Exercises

1. According to classical physics, a non-relativistic electron whose instantaneous acceleration is of magnitude $a$ radiates electromagnetic energy at the rate
   $$P = \frac{e^2\,a^2}{6\pi\,\epsilon_0\,c^3},$$
   where $e$ is the magnitude of the electron charge, $\epsilon_0$ the permittivity of the vacuum, and $m_e$ the electron mass. Consider a classical electron in a circular orbit of radius $r$ around a proton. Demonstrate that the radiated energy would cause the orbital radius to decrease in time according to
   $$\frac{d}{dt}\left(\frac{r}{a_0}\right)^3 =- \frac{1}{\tau},$$
   where $a_0=4\pi\,\epsilon_0\,\hbar^2/(m_e\,e^2)$ is the Bohr radius, $\hbar$ the reduced Planck constant, and
   $$\tau=\frac{a_0}{4\,\alpha^4\,c}.$$
   Here, $c$ is the velocity of light in a vacuum, and $\alpha=e^2/(4\pi\,\epsilon_0\,\hbar\,c)$ the fine structure constant. Deduce that the classical lifetime of a hydrogen atom is $\tau\simeq 1.6\times 10^{-11}\,{\rm s}$ .

2. Demonstrate that
   $$\langle B\vert A \rangle = \langle A\vert B \rangle^\ast$$
   in a finite dimensional ket space.

3. Demonstrate that in a finite dimensional ket space:

   1. $$\langle B\vert\,X^{\dag }\,\vert A\rangle = \langle A\vert\,X\, \vert B\rangle^\ast.$$

   2. $$(X\, Y)^{\dag } = Y^{\dag }\, X^{\dag }.$$

   3. $$(X^\dag )^\dag = X.$$

   4. $$(\vert B\rangle\langle A\vert)^\dag = \vert A\rangle\langle B\vert.$$

   Here, $X$ , $Y$ are general operators.

4. If $A$ , $B$ are Hermitian operators then demonstrate that $A\,B$ is only Hermitian provided $A$ and $B$ commute. In addition, show that $(A+B)^n$ is Hermitian, where $n$ is a positive integer.

5. Let $A$ be a general operator. Show that $A+A^\dag$ , ${\rm i}\,(A-A^\dag )$ , and $A\,A^\dag$ are Hermitian operators.

6. Let $H$ be an Hermitian operator. Demonstrate that the Hermitian conjugate of the operator $\exp(\,{\rm i}\,H)\equiv\sum_{n=0,\infty} ({\rm i}\,H)^n/n!$ is $\exp(-{\rm i}\,H)$ .

7. Let the $\vert\xi'\rangle$ be the eigenkets of an observable $\xi$ , whose corresponding eigenvalues, $\xi'$ , are discrete. Demonstrate that
   $$\sum_{\xi'} \vert\xi'\rangle \langle \xi'\vert = 1,$$
   where the sum is over all eigenvalues, and $1$ denotes the unity operator.

8. Let the $\vert\xi_i'\rangle$ , where $i=1,N$ , and $N>1$ , be a set of degenerate eigenkets of some observable $\xi$ . Suppose that the $\vert\xi_i'\rangle$ are not mutually orthogonal. Demonstrate that a set of mutually orthogonal (but unnormalized) degenerate eigenkets, $\vert\xi''_i\rangle$ , for $i=1,N$ , can be constructed as follows:
   $$\vert\xi''_i\rangle = \vert\xi'_i\rangle - \sum_{j=1,i-1}\frac{\l... ...'\vert\xi_i'\rangle}{\langle \xi_j''\vert\xi_j''\rangle}\,\vert\xi_j''\rangle.$$
   This process is known as Gram-Schmidt orthogonalization.

9. Demonstrate that the expectation value of a Hermitian operator is a real number. Show that the expectation value of an anti-hermitian operator is an imaginary number.

10. Let $H$ be an Hermitian operator. Demonstrate that $\langle H^{\,2}\rangle \geq 0$ .

11. Consider an Hermitian operator, $H$ , that has the property that $H^{\,4}=1$ , where $1$ is the unity operator. What are the eigenvalues of $H$ ? What are the eigenvalues if $H$ is not restricted to being Hermitian?

12. Let $\xi$ be an observable whose eigenvalues, $\xi'$ , lie in a continuous range. Let the $\vert\xi'\rangle$ , where
    $$\langle \xi'\vert\xi''\rangle = \delta(\xi'-\xi''),$$
    be the corresponding eigenkets. Demonstrate that
    $$\int d\xi'\, \vert\xi'\rangle\langle \xi'\vert= 1,$$
    where the integral is over the whole range of eigenvalues, and $1$ denotes the unity operator.