Exercises

1. According to the Larmor formula of 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 $c$ the velocity of light in vacuum [49]. 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, $m_e$ the electron mass, $\hbar$ the reduced Planck constant, and
   $$\tau=\frac{a_0}{4\,\alpha^{\,4}\,c}.$$
   Here, $\alpha=e^{\,2}/(4\pi\,\epsilon_0\,\hbar\,c)$ is the fine structure constant. Deduce that the classical lifetime of a ground-state electron in a hydrogen atom is $\tau\simeq 1.6\times 10^{-11}\,{\rm s}$ .

2. Let the $\vert i\rangle$ , for $i=1,N$ , be a set of orthonormal kets that span an $N$ -dimensional ket space. By orthonormal, we mean that the kets are mutually orthogonal, and have unit norms, so that
   $$\langle i\vert j\rangle = \delta_{ij}$$
   for $i,j = 1,N$ . Show that
   $$\sum_{i=1,N} \vert i\rangle\langle i\vert = 1.$$

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

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

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

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

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

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

5. 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. [53]

6. Let $A$ be a general operator. Show that $A+A^\dag$ , ${\rm i}\,(A-A^\dag )$ , and $A\,A^\dag$ are Hermitian operators. [53]

7. 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 $\exp(-{\rm i}\,H)$ . [53]

8. Suppose that $A$ and $B$ are two commuting operators. Demonstrate that
   $$\exp(A)\,\exp(B) = \exp(A+B).$$

9. Let the $\vert\xi'\rangle$ be the normalized 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.

10. Let the $\vert\xi_i'\rangle$ , where $i=1,N$ , and $N>1$ , be a set of degenerate unnormalized 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.

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

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

13. Consider an Hermitian operator, $H$ , that has the property that $H^{\,4}=1$ . What are the eigenvalues of $H$ ? What are the eigenvalues if $H$ is not restricted to being Hermitian? [53]

14. An operator $U$ is said to be unitary if
    $$U\,U^\dag = U^\dag\,U = 1.$$
    Show that if $\langle A\vert A\rangle =1$ and $\vert B\rangle = U\,\vert A\rangle$ then $\langle B\vert B\rangle =1$ . [53]

15. Show that if $H$ is Hermitian then $\exp(\,{\rm i}\,H)$ is unitary. [53]

16. Show that if the $\vert u_i\rangle$ , for $i=1,N$ , form a complete orthonormal set, so that
    $$\langle u_i\vert u_j\rangle =\delta_{ij},$$
    then the $\vert v_i\rangle = U\,\vert u_i\rangle$ , for $i=1,N$ , where $U$ is unitary, are also orthonormal. [53]

17. The eigenstates of some operator $A$ acting in an $N$ -dimensional ket space are written
    $$A\,\vert i\rangle = a_i\,\vert i\rangle,$$
    for $i=1,N$ , where all of the $a_i$ are real. Suppose that the $\vert i\rangle$ are orthonormal, and span the ket space. Deduce that $A$ is Hermitian.

18. 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.

19. Show that

    $$\delta(-x) = \delta(x),$$

    $$\delta(a\,x) = \frac{1}{a}\,\delta(x),$$

    
    
    where $\delta(x)$ is a Dirac delta function, and $a$ a constant.