Height and depth of every node in Path Compression
I assume that the height of a node $\small x$, denoted as $\small h(x)$, is recursively defined as:
$$
\small
h(x) = \begin{cases}
0 &amp; \text{ if $x$ is a leaf} \\
1+\max\{h(y) \mid y\text{ is child of } x\} &amp; \text{ otherwise }
\end{cases}
$$
Let the value of the potential function before calling $\small \texttt{Find}(x)$ be $\small \Phi$ and the value after the call be $\small \bar{\Phi}$. Similarly we can define $\small h(x)$ and $\small \bar{h}(x)$ respectively. Denote as $\small P_x$ the path from $\small x$ to the root $\small r$ of the corresponding tree. By path compression, nodes in $\small P_x$ (except $\small r$) are all set to point to $\small r$. Therefore, generally $\small \bar{h}(y) \leq h(y)$ for $\small y \in P_x$, since $\small y$ may have one child removed. 

The amortized cost of $\small \texttt{Find}(x)$ is:
$$
\small
\mathcal{O}(|P_x|) + \bar{\Phi} - {\Phi} = \mathcal{O}(|P_x|) + \sum_{y \in P_x}\{\bar h(y) - {h}(y)\}
$$
where $\small |P_x|$ is the # of nodes in $\small P_x$ and $\small \mathcal{O}(|P_x|)$ is the actual cost of $\small \texttt{Find}(x)$. Now look at term $\small \bar h(y) - {h}(y)$, by the discussion above, its value is no greater than $\small 0$. Unfortunately, there are special cases where $\small \bar h(y) = h(y)$ for all $\small y \in P_x$. Therefore, what we can say about the amortized cost, using sum of heights as potential function, is the amortized cost is $\small \mathcal{O}(|P_x|)$, which is the same as the worse-case bound.