary transformations are indispensable because they ensure that the fundamental postulates of quantum mechanics—especially the preservation of probability—remain valid during the evolution of quantum states. Let’s explore...
10 Interference
Let’s break down quantum interference in a way that connects the math to physical intuition. In quantum mechanics, rather than directly adding probabilities (as in classical systems), we add complex...