right align in align environment

The following example uses aligned to align the second and right line to the right side.

\documentclass{article}
\usepackage{amsmath}
\newcommand*{\epsO}{\epsilon_0}

\begin{document}
\begin{align*}
\hat{H} \Psi(r,\theta) & = \left\lbrace\frac{-\hbar^2}{2m_er^2} \left[
\frac{\partial}{\partial r} \left(r^2\frac{\partial }{\partial r}\right)
+ \frac{1}{\sin\theta}\frac{\partial}{\partial \theta}
\left(\sin\theta\frac{\partial}{\partial\theta}\right)
+ \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\varphi^2}\right]
- \frac{e^2}{4\pi \epsO r} \right\rbrace \Psi(x,\theta) \\
&
  \begin{aligned}
  {}= \frac{-\hbar^2}{2m_er^2} \left[
  \frac{\partial}{\partial r} \left(r^2\frac{\partial }{\partial r}\right)\Psi(x,\theta)
  + \frac{1}{\sin\theta}\frac{\partial}{\partial \theta}
  \left(\sin\theta\frac{\partial}{\partial\theta}\right)\Psi(x,\theta)
  + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\varphi^2}\Psi(x,\theta)\right]\\
  - \frac{e^2}{4\pi \epsO r}\Psi(x,\theta)
  \end{aligned} \\
\end{align*}
\end{document}

Rather than forcing a tiny bit of the full expression of the second full line to dangle by itself on the far right of the third line, I recommend you take a bit more of that line and place it (together with a the "tiny bit") on the third line, only moderately indented. I also suggest you employ an \hphantom to indent the start of the second line a bit so that its start lines up vertically with the material in the preceding line.

\documentclass{article}
\newcommand\eps\varepsilon
\usepackage{amsmath}
\begin{document}
\begin{align*}
\hat{H} \Psi(r,\theta) 
&= \left\lbrace\frac{-\hbar^2}{2m_er^2} 
\left[ 
\frac{\partial}{\partial r} \left(r^2\frac{\partial }{\partial r}\right)
+ \frac{1}{\sin\theta}\frac{\partial}{\partial \theta}
\left(\sin\theta\frac{\partial}{\partial\theta}\right)
+ \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\varphi^2}\right] 
- \frac{e^2}{4\pi \eps_0 r} \right\rbrace \Psi(x,\theta) \\
&= \hphantom{\bigg\lbrace} \frac{-\hbar^2}{2m_er^2} 
\left[ 
\frac{\partial}{\partial r} \left(r^2\frac{\partial }{\partial r}\right)\Psi(x,\theta)
+ \frac{1}{\sin\theta}\frac{\partial}{\partial \theta}
\left(\sin\theta\frac{\partial}{\partial\theta}\right)\Psi(x,\theta)\right.\\
&\qquad\qquad+ \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\varphi^2}\Psi(x,\theta)\biggr] 
 - \frac{e^2}{4\pi \eps_0 r}\Psi(x,\theta) 
\end{align*}
\end{document}