2.6 Hermitian Operators

Most operators in quantum mechanics are of a special kind called “Hermitian”. This section lists their most important properties.

An operator is called Hermitian when it can always be flipped over to the other side if it appears in a inner product:

$$\langle f \vert A g\rangle = \langle Af \vert g\rangle \mbox{ always iff $A$\ is Hermitian.}$$ (2.15)

That is the definition, but Hermitian operators have the following additional special properties:

• They always have real eigen­values, not involving ${\rm i}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sqrt{-1}$. (But the eigen­functions, or eigen­vectors if the operator is a matrix, might be complex.) Physical values such as position, momentum, and energy are ordinary real numbers since they are eigen­values of Hermitian operators {N.3}.
• Their eigen­functions can always be chosen so that they are normalized and mutually orthogonal, in other words, an ortho­normal set. This tends to simplify the various mathematics a lot.
• Their eigen­functions form a “complete” set. This means that any function can be written as some linear combin­ation of the eigen­functions. (There is a proof in derivation {D.8} for an important example. But see also {N.4}.) In practical terms, it means that you only need to look at the eigen­functions to completely understand what the operator does.

In the linear algebra of real matrices, Hermitian operators are simply symmetric matrices. A basic example is the inertia matrix of a solid body in Newtonian dynamics. The ortho­normal eigen­vectors of the inertia matrix give the directions of the principal axes of inertia of the body.

An ortho­normal complete set of eigen­vectors or eigen­functions is an example of a so-called “basis.” In general, a basis is a minimal set of vectors or functions that you can write all other vectors or functions in terms of. For example, the unit vectors ${\hat\imath}$, ${\hat\jmath}$, and ${\hat k}$ are a basis for normal three-di­mensional space. Every three-di­mensional vector can be written as a linear combin­ation of the three.

The following properties of inner products involving Hermitian operators are often needed, so they are listed here:

$$\mbox{If $A$\ is Hermitian: }\quad \langle g \vert A f\r... ...e^*, \quad \langle f \vert A f\rangle \mbox{ is real.} %$$ (2.16)

The first says that you can swap $f$ and $g$ if you take the complex conjugate. (It is simply a reflection of the fact that if you change the sides in an inner product, you turn it into its complex conjugate. Normally, that puts the operator at the other side, but for a Hermitian operator, it does not make a difference.) The second is important because ordinary real numbers typically occupy a special place in the grand scheme of things. (The fact that the inner product is real merely reflects the fact that if a number is equal to its complex conjugate, it must be real; if there was an ${\rm i}$ in it, the number would change by a complex conjugate.)

Key Points

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Hermitian operators can be flipped over to the other side in inner products.

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Hermitian operators have only real eigen­values.

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Hermitian operators have a complete set of ortho­normal eigen­functions (or eigen­vectors).

2.6 Review Questions

1. A matrix $A$ is defined to convert any vector ${\skew0\vec r}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $x{\hat\imath}+y{\hat\jmath}$ into ${\skew0\vec r}_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $2x{\hat\imath}+4y{\hat\jmath}$. Verify that ${\hat\imath}$ and ${\hat\jmath}$ are ortho­normal eigen­vectors of this matrix, with eigen­values 2, respectively 4.

   Solution herm-a

2. A matrix $A$ is defined to convert any vector ${\skew0\vec r}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $(x,y)$ into the vector ${\skew0\vec r}_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $(x+y,x+y)$. Verify that $(\cos 45^\circ ,\sin 45^\circ)$ and $(\cos 45^\circ ,-\sin 45^\circ)$ are ortho­normal eigen­vectors of this matrix, with eigen­values 2 respectively 0. Note: $\cos 45^\circ$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sin 45^\circ$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac 12\sqrt{2}$.

   Solution herm-b

3. Show that the operator $\widehat 2$ is a Hermitian operator, but $\widehat{\rm i}$ is not.

   Solution herm-c

4. Gener­alize the previous question, by showing that any complex constant $c$ comes out of the right hand side of an inner product unchanged, but out of the left hand side as its complex conjugate;
   
   $$\langle f\vert cg\rangle = c \langle f\vert g\rangle\qquad\langle c f\vert g\rangle = c^* \langle f\vert g\rangle .$$
   
   
   As a result, a number $c$ is only a Hermitian operator if it is real: if $c$ is complex, the two expressions above are not the same.

