From f0095fcdaaca2158d3cca024fb05ee34817b7dfd Mon Sep 17 00:00:00 2001
From: Christian Carver <christianjcarv@gmail.com>
Date: Wed, 13 Dec 2023 12:10:46 -0700
Subject: [PATCH] lowercase titles and quick fixes

---
 docs/tutorials/quantum.ipynb | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/docs/tutorials/quantum.ipynb b/docs/tutorials/quantum.ipynb
index 6980d585..cc21d367 100644
--- a/docs/tutorials/quantum.ipynb
+++ b/docs/tutorials/quantum.ipynb
@@ -23,7 +23,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Quantum States\n",
+    "## Quantum states\n",
     "\n",
     "Quantum states are mathematical representations of the state of a quantum system. These quantum systems can be described using the schrodinger equation, which is a partial differential equation that describes how the quantum state of a physical system changes with time. The schrodinger equation is given by:\n",
     "\n",
@@ -62,7 +62,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Quantum Harmonic Oscillator\n",
+    "## Quantum harmonic oscillator\n",
     "\n",
     "The quantum harmonic oscillator is a quantum system that is described by the schrodinger equation. The Hamiltonian operator for the quantum harmonic oscillator is given by:\n",
     "\n",
@@ -98,7 +98,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Gaussian States\n",
+    "## Gaussian states\n",
     "\n",
     "To understand how simphony simulates quantum state evolution, let us consider a bosonic system with mode operators $\\hat{a}_k$ and $\\hat{a}_k^{\\dagger}$ where $k=1,...,n$ that satisfy the commutation relation $[\\hat{a}_k, \\hat{a}_l^{\\dagger}] = \\delta_{kl}$. The quadrature operators $\\hat{x}$ and $\\hat{p}$ are defined for the $k$-th mode as follows:\n",
     "\n",
@@ -332,7 +332,7 @@
    "source": [
     "```{note} \n",
     "\n",
-    "Simphony adopts the convention of $\\hbar=1/2$ such that $[\\hat{x}, \\hat{p}] = i\\hbar = i/4$ where $\\hat{x}$ is the position operator, $\\hat{p}$ is the momentum operator, and $\\hbar$ is the reduced Planck constant. Which corresponds to the uncertainty of the vacuum state $\\langle (\\Delta \\hat{X})^2\\rangle=\\frac{1}{4}$ This convention is used in the definition of the covariance matrix and used in {cite:p}`brask2021gaussian`.\n",
+    "Simphony adopts the convention of $\\hbar=1/2$ such that $[\\hat{x}, \\hat{p}] = i\\hbar = i/4$ where $\\hat{x}$ is the position operator, $\\hat{p}$ is the momentum operator, and $\\hbar$ is the reduced Planck constant. Which corresponds to the uncertainty of the vacuum state $\\langle (\\Delta \\hat{X})^2\\rangle=\\frac{1}{4}$ This convention is used in the definition of the covariance matrix and used in {cite:p}`gerry_knight_2004`.\n",
     "```"
    ]
   },