1.6 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:

$$\langle f \vert A g\rangle = \langle Af \vert g\rangle \mbox{ always iff $A$\ is Hermitian.}$$ (1.15)

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

• They always have real eigenvalues, not involving ${\rm i}=\sqrt{-1}$. (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 {A.7}.
• 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 note {A.5} for an important example. But see also {A.8}.) 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:

$$\mbox{If $A$\ is Hermitian: }\quad \langle g \vert A f\r... ...e^*, \quad \langle f \vert A f\rangle \mbox{ is real.} %$$ (1.16)

The first says that you can swap $f$ and $g$ if you take 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

$\diamond$

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

$\diamond$

Hermitian operators have only real eigenvalues.

$\diamond$

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

1.6 Review Questions

1

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

2

A matrix $A$ is defined to convert any vector ${\skew0\vec r}=(x,y)$ into the vector ${\skew0\vec r}_2=(x+y,x+y)$. 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: $\cos 45^\circ =\sin 45^\circ =\frac 12\sqrt{2}$ Answer

3

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

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. Answer

5

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

6

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

7

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

8

A complete set of orthonormal eigenfunctions of $-{\rm d}^2/{\rm d}x^2$ on the interval $0\le x\le\pi$ that are zero at the end points are the infinite set of functions

$$\frac{\sin(x)}{\sqrt{\pi /2}}, \frac{\sin(2x)}{\sqrt{\pi /2... ...n(3x)}{\sqrt{\pi /2}}, \frac{\sin(4x)}{\sqrt{\pi /2}}, \ldots$$

Check that these functions are indeed zero at $x=0$ and $x=\pi$, that they are indeed orthonormal, and that they are eigenfunctions of $-{\rm d}^2/{\rm d}x^2$ 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. Answer

9

A complete set of orthonormal eigenfunctions of the operator ${\rm i}{\rm d}/{\rm d}x$ that are periodic on the interval $0\le x\le 2\pi$ are the infinite set of functions

$$\ldots , \frac{e^{-3{\rm i}x}}{\sqrt{2\pi}}, \frac{e^{-2{\r... ...i}x}}{\sqrt{2\pi}}, \frac{e^{3{\rm i}x}}{\sqrt{2\pi}}, \ldots$$

Check that these functions are indeed periodic, orthonormal, and that they are eigenfunctions of ${\rm i}{\rm d}/{\rm d}x$ 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. Answer