3.3 The Operators of Quantum Mechanics

The numerical quantities that the old Newtonian physics uses, (position, momentum, energy, ...), are just shadows of what really describes nature: operators. The operators described in this section are the key to quantum mechanics.

As the first example, while a mathematically precise value of the position $x$ of a particle never exists, instead there is an $x$-​position operator ${\widehat x}$. It turns the wave function $\Psi$ into $x\Psi$:

\begin{displaymath}
\Psi(x,y,z,t)
\quad
\begin{picture}(100,10)
\put(50,...
...2){\vector(1,0){100}}
\end{picture}
\quad
x \Psi(x,y,z,t)
\end{displaymath} (3.3)

The operators ${\widehat y}$ and ${\widehat z}$ are defined similarly as ${\widehat x}$.

Instead of a linear momentum $p_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $mu$, there is an $x$-​momentum operator

\begin{displaymath}
\fbox{$\displaystyle
{\widehat p}_x = \frac{\hbar}{{\rm i}} \frac{\partial}{\partial x}
$}
\end{displaymath} (3.4)

that turns $\Psi$ into its $x$-​derivative:
\begin{displaymath}
\Psi(x,y,z,t)
\quad
\begin{picture}(100,25)
\put(50,...
...
\end{picture}
\quad
\frac{\hbar}{{\rm i}}\Psi_x(x,y,z,t)
\end{displaymath} (3.5)

The constant $\hbar$ is called “Planck's constant.” (Or rather, it is Planck's original constant $h$ divided by $2\pi$.) If it would have been zero, all these troubles with quantum mechanics would not occur. The blobs would become points. Unfortunately, $\hbar$ is very small, but nonzero. It is about 10$\POW9,{-34}$ kg m$\POW9,{2}$/s.

The factor ${\rm i}$ in ${\widehat p}_x$ makes it a Hermitian operator (a proof of that is in derivation {D.9}). All operators reflecting macroscopic physical quantities are Hermitian.

The operators ${\widehat p}_y$ and ${\widehat p}_z$ are defined similarly as ${\widehat p}_x$:

\begin{displaymath}
\fbox{$\displaystyle
{\widehat p}_y = \frac{\hbar}{{\rm ...
...p}_z = \frac{\hbar}{{\rm i}} \frac{\partial}{\partial z}
$}
\end{displaymath} (3.6)

The kinetic energy operator ${\widehat T}$ is:

\begin{displaymath}
{\widehat T}= \frac{{\widehat p}_x^2 + {\widehat p}_y^2 + {\widehat p}_z^2}{2 m}
\end{displaymath} (3.7)

Its shadow is the Newtonian notion that the kinetic energy equals:

\begin{displaymath}
T = \frac12 m \left( u^2 + v^2 + w^2 \right)
= \frac{(mu)^2 + (mv)^2 + (mw)^2}{2m}
\end{displaymath}

This is an example of the “Newtonian analogy”: the relationships between the different operators in quantum mechanics are in general the same as those between the corresponding numerical values in Newtonian physics. But since the momentum operators are gradients, the actual kinetic energy operator is, from the momentum operators above:
\begin{displaymath}
{\widehat T}= - \frac{\hbar^2}{2m}
\left(
\frac{\parti...
...{\partial y^2} +
\frac{\partial^2}{\partial z^2}
\right).
\end{displaymath} (3.8)

Mathematicians call the set of second order derivative operators in the kinetic energy operator the Laplacian, and indicate it by $\nabla^2$:

\begin{displaymath}
\fbox{$\displaystyle
\nabla^2 \equiv
\frac{\partial^2}...
...ial^2}{\partial y^2} +
\frac{\partial^2}{\partial z^2}
$}
\end{displaymath} (3.9)

In those terms, the kinetic energy operator can be written more concisely as:
\begin{displaymath}
\fbox{$\displaystyle
{\widehat T}= - \frac{\hbar^2}{2m} \nabla^2
$}
\end{displaymath} (3.10)

Following the Newtonian analogy once more, the total energy operator, indicated by $H$, is the the sum of the kinetic energy operator above and the potential energy operator $V(x,y,z,t)$:

\begin{displaymath}
\fbox{$\displaystyle
H = -\frac{\hbar^2}{2m} \nabla^2 + V
$}
\end{displaymath} (3.11)

This total energy operator $H$ is called the Hamiltonian and it is very important. Its eigenvalues are indicated by $E$ (for energy), for example $E_1$, $E_2$, $E_3$, ...with:

\begin{displaymath}
H \psi_n = E_n \psi_n \quad\mbox{for } n = 1, 2, 3, ...
\end{displaymath} (3.12)

where $\psi_n$ is eigenfunction number $n$ of the Hamiltonian.

It is seen later that in many cases a more elaborate numbering of the eigenvalues and eigenvectors of the Hamiltonian is desirable instead of using a single counter $n$. For example, for the electron of the hydrogen atom, there is more than one eigenfunction for each different eigenvalue $E_n$, and additional counters $l$ and $m$ are used to distinguish them. It is usually best to solve the eigenvalue problem first and decide on how to number the solutions afterwards.

(It is also important to remember that in the literature, the Hamiltonian eigenvalue problem is commonly referred to as the time-independent Schrö­din­ger equation. However, this book prefers to reserve the term Schrö­din­ger equation for the unsteady evolution of the wave function.)


Key Points
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Physical quantities correspond to operators in quantum mechanics.

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Expressions for various important operators were given.

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Kinetic energy is in terms of the so-called Laplacian operator.

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The important total energy operator, (kinetic plus potential energy,) is called the Hamiltonian.