Problems

1. Suppose the Dirac wavefunctions are normalized in the same way as Klein-Gordon wavefunctions:

   
   

   $$i\int d^3x \overline{\psi}_\alpha(x) \stackrel{\leftrightarrow}{\partial}_0 \psi_\beta(x) = \delta_{\alpha\beta} .$$
   
   

   Use the Dirac equation to show that these wavefunctions differ from the ones normalized via

   
   

   $$\int d^3x \psi^\dagger_\alpha(x) \psi_\beta(x) = \delta_{\alpha\beta}$$
   
   

   by a factor of $\sqrt{2m}$.

2. Show that the current density for a free-particle wavefunction agrees with the corresponding nonrelativistic expression in the proper limit.

3. Re-derive current conservation using the Lorentz invariant form of the Dirac equation.

4. Show that

   
   

   $$S^{-1} = \gamma_0 S^\dagger \gamma_0$$
   
   

   for a general Lorentz transformation, where $S$ is a $4\times 4$ matrix which is a function of the parameters of the Lorentz transformation and operates upon the four components of the column vector of the wavefunction satisfying the Dirac equation.

5. 1. Suppose $\psi(x)$ is the Dirac wavefunction in a frame $\mathcal{O}$. What is the wavefunction $\psi^\prime(x^\prime)$ in a frame $\mathcal{O^\prime}$ which is obtained from $\mathcal{O}$ by a rotation about the $x$-axis through $60^\circ$?
   2. If a spin one-half particle at rest has its spin in the $z$-direction, what is the probability that its spin will be observed to be in a direction $60^\circ$ from the $z$-axis.
   3. Consider a Lorentz transformation to a frame moving in the $+z$-direction with speed $v$. Obtain the transformation matrix $S$.
   4. Use the results of part (c) to get the wavefunction of a spin one-half particle whose spin is in the $+z$-direction and whose velocity is $v$ in the $-z$-direction. Assume the wavefunction in the reset frame is $u=(1,0,0,0)$ and apply the boost of part (c).
   5. Obtain the transformation matrix for the boost of part (c) followed by the rotation of part (a).
   6. In this new frame in which the particle has negative helicity and a velocity $60^\circ$ from the $z$-axis, calculate the probability that it will be observed to have its spin in the new $+z$-direction.

6. Show that $\gamma_5$ commutes with $S$ for proper Lorentz transformations, $a$, but anti-commutes with $S$ for improper Lorentz transformations, $a$, and hence

   
   

   $$S(a)\gamma_5 = \gamma_5S(a)\textrm{det}\vert a\vert .$$
   
   

7. Show how the bilinears $\overline{\psi}(x)\Gamma^n\psi(x)$ transform under a general Lorentz transformation for $\Gamma^n=1, \gamma_\mu, \sigma_{\mu\nu}, \gamma_5$, and $\gamma_5\gamma_\mu$.

8. Given a free-particle spinor $u(p)$, construct $u(p+q)$ for $q_\mu\rightarrow 0$ and $p\cdot q\rightarrow 0$, in terms of $u(p)$ by means of a Lorentz transformation.

9. Prove the completeness relation

   
   

   $$\sum_{r=1}^4 w_\alpha^r(\epsilon^r\vec{p}) w_\beta^{r\dagger}(\epsilon^r\vec{p}) = \frac{E}{m} \delta_{\alpha\beta} .$$
   
   

10. Derive the following completeness relations

    
    

    $$\sum_{\pm s} u_\alpha(p,s)\overline{u}_\beta(p,s) = [\Lambda_+(p)]_{\alpha\beta} ,$$
    
    

    
    

    $$\sum_{\pm s} v_\alpha(p,s)\overline{v}_\beta(p,s) = -[\Lambda_-(p)]_{\alpha\beta} .$$
    
    

11. The operators

    
    

    $$P_L = \frac{1}{2} (1 + \gamma_5) \quad P_R = \frac{1}{2} (1 - \gamma_5)$$
    
    

    are projection operators which are said to identify states of definite chirality (handedness).

