Problems

1. Show that in the nonrelativistic limit, the free-particle Klein-Gordon equation becomes the free-particle Schrödinger equation.

2. The Lorentz-invariant step function $\theta(p)\equiv\theta(p_0)$ is defined by

   
   

   $$\theta(p) = \left\{ \begin{array}{c} 1\quad\textrm{for}\quad p_0>0 , \\ 0\quad\textrm{for}\quad p_0<0 . \end{array} \right.$$
   
   

   It is said that this step function is Lorentz invariant because it only distinguishes between the past and the future, which is a Lorentz invariant concept. Show that this step function is Lorentz invariant.

   Hint: I believe this is true only if $p$ is restricted to be a time-like vector.

3. Use explicit plane-wave solutions, $f^{(\pm)}_{\vec{p}}(x)$, to establish the following normalization and orthogonality relationships:
   $$\begin{eqnarray*} \int d^3x {f^{(\pm)}_{\vec{p}^{\:\prime}}}^*(x) i\stackrel{\le... ...ckrel{\leftrightarrow}\partial_0 f^{(\mp)}_{\vec{p}}(x) & = & 0. \end{eqnarray*}$$
   
   
   
   
   

4. Derive the conserved current for a scalar field interacting with an electromagnetic field with minimal coupling.

5. Show that

   
   

   $$Q = \int d^3x {\phi^{(\pm)}}^*(x) i\stackrel{\leftrightarrow}{\partial}_0 \phi^{(\pm)}(x) < 0$$
   
   

   is satisfied for a solution to the Klein-Gordon equation, $\phi^{(-)}(x)$, that is a superposition of negative energy plane-wave solutions.

6. Show that

   
   

   $$\rho = \frac{i\hbar}{2mc^2} \left( \phi^* \frac{\partial \phi}{\partial t} - \phi \frac{\partial \phi^*}{\partial t} \right)$$
   
   

   reduces to the proper nonrelativistic expression in the nonrelativistic limit.

7. The positive and negative energy two-component solutions of the Klein-Gordon equation in the Schrödinger form were define as

   
   

   $$\chi^{(\pm)}(\vec{p}) e^{\mp iEt + i\vec{p}\cdot\vec{x}} \... ...m\mp E\end{array} \right) e^{\mp iEt + i\vec{p}\cdot\vec{x}} .$$
   
   

   Show that the $\chi^{(\pm)}(\vec{p})$ are orthonormalized.

8. Show that in the nonrelativistic limit

   
   

   $$\chi^{(-)}(\vec{p}) \approx \left( \begin{array}{c} 0 \\ 1\end{array}\right) .$$
   
   

9. By completeness, any wavepacket can be expanded in terms of a linear combination of positive and negative energy solutions:

   
   

   $$\phi(\vec{x},t) = \int\frac{d^3p}{(2\pi)^3} \left[ a_{\vec{... ...-)}(t) \chi^{(-)}(\vec{p}) e^{-i\vec{p}\cdot\vec{x}} \right] .$$
   
   

   The $\chi^{(\pm)}(\vec{p})$ are defined as

   
   

   $$\chi^{(\pm)}(\vec{p}) e^{\mp iEt + i\vec{p}\cdot\vec{x}} \... ...m\mp E\end{array} \right) e^{\mp iEt + i\vec{p}\cdot\vec{x}} .$$
   
   

   and the $a_{\vec{p}}^{(\pm)}(t) = e^{\mp iEt} f^{(\pm)}(\vert\vec{p}\vert)$, where $f^{(\pm)}(\vert\vec{p}\vert)$ are general scalar functions of the magnitude of $\vec{p}$. Derive the normalization requirement for $\langle\phi\vert\phi\rangle = \pm 1$.

10. Invert

    
    

    $$\phi(\vec{x},t) = \int\frac{d^3p}{(2\pi)^3} \left[ a_{\vec{... ...-)}(t) \chi^{(-)}(\vec{p}) e^{-i\vec{p}\cdot\vec{x}} \right] .$$
    
    

    to obtain experessions for $a^{(+)}_{\vec{p}}(t)$ and $a^{(-)}_{\vec{p}}(t)$.

11. Derive the expectation value of the position operator for a wavepacket that is a mixture of positive and negative energy components: $\langle\phi\vert\hat{x}\vert\phi\rangle$. Show that your result contains a piece that represents the expect uniform velocity motion of the wavepacket and a piece that represents a rapid wiggling (Zitterbewegung) of the position of the particle about its central location, due to the interference of positive and negative energy components.

12. Solve the coulomb potential problem. In lectures we obtained

    
    

    $$\frac{1}{\rho^2}\frac{d}{d\rho} \left(\rho^2\frac{dR}{d\rho}... ... \frac{1}{4} - \frac{l(l+1)-\gamma^2}{\rho^2} \right) R = 0 ,$$
    
    

    where $\gamma\equiv\frac{Ze^2}{\hbar c}$, $\lambda\equiv\frac{2E\gamma}{\hbar c\alpha}$, $\alpha^2\equiv\frac{4(m^2c^4-E^2)}{\hbar^2c^2}$, and $\rho\equiv\alpha r$.