   Solution herm-d

5. Show that an operator such as ${\widehat{x}}^2$, corre­sponding to multi­plying by a real function, is an Hermitian operator.

   Solution herm-e

6. Show that the operator ${\rm d}$$\raisebox{.5pt}{$/$}$${\rm d}{x}$ is not a Hermitian operator, but ${\rm i}{\rm d}$$\raisebox{.5pt}{$/$}$${\rm d}{x}$ is, assuming that the functions on which they act vanish at the ends of the inter­val $a$ $\raisebox{-.3pt}{$\leqslant$}$ $x$ $\raisebox{-.3pt}{$\leqslant$}$ $b$ on which they are defined. (Less restrictively, it is only required that the functions are “periodic”; they must return to the same value at $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $b$ that they had at $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $a$.)

   Solution herm-f

7. Show that if $A$ is a Hermitian operator, then so is $A^2$. As a result, under the conditions of the previous question, $\vphantom0\raisebox{1.5pt}{$-$}$${\rm d}^2$$\raisebox{.5pt}{$/$}$${\rm d}{x}^2$ is a Hermitian operator too. (And so is just ${\rm d}^2$$\raisebox{.5pt}{$/$}$${\rm d}{x}^2$, of course, but $\vphantom0\raisebox{1.5pt}{$-$}$${\rm d}^2$$\raisebox{.5pt}{$/$}$${\rm d}{x}^2$ is the one with the positive eigen­values, the squares of the eigen­values of ${\rm i}{\rm d}$$\raisebox{.5pt}{$/$}$${\rm d}{x}$.)

   Solution herm-g

8. A complete set of ortho­normal eigen­functions of $\vphantom0\raisebox{1.5pt}{$-$}$${\rm d}^2$$\raisebox{.5pt}{$/$}$${\rm d}{x}^2$ on the inter­val 0 $\raisebox{-.3pt}{$\leqslant$}$ $x$ $\raisebox{-.3pt}{$\leqslant$}$ $\pi$ that are zero at the end points is the infinite set of functions
   
   $$\frac{\sin(x)}{\sqrt{\pi /2}}, \frac{\sin(2x)}{\sqrt{\pi /2}... ...in(3x)}{\sqrt{\pi /2}}, \frac{\sin(4x)}{\sqrt{\pi /2}}, \ldots$$
   
   

   Check that these functions are indeed zero at $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 and $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\pi$, that they are indeed ortho­normal, and that they are eigen­functions of $\vphantom0\raisebox{1.5pt}{$-$}$${\rm d}^2$$\raisebox{.5pt}{$/$}$${\rm d}{x}^2$ with the positive real eigen­values
   

   $$1, 4, 9, 16, \ldots$$
   
   

   Complete­ness is a much more difficult thing to prove, but they are. The complete­ness proof in the notes covers this case.

   Solution herm-h

9. A complete set of ortho­normal eigen­functions of the operator ${\rm i}{\rm d}$$\raisebox{.5pt}{$/$}$${\rm d}{x}$ that are periodic on the inter­val 0 $\raisebox{-.3pt}{$\leqslant$}$ $x$ $\raisebox{-.3pt}{$\leqslant$}$ $2\pi$ are the infinite set of functions
   
   $$\ldots , \frac{e^{-3{\rm i}x}}{\sqrt{2\pi}}, \frac{e^{-2{\rm... ... i}x}}{\sqrt{2\pi}}, \frac{e^{3{\rm i}x}}{\sqrt{2\pi}}, \ldots$$
   
   

   Check that these functions are indeed periodic, ortho­normal, and that they are eigen­functions of ${\rm i}{\rm d}$$\raisebox{.5pt}{$/$}$${\rm d}{x}$ with the real eigen­values
   

   $$\ldots , 3, 2, 1, 0 , -1, -2, -3, \ldots$$
   
   

   Complete­ness is a much more difficult thing to prove, but they are. The complete­ness proof in the notes covers this case.

   Solution herm-i