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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

    $\displaystyle 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

    $\displaystyle \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

    $\displaystyle \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

    $\displaystyle \langle i\vert j\rangle = \delta_{ij}
$

    for $ i,j = 1,N$ . Show that

    $\displaystyle \sum_{i=1,N} \vert i\rangle\langle i\vert = 1.
$

  3. Demonstrate that

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

    2. $\displaystyle (X^\dag )^\dag = X.
$

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

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

    5. $\displaystyle (\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

    $\displaystyle \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

    $\displaystyle \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:

    $\displaystyle \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

    $\displaystyle 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

    $\displaystyle \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

    $\displaystyle 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

    $\displaystyle \langle \xi'\vert\xi''\rangle = \delta(\xi'-\xi''),$

    be the corresponding eigenkets. Demonstrate that

    $\displaystyle \int d\xi'\, \vert\xi'\rangle\langle \xi'\vert= 1,
$

    where the integral is over the whole range of eigenvalues.

  19. Show that

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

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


next up previous
Next: Position and Momentum Up: Fundamental Concepts Previous: Continuous Spectra
Richard Fitzpatrick 2016-01-22