Time-evolution operator The time-evolution operator $U(t,t_0)$ is defined as the operator which acts on the ket at time $t_0$ to produce the ket at some other time $t$: $ \ket{\Psi(t)}=\hat{U}(...

Prerequisites Like a photon, the momentum and energy of a quantum-like particle with wave-like properties and particle-like properties(i.e., duality) are summarized as follows. (Duality derived by...

Commutator For operators $\hat{A}$ and $\hat{B}$ acting on wavefunctions, the following operator is called the commutator of $\hat{A}$ and $\hat{B}$. $ \com{\hat{A}}{\hat{B}} \coloneqq \hat{A}...

Matrix representation of state vectors Any state vector can be represented by the sum of eigenvectors for a Hermitian operator $L$. $ \dps \ket{\psi}=\sum_n c_n\ket{\psi_n} $ Then, we can use...

Differentiable map Given an $m$-dimensional differentiable manifold $M$ and an $n$-dimensional differentiable manifold $N$, we can think of a function $f$ from $M$ to $N$. If the function from a lo...

Hermitian operator A Hermitian operator, or self-adjoint operator is an operator that is equal to its own hermitian conjugate. $ \hat{A}=\hat{A}^\dag $ Since operators with such properties ar...

Bras & kets A physical system is represented by a state vector $\ket{\Psi}$, an element of $L^2$ complex Hilbert space $\mathcal{H}$. (In fact, it doesn’t have to be $L^2$. See $C^*$ formalism...

Introduction Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. Classical physics, ...

Differentiable manifold Given a manifold $M$ and its atlas $\mathcal{A}$, let’s think of transition maps. If all transition maps are differentiable, $M$ is a differentiable manifold. Furthermore, ...

Introduction Topological manifold is one of topological spaces which is locally euclidean. Thus, we can think of coordinates which represents the neighborhood of a point of the topological manifol...