我的代码中的 `\center` 和 `\frac` 的大小

Question

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath}
\usepackage{fullpage}
\usepackage{secdot}
\usepackage{setspace}

\setlength\parindent{0pt}

\title{\vspace{-3em} Quantum Mechanics}
\author{David Ge}

\begin{document}
\maketitle

\section{Operators In Schroedinger Equation}
\onehalfspacing
In quantum mechanics, a question that concerns us all is whether light is particles or waves. Let's first assume light is wave, so it must satisfy the wave equation:

\vspace{-1em}
\begin{align}
    \left (\frac{\partial^2 \psi}{\partial x^2}+\frac{\partial^2 \psi}{\partial y^2}+\frac{\partial^2 \psi}{\partial z^2} \right)-
        \frac{1}{c^2}  \frac{\partial^2 \psi}{\partial t^2} & = 0 \\
    \nabla^2 \psi - \frac{1}{c^2}  \frac{\partial^2 \psi}{\partial t^2} & = 0
\end{align}

Naturally, we construct the solution of this wave equation:

\begin{center}
    $ \psi(x,y,z;t) = \psi_{0} \, e^{i \, [\, k (x+y+z)- \omega t + \phi_{0}\, ]}$
\end{center}

Here, we can set the initial condition $\phi_{0} = 0$, and we would like to explore the 1D case:
\begin{center}
    $ \psi(x,t) = \psi_{x,0} \, e^{i \, (kx- \omega t)}    $
\end{center}

Now, we lay down all the foundation, and we want to express the momentum and energy operators, by using the solution of the wave function. By observing the equation:

\begin{center}
$ E = \frac{p^2}{2m} + V  $
\end{center}
\end{document}

以上是我的代码，有两个问题想请教一下：

1. \begin{center} \end{center}每次输入公式时我都觉得很烦。我想知道有没有更简单的方法来实现这一点？

2. 看到了$ E = \frac{p^2}{2m} + V $。这里，分数的大小看起来很难看，有人能帮帮我吗，因为我不知道如何问这个问题。

Answer 1

代替

Naturally, we construct the solution of this wave equation:

\begin{center}
    $ \psi(x,y,z;t) = \psi_{0} \, e^{i \, [\, k (x+y+z)- \omega t + \phi_{0}\, ]}$
\end{center}

使用

Naturally, we construct the solution of this wave equation: 
\[ \psi(x,y,z;t) = \psi_{0} \, e^{i \, [\, k (x+y+z)- \omega t + \phi_{0}\, ]}\]

要内联输入分数并获得与显示相同的输出，请使用\dfrac{}{}。但如果要在里面插入分数，\[ \]则没有必要。只需使用\frac{}{}。