    1. Show that $P_L$, $P_R$ are legitimate projection operators in that

       
       

       $$P_L^2 = P_L, \quad P_R^2 = P_R, \quad P_LP_R = P_RP_L = 0 .$$
       
       

    2. Demonstrate that in the limit of high energy $E/m\gg 1$ - or equivalently in the massless limit - that the Dirac spinors for positive helicity (right-handed) and negative helicity (left-handed) states of momentum $\vec{p}$ are given by

       
       

       $$u_{\pm}(\vec{p}) = \sqrt{\frac{1}{2}} \left(\begin{array}{r} \chi_{\pm\hat{p}} \\ \pm\chi_{\pm\hat{p}} \end{array}\right)$$
       
       

       where $\chi_{\pm\hat{p}}$ are spinors such that

       
       

       $$\vec{\sigma}\cdot\hat{p} \chi_{\pm\hat{p}} = \pm\chi_{\pm\hat{p}} .$$
       
       

       Note: In order to conveniently deal with massless particles, it is important to use the normalization $u^\dagger(p)u(p)=1$. The appropriate Dirac spinors can then be found by multiplying the usual forms by the factor $\sqrt{m/E}$. Demonstrate this.

    3. Show that

       
       

       $$P_Lu_-(p) = u_-(p), \quad P_Ru_+(p) = u_+(p), \quad P_Lu_+(p) = P_Ru_-(p) = 0$$
       
       

       so that the chirality operator is equivalent to the helicity operator in this limit.

12. Examine in detail the influence of the transformation $\psi_C = C\overline{\psi}^T = i\gamma^2\psi^*$ on the eigenfunctions of an electron at rest with negative energy.

13. Find out how $\overline{\psi}_\alpha\gamma_\mu\gamma_\nu\psi_\beta$, $\overline{\psi}_\alpha\gamma_\mu\gamma_5\psi_\beta$, and $\overline{\psi}_\alpha\gamma_5\psi_\beta$ and $\overline{\psi}\sigma^{\mu\nu}\gamma^5\psi$ transform under time reversal.

14. If $\psi_1$ and $\psi_2$ are two arbitrary solutions of the free Dirac equation, prove the Gordon decomposition

    
    

    $$c\overline{\psi}_2\gamma^\mu\psi_1 = \frac{1}{2m} \left[\ove... ...c{i}{2m} \hat{p}_\nu (\overline{\psi}_2\sigma^{\mu\nu}\psi_1)$$
    
    

    It expresses the Dirac current as the sum of a convection current similar to the nonrelativistic one, and a spin current.

15. Calculate the current

    
    

    $$j^k = c\int d^3x \psi^\dagger(\vec{x},t) \alpha_k \psi(\vec{... ... \int d^3x \overline{\psi}(\vec{x},t) \gamma^k \psi(\vec{x},t)$$
    
    

    for the general wave packet which contains both positive and negative energy plane waves.

16. At time $t=0$ the following wave packet with Gaussian density distribution is defined as

    
    

    $$\psi^\prime(\vec{x},0,s) = \frac{1}{(\pi d^2)^{3/4}} e^{-\vert x\vert^2/2d^2} w^1(0) .$$
    
    

    Determine the wave packet at time $t$ developed from the above. Consider the intensity of the negative energy solutions in the wave packet. What does one learn in general about the applicability of the one-particle interpretation of the Dirac equation?

17. Solve the Dirac equation for an attractive square well potential of depth $V_0$ and radius $a$, after determining the continuity conditions at $r=a$. Obtain an explicit expression for the minimum $V_0$ with given $a$ that just binds a particle of mass $m$.

18. The Dirac equation describing the interaction of a proton or neutron with an applied external electromagnetic field has an additional term

    
    

    $$\left( i\not{\nabla} -Q_i\not{\!\!A}+ \frac{\kappa_i\vert e\vert}{4m} \sigma_{\mu\nu} F^{\mu\nu} - m\right) \psi(x) = 0$$
    
    

    involving the so-called anomalous magnetic moment. (For the proton, of course, $Q_i=\vert e\vert$ and for the neutron $Q_i=0$.)

    1. Verify that the choice

       
       

       $$\kappa_p = 1.79 \quad\textrm{and}\quad \kappa_n = -1.91$$
       
       

       corresponds to the observed magnetic moments of these particles, and

    2. Show that the additional interaction disturbs neither the Lorentz covariance of the equation nor the hermiticity of the Hamiltonian.

