quantum mechanics - Really how can an observable quantity be equal to an operator?

A wave-function can be written as

$$\Psi = Ae^{-i(Et - px)/\hbar}$$ where $E$ & $p$ are the energy & momentum of the particle. Now, differentiating $\Psi$ w.r.t. $x$ and $t$ respectively, we get $$\begin{align} \frac{\partial \Psi}{\partial x} &= \frac{i}{\hbar} p\Psi \\ \frac{\partial \Psi}{\partial t} &= -\frac{i}{\hbar}E\Psi \, . \end{align}$$ The above equations can be written in suggestive forms $$\begin{align} p \Psi &= \left( \frac{\hbar}{i} \frac{\partial}{\partial x} \right) \Psi \\ E \Psi &= \left( i \hbar \frac{\partial}{\partial t} \right) \Psi \, . \end{align}$$ Evidently the dynamical quantities momentum and energy are equivalent to the operators $$\begin{align} p &=\frac{\hbar}{i} \frac{\partial}{\partial x} \\ E &= i \hbar \frac{\partial}{\partial t} \, . \end{align}$$

quoted from Arthur Beiser's Concept of Modern Physics

How can observable quantities be equal to operators? You can't measure an "operator". Can anyone intuitively explain how momentum and energy are equal to operators?

Answer

Not equal, but equivalent, in the sense that they have the same effect on the wavefunction in question.

More precisely, the book is using a slight abuse of terminology. Taking momentum as an example, it's not really the case that the dynamical quantity of momentum is equivalent to the operator $\frac{\hbar}{i}\frac{\partial}{\partial x}$, because a number can't actually be equivalent to an operator. A number is a thing all on its own, whereas an operator is something that needs to be applied to something else to have any meaning. But the operation of multiplying the wavefunction by the wave's momentum (a number) is equivalent to the operation of taking the derivative and multiplying by $\frac{\hbar}{i}$. In mathematical language:

$$\frac{\hbar}{i}\frac{\partial}{\partial x}\psi = p\psi\tag{1}$$ whereas, strictly speaking, we can't say this: $$\frac{\hbar}{i}\frac{\partial}{\partial x} = p$$ At least, not in the way you're thinking about the notation.

What we normally do in quantum physics is always think about quantities as operators. That is, if you write just $p$ in an equation, it's implicitly understood that this should be applied to some wavefunction. For example, if you write

$$H = \frac{p^2}{2m}$$ what you really mean is $$H\psi = \frac{1}{2m}p (p \psi)$$