Operators in Quantum Mechanics

In quantum mechanics (QM), operators represent physical processes such as the evolution of states over time. There are two most "famous" operators, momentum ($\hat{p}$) and position ($\hat{x}$). An operator acts on a function and produces another function.

$$ \begin{equation} \label{ope1} f(x) \hookrightarrow \hat{A} \longrightarrow g(x) . \end{equation} $$

It is in accordance with the fact that a function acts on a value in order to produce another value.

$$ \begin{equation} \label{ope2} x \hookrightarrow f(x) \longrightarrow y . \end{equation} $$

Below is a summary of the general properties of an operator.

So, why do we need operators in QM? The answer is that to retrieve information encoded within a Wavefunction (WF), we need operators that can act on the Wavefunction. Since quantum particles exist in several places simultaneously, WF provides us with a mathematical model of an isolated quantum system.

$$ \begin{equation} \label{ope5} P(x) = \Psi(x)^* \Psi (x) = |\Psi(x)|^2 \end{equation} $$

In eq. (\ref{ope5}), $P(x)$ is the probability distribution function. And calculating the modulus squared of the WF yields the probability that a particle will be found at a particular location. Note that it is a \textit{probability} that we receive when trying to determine a particle's position or momentum. According to QM, the position and momentum operators acting on a WF can be expressed in the following way.

$$ \begin{equation} \label{ope6} \hat{x}\Psi(x) = x\Psi(x) , \end{equation} $$

$$ \begin{equation} \label{ope7} \hat{p}\Psi(x) = -i\hbar\frac{\partial }{\partial x} \Psi(x) . \end{equation} $$

If we consider a WF like $\Psi = e^{i(kx-\omega t)}$, the position operator results in,

$$ \begin{equation} \label{ope8} \hat{x}\Psi(x) = x\Psi(x) , \end{equation} $$

and the momentum operator results in,

$$ \begin{equation} \label{ope9} \hat{p}\Psi(x) = -i\hbar\frac{\partial }{\partial x} \Psi(x) , \end{equation} $$

$$ \begin{equation} \label{ope10} \hat{p}\Psi(x) = -i\hbar\left(ike^{i(kx-\omega t)}\right) = \hbar k \Psi . \end{equation} $$

The eigenvalues of equations eq. (\ref{ope8}) and eq. (\ref{ope10}) are $x$ and $\hbar k$. It is important to note that $\hat{x}$ and $\hat{p}$ are *Hermitian*. In other words, these operators have real eigenvalues. This is understandable since positions and momentums cannot contain imaginary numbers. This can be seen in the 3rd property listed above, which states that the complex conjugate of an operator ($\hat{A}^{\dagger}$) should equal itself ($\hat{A}$). In addition, the fourth property implies that there is no commutation between the operators. The implication is that, in general, the order of operators should be strictly adhered to in a calculation.

As I mentioned earlier, the operators do not commute! So what is $\left[\hat{x},\hat{p}\right]\Psi$ in this case?

$$ \begin{equation} \label{ope11} \left[\hat{x},\hat{p}\right]\Psi = \hat{x}\hat{p}\Psi - \hat{p}\hat{x}\Psi \end{equation} $$

$$ \begin{equation} \label{ope12} \left[\hat{x},\hat{p}\right]\Psi = \hat{x}\left(-i\hbar\frac{\partial \Psi}{\partial x}\right) - \hat{p}\hat{x}\Psi \end{equation} $$

$$ \begin{equation} \label{ope13} \left[\hat{x},\hat{p}\right]\Psi = -\hat{x}i\hbar\left(ik\right)\Psi - \hat{p}\hat{x}\Psi \end{equation} $$

$$ \begin{equation} \label{ope14} \left[\hat{x},\hat{p}\right]\Psi = \hat{x}\hbar k\Psi - \hat{p}\hat{x}\Psi \end{equation} $$

$$ \begin{equation} \label{ope15} \left[\hat{x},\hat{p}\right]\Psi = x\hbar k\Psi - \hat{p}\hat{x}\Psi \end{equation} $$

$$ \begin{equation} \label{ope16} \left[\hat{x},\hat{p}\right]\Psi = x\hbar k\Psi - \hat{p}x\Psi \end{equation} $$

$$ \begin{equation} \label{ope17} \left[\hat{x},\hat{p}\right]\Psi = x\hbar k\Psi - \left(-i\hbar\frac{\partial (x\Psi)}{\partial x}\right) \end{equation} $$

$$ \begin{equation} \label{ope18} \left[\hat{x},\hat{p}\right]\Psi = x\hbar k\Psi - \left(-i\hbar(\Psi + xik \Psi)\right) \end{equation} $$

$$ \begin{equation} \label{ope19} \left[\hat{x},\hat{p}\right]\Psi = x\hbar k\Psi - \left(-i\hbar\Psi + x\hbar k \Psi\right) \end{equation} $$

$$ \begin{equation} \label{ope20} \left[\hat{x},\hat{p}\right]\Psi = x\hbar k\Psi + i\hbar\Psi - x\hbar k \Psi \end{equation} $$

$$ \begin{equation} \label{ope21} \left[\hat{x},\hat{p}\right]\Psi = i\hbar\Psi \end{equation} $$

Eq. (\ref{ope21}) indicates that $\hat{x}$ and $\hat{p}$ are conjugate operators which do not commute ($\left[\hat{x},\hat{p}\right] = i\hbar$). In other words, there is a limitation to the precision/accuracy with which these quantities can be measured. Thus the Heisenberg uncertainty principle $\Delta x \Delta p \geq \frac{\hbar}{2}$.