    Look for solutions that are finite at $\rho=0$ and $\infty$ and show that

    
    

    $$\lambda=n^{\prime}+s+1 ,$$
    
    

    where $n^{\prime}$ is $0$ or a positive integer and $s$ is the non-negative solution of

    
    

    $$s(s+1) = l(l+1)-\gamma^2 .$$
    
    

    Show the energy can be expanded in powers of $\gamma^2$ and to order $\gamma^4$ is

    
    

    $$E = mc^2\left[ 1 - \frac{\gamma^2}{2n^2} - \frac{\gamma^4}{2... ...left( \frac{n}{l+\frac{1}{2}} - \frac{3}{4} \right) \right] ,$$
    
    

    where $n=n^{\prime}+l+1$ is the total quantum number and can take on positive integer values. Identify the rest energy, the energy in the nonrelativistic theory and the fine-structure energy. Calculate the spread of the fine-structure levels for a given $n$. (Note: they are much larger than obeserved experimentally in the hydrogen spectrum.)

13. The homogeneous Green function can be written as

    
    

    $$\Delta^{\pm}(x-x^\prime) = \frac{i}{(2\pi)^3}\int d^4p\Theta(\pm p)\delta(p^2-m^2)C_{\pm} c^{ip\cdot(x-x^\prime)} .$$
    
    

    Show that without loss of generality $C_{\pm}=\pm 1$.

14. Show that
    1. $\Delta (\vec{x},0) = 0$,
    2. $(\partial_0\Delta)(\vec{x},0) = \delta(\vec{x})$,
    3. $(\Box +m^2)\Delta(x) = 0$,
    4. $\Delta^*(x) = \Delta(x)$,
    5. $\Delta(-x) = -\Delta(x)$,
    6. $\Delta(\vec{x},x_0) = -\Delta(x)$,
    7. $\Delta(-\vec{x},x_0) = \Delta(x)$,
    8. $(\partial_0\Delta_{\pm})(\vec{x},0) = \frac{1}{2}\delta(\vec{x})$.

15. Show that

    
    

    $$\int d^3y\Delta(x-y)\stackrel{\leftrightarrow}\partial_0\Delta(y-z) = \Delta(x-z) .$$
    
    

16. Prove that

    
    

    $$\partial_0\theta(x) = \delta(x_0)$$
    
    

    and thus show that

    
    

    $$(\Box + m^2)\triangle_C(x) = -\delta(x) .$$
    
    

17. Find the solution of the Klein-Gordon equation for the $\pi$-meson in a Coulomb potential and discusss the energy eigenvalues. The pion has a mass $m_{\pi} c^2=139.577$ MeV and spin 0.

18. Starting with a positive energy wavepacket only, show that there is a minimum width to the wavepacket.

19. Sarting with a positive energy wavepacket only, show that they can not be localized within a distance smaller than the Compton wavelength.

20. We have seen that the energy levels of a mesonic atom are given by

    
    

    $$E = m\left[ 1 + \frac{Z^2\alpha^2}{(n-l-\frac{1}{2}+\sqrt{(l+\frac{1}{2})^2 - Z^2\alpha^2})^2} \right]^{-\frac{1}{2}}$$
    
    
    1. Show that the ground state energy for any mesonic atom heavier than $Z=69$ is complex. Explain what this complex energy means.

    2. Mesonic atoms have been well studied at places like Los Alamos and it has been found that the ground states of atoms as heavy as lead ($Z=82$) or uranium ($Z=92$) are quite stable. How do you reconcile this fact with the result obtained above? Be as quantitative as you can.

21. A rapidly varying electric field can lead to the creation of particle-antiparticle pairs. Calculate to lowest order in $\alpha$ the probability per unit volume per unit time of producing such pairs in the presence of an external electric field.

    
    

    $$\vec{E}(t) = \hat{\epsilon}_x a \cos\omega t$$
    
    

    and show that

    
    

    $$prob. = VT\frac{\alpha a^2}{6} \left( 1 - \frac{4m^2}{\omega^2} \right)^{\frac{3}{2}} \theta(\omega - 2m)$$
    
    

    Suggestion: Use as an interaction potential the usual form

    
    

    $$H_{int} = e \int d^3x j_{\mu} A^\mu$$
    
    

    where

    
    

    $$j_{\mu} = \frac{i}{2m}(\phi^*\partial_{\mu}\phi - \partial\phi^*\phi)$$
    
    

    
    

    $$\vec{A}(t) = - \hat{\epsilon}_x \frac{a}{\omega} \sin\omega t$$
    
    

    Utilize normalized plane waves solutions of the Klein-Gordon equation

    
    

    $$\phi(x) = \frac{m}{E} \exp(i\vec{p}\cdot\vec{x} - iEt) \quad\textrm{with}\quad E = \sqrt{\vec{p}^2 + m^2}$$
    
    

    and simple first order perturbation theory

    
    

    $$amp = -i \int_{-T/2}^{T/2} \langle f\vert H_{int}(t)\vert \rangle dt$$
    
    

22. Consider a free charge Klein-Gordon particle of mass $m$ and charge $e$ immersed in a uniform magnetic field $B$ in the $z$-direction. Using the gauge $\vec{A} = 1/2(\vec{B}\times\vec{r})$ show that motion is quantized with energy

    
    

    $$E_n = \sqrt{m^2 + p_z^2 + eB(2n + 1)} \quad n= 0,1,2,\dots$$
    
    

23. Develop the wave equation for the nonrelativistic Hamiltonian developed from

    
    

    $$E^2 = \left(\frac{\vec{p}^{\ 2}}{2m}\right)^2$$
    
    

    and show that you get positive- and negative-energy solutions.

24. Show that the Schrödinger equation is not Lorentz invariant.