Another family of examples, now parametrized by an arbitrary function in a certain general class of functions:

Let $g\colon\mathbb R\to\mathbb R$ be any bounded continuously differentiable function with a bounded derivative such that $g(0)\ne0$ and $g'(u)^2\ge1$ if $|u|\le1$. Let us then show that a function $\phi$ of the form $f_c$ for some real $c>0$ will do, where $$f_c(x):=g(cx)$$ for all real $x$; that is, for some real $c>0$ we will have $$Eg(cZ)^2=c^2Eg'(cZ)^2, \tag{1}$$ where $Z\sim N(0,1)$.

Let $$D(c):=Eg(cZ)^2-c^2Eg'(cZ)^2.$$ Then, by dominated convergence, $D(c)$ is continuous in $c\ge0$. Also, $$D(0)=g(0)^2>0.$$

On the other hand, for $c\to\infty$ $$Eg'(cZ)^2\ge Eg'(cZ)^2 1_{|Z|\le1/c}\ge E1_{|Z|\le1/c}=P(|Z|\le1/c)\gtrsim\frac2{c\sqrt{2\pi}},$$ whereas $Eg(cZ)^2$ stays bounded (since $g$ is bounded). So, $D(c)\to-\infty$ as $c\to\infty$.

Therefore and because $D(c)$ is continuous in $c\ge0$, we see that $D(c)=0$ for some real $c>0$, which indeed yields (1).

Clearly, the condition $g'(u)^2\ge1$ if $|u|\le1$ here can be relaxed just to $g'(0)\ne0$.

Another family of examples, now parametrized by an arbitrary function in a certain general class of functions:

Let $g\colon\mathbb R\to\mathbb R$ be any bounded continuously differentiable function with a bounded derivative such that $g(0)\ne0$ and $g'(u)^2\ge1$ if $|u|\le1$. Let us then show that a function $\phi$ of the form $f_c$ for some real $c>0$ will do, where $$f_c(x):=g(cx)$$ for all real $x$; that is, for some real $c>0$ we will have $$Eg(cZ)^2=c^2Eg'(cZ)^2, \tag{1}$$ where $Z\sim N(0,1)$.

Let $$D(c):=Eg(cZ)^2-c^2Eg'(cZ)^2.$$ Then, by dominated convergence, $D(c)$ is continuous in $c\ge0$. Also, $$D(0)=g(0)^2>0.$$

On the other hand, for $c\to\infty$ $$Eg'(cZ)^2\ge Eg'(cZ)^2 1_{|Z|\le1/c}\ge E1_{|Z|\le1/c}=P(|Z|\le1/c)\gtrsim\frac2{c\sqrt{2\pi}},$$ whereas $Eg(cZ)^2$ stays bounded (since $g$ is bounded). So, $D(c)\to-\infty$ as $c\to\infty$.

Therefore and because $D(c)$ is continuous in $c\ge0$, we see that $D(c)=0$ for some real $c>0$, which indeed yields (1).

Clearly, the condition $g'(u)^2\ge1$ if $|u|\le1$ here can be relaxed just to $g'(0)\ne0$.

E.g., we may take $g(x)=1+\sin x$ for all real $x$, and then $D(c)=\frac{3}{2}-\frac{1}{2} e^{-2 c^2}-e^{-c^2} c^2 \cosh \left(c^2\right)$, and the equation $D(c)=0$ has two roots, $c\approx\pm1.73$.

Another family of examples, now parametrized by an arbitrary function in a certain general class of functions:

Let $g\colon\mathbb R\to\mathbb R$ be any bounded continuously differentiable function with a bounded derivative such that $g(0)\ne0$ and $g'(u)^2\ge1$ if $|u|\le1$. Let us then show that a function $\phi$ of the form $f_c$ for some real $c>0$ will do, where $$f_c(x):=g(cx)$$ for all real $x$; that is, for some real $c>0$ we will have $$Eg(cZ)^2=c^2Eg'(cZ)^2, \tag{1}$$ where $Z\sim N(0,1)$.

Let $$D(c):=Eg(cZ)^2-c^2Eg'(cZ)^2.$$ Then, by dominated convergence, $D(c)$ is continuous in $c\ge0$. Also, $$D(0)=g(0)^2>0.$$

On the other hand, for $c\to\infty$ $$Eg'(cZ)^2\ge Eg'(cZ)^2 1_{|Z|\le1/c}\ge E1_{|Z|\le1/c}=P(|Z|\le1/c)\gtrsim\frac2{c\sqrt{2\pi}},$$ whereas $Eg(cZ)^2$ stays bounded (since $g$ is bounded). So, $D(c)\to-\infty$ as $c\to\infty$.

Therefore and because $D(c)$ is continuous in $c\ge0$, we see that $D(c)=0$ for some real $c>0$, which indeed yields (1).

Another family of examples, now parametrized by an arbitrary function in a certain general class of functions:

Let $g\colon\mathbb R\to\mathbb R$ be any bounded continuously differentiable function with a bounded derivative such that $g(0)\ne0$ and $g'(u)^2\ge1$ if $|u|\le1$. Let us then show that a function $\phi$ of the form $f_c$ for some real $c>0$ will do, where $$f_c(x):=g(cx)$$ for all real $x$; that is, for some real $c>0$ we will have $$Eg(cZ)^2=c^2Eg'(cZ)^2, \tag{1}$$ where $Z\sim N(0,1)$.

Let $$D(c):=Eg(cZ)^2-c^2Eg'(cZ)^2.$$ Then, by dominated convergence, $D(c)$ is continuous in $c\ge0$. Also, $$D(0)=g(0)^2>0.$$

On the other hand, for $c\to\infty$ $$Eg'(cZ)^2\ge Eg'(cZ)^2 1_{|Z|\le1/c}\ge E1_{|Z|\le1/c}=P(|Z|\le1/c)\gtrsim\frac2{c\sqrt{2\pi}},$$ whereas $Eg(cZ)^2$ stays bounded (since $g$ is bounded). So, $D(c)\to-\infty$ as $c\to\infty$.

Therefore and because $D(c)$ is continuous in $c\ge0$, we see that $D(c)=0$ for some real $c>0$, which indeed yields (1).

Clearly, the condition $g'(u)^2\ge1$ if $|u|\le1$ here can be relaxed just to $g'(0)\ne0$.