Update README.md

README.md

@@ -65,7 +65,7 @@ There are other examples in the examples folder: [Speed Tests: Numba & Cython](h
 
 ## 🌊 The Wavefunction
 
-The wavefunction, $\Psi(x,t)$, is a fundamental concept in quantum mechanics that describes the quantum state of a particle or system. Its absolute square, $|\Psi(x,t)|^2$, represents the probability density of finding the particle at position $\mathbf{x}$ and time $\mathbf{t}$. Due to the normalization property: $\int_{-\infty}^{\infty} |\Psi(x,t)|^2 dx = 1$ it's guaranteed that, for a given time $\mathbf{t}$, the total probability of finding the particle somewhere in space is unity [[5](#-ref)].
+The wavefunction, $\Psi(x,t)$, is a fundamental concept in quantum mechanics that describes the quantum state of a particle or system. Its absolute square, $|\Psi(x,t)|^2$, represents the probability density of finding the particle at position $\mathbf{x}$ and time $\mathbf{t}$. Due to the normalization property: $\int_{-\infty}^{\infty} |\Psi(x,t)|^2 dx = 1$ it's guaranteed that for a given time $\mathbf{t}$, the total probability of finding the particle somewhere in space is unity [[5](#-ref)].
 
 ###  Schrödinger Equation
 
@@ -79,7 +79,7 @@ where $\mathbf{\hbar}$ is the reduced Planck constant, $\mathbf{m}$ is the mass
 
 $$ f(t) = Ce^{-iEt/\hbar} \quad \mathbf{(2)}$$
 
-where $\mathbf{C}$ may be taken to be an arbitrary complex constant and $\mathbf{E}$, the system separation constant, can be interpreted as the energy of the system.Substituting into the wavefunction we have [[5](#-ref)]:
+where $\mathbf{C}$ may be considered an arbitrary complex constant and $\mathbf{E}$, the system separation constant can be interpreted as the system's energy. Substituting into the wavefunction we have [[5](#-ref)]:
 
 $$ \Psi(x,t) = Ce^{-iEt/\hbar}\psi(x) \quad \mathbf{(3)}$$
 
@@ -95,10 +95,10 @@ $$
 By solving equation **(4)**, we obtain a family of energy eigenfunctions defined as follows [[5](#-ref)]:
 
 $$
-\psi_n(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \frac{1}{\sqrt{2^n n!}} H_n\left(\sqrt{\frac{m\omega}{\hbar}}x\right) e^{-m\omega x^2/2\hbar} , \quad  n \in \mathbb{N_{0}} \quad \mathbf{(3)}
+\psi_n(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \frac{1}{\sqrt{2^n n!}} H_n\left(\sqrt{\frac{m\omega}{\hbar}}x\right) e^{-m\omega x^2/2\hbar} , \quad  n \in \mathbb{N}_{0} \quad \mathbf{(3)}
 $$
 
-where $\mathbf{n}$ represents a non-negative integer corresponding to the different energy states of the system, with energies given by $E_n = \big(n + \frac{1}{2}\big)\hbar \omega$. The term \( \mathbf{H_n} \) denotes the Hermite polynomial of degree $\mathbf{n}$; thus, for each energy state $\mathbf{n}$, there is an associated Hermite polynomial of degree $\mathbf{n}$ within its eigenfunction. The energy eigenfunction for an energy state $\mathbf{n}$ is understood as the wavefunction for an energy state $\mathbf{n}$ of a Quantum Harmonic Oscillator. From this definition, we can then represent the wave function $\Psi(x,t)$ as a series expansion of its family of energy eigenfunctions $\{\psi_{n}(x)\}$ [[5](#-ref)]:
+where $\mathbf{n}$ represents a non-negative integer corresponding to the different energy states of the system, with energies given by $E_n = \big(n + \frac{1}{2}\big)\hbar \omega$. The term \( \mathbf{H_n} \) denotes the Hermite polynomial of degree $\mathbf{n}$; thus, for each energy state $\mathbf{n}$, there is an associated Hermite polynomial of degree $\mathbf{n}$ within its eigenfunction. The energy eigenfunction for an energy state $\mathbf{n}$ is the wavefunction for an energy state $\mathbf{n}$ of a Quantum Harmonic Oscillator. From this definition, we can then represent the wave function $\Psi(x,t)$ as a series expansion of its family of energy eigenfunctions $\{\psi_{n}(x)\}$ [[5](#-ref)]:
 
 $$
 \Psi(x,t) = \sum_{n=0}^{\infty} c_{n}\psi_{n}(x)e^{-\mathbf{i}E_{n}t/\hbar} \quad \mathbf{(4)}
@@ -110,7 +110,7 @@ In this equation, $\mathbf{c_{n}}$ are complex constants that determine the cont
 
 In essence, Mr Mustard's strategy is to use the [Renormalized Hermite Polynomial](https://mrmustard.readthedocs.io/en/stable/code/api/mrmustard.math.hermite_renormalized.html) [[3, 4](#-ref)] for the computation of the wavefunction of a quantum harmonic oscillator. Below, we show the recurrence for calculating the Renormalized Hermite Polynomial, as well as the method for calculating it using the traditional Hermite polynomial:
 
-$$H_{n+1}^{\; re}(x) = \displaystyle\frac{2}{\sqrt{n+1}}\bigg[xH_{n}^{\; re}(x) - H_{n-1}^{\; re}(x)\sqrt{n-1}\bigg] \quad \mathbf{(4)} $$ 
+$$H_{n+1}^{re}(x) = \displaystyle\frac{2}{\sqrt{n+1}}\bigg[xH_{n}^{\; re}(x) - H_{n-1}^{\; re}(x)\sqrt{n-1}\bigg] \quad \mathbf{(4)} $$ 
 
 $$H_{n}^{re}(x) = \displaystyle\frac{H_{n}(x)}{\sqrt{n!}} \quad \mathbf{(5)} $$