19. Suppose that the electron had a static electric dipole moment $d$ analogous to its magnetic moment.
    1. Show that this could be accommodated by modifying the Dirac equation to become

       
       

       $$(i\not{\nabla} - e\not{\!\!A}- i\frac{ed}{4m} \sigma_{\mu\nu} \gamma_5 F^{\mu\nu}- m) \psi(x) = 0$$
       
       

    2. Demonstrate that this equation is covariant but not invariant under a parity transformation.

20. Consider a positive energy spin-1/2 particle at rest. Suppose that at $t = 0$ we apply an external (classical) vector potential

    
    

    $$\vec{A} = -\hat{e}_x \frac{a}{\omega} \sin\omega t$$
    
    

    which corresponds to an electric field of the form

    
    

    $$\vec{E}= \hat{e}_x a \cos\omega t$$
    
    

    Show that for $t > 0$ there exists a finite probability of finding the particle in a negative energy state if such negative energy states are assumed to be originally empty. In particular, work out quantitatively the two cases: $\omega \ll 2m$ and $\omega \approx 2m$ and comment.

21. As we saw in the previous section, a rapidly varying electric field can lead to the creation of particle-antiparticle pairs. Calculate to lowest order the probability per unit volume per unit time of producing fermion pairs in the presence of an external electric field

    
    

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

    and show that

    
    

    $$prob = VT \frac{e^2a^2}{24\pi} \left( 1 - \frac{4m^2}{\omega... ...right)^{\frac{1}{2}} \left( 1 + \frac{2m^2}{\omega^2} \right)$$
    
    

    Suggestion: Utilize normalized plane wave solutions of the Dirac equation

    
    

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

    and simple first order perturbation theory

    
    

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

    with $H_{int} = e\int d^3x j_\mu A^\mu$ as in the Klein-Gordon case.

22. Find the energy levels of a Dirac particle in a one-dimensional box of depth $V_0$ and width $a$.

23. In order that $\mathcal{T}$ be a symmetry operation of the Dirac theory, the rules of interpretation of the wave function $\psi^\prime(t^\prime)$ must be the same as those of $\psi(t)$. This means that observables composed of forms bilinear in $\psi^\prime$ and $\psi^{\prime^\dagger}$ must have the same interpretation (within a sign, appropriate to the observable) as those of $\psi$.
    1. Verify that this is so for the current:
       
       $$j^\prime_\mu(x^\prime) = j^\mu(x)$$
       
       
       and also
       
       $$\langle\vec{r}\rangle^\prime = \langle\vec{r}\rangle \quad \langle\vec{p}\rangle^\prime = -\langle\vec{p}\rangle$$
       
       

    2. Repeat these calculations for the charge-conjugation transformation $\mathcal{C}$. In particular, show
       
       $$\overline{\psi}_C(x)\gamma_\mu\psi_C(x) = +\overline{\psi}(x)\gamma_\mu\psi(x)$$
       
       
       and interpret using the hole theory.

24. Obtain explicit representations for the matrix elements of the operators $\mathcal{O}_i = \gamma^\mu, \gamma^\mu\gamma^5$, $\gamma^5$, $\sigma^{\mu\nu}$ between spinors $u(\vec{p},s)$ and $u(\vec{q},r)$. Analyse in detail the case that $\vec{p}=(E,0,0,p)$ and $\vec{q}=(E,0,0,-p)$. That is, computer $\overline{u}(\vec{p},s)\mathcal{O}_i u(\vec{q},r)$.

25. In the Majorana representation all the $\gamma^\mu$ are pure imaginary. Find an explicit form of the Majorana representation.

26. Find the exact energy eigenvalues and eigenfunctions for an electron in a uniform magnetic field.

27. Compute
    1. $\gamma_\mu\gamma^\mu$, $\gamma_\mu\gamma_\alpha\gamma^\mu$, $\gamma_\mu\gamma_\alpha\gamma_\beta\gamma^\mu$
    2. Simplify $\not{\;\!\!\!p}\not{\;\!\!\!p}$, $\not{\;\!\!\!p}\gamma^\mu\not{\;\!\!\!p}$, $\not{\;\!\!\!p}\gamma^\mu\gamma^\nu\not{\;\!\!\!p}$.

28. For a Dirac electron of mass $m$ in an attractive electrostatic potential

    
    

    $$V(z) = \left\{ \begin{array}{cl} 0 & z<0, z>a \\ -V_0 & 0<z<a \end{array} \right.$$
    
    
    1. Find the energy levels.
    2. Solve the problem of scattering of such an electron with momentum $\vec{p}$ off this potential.

29. Show that the Dirac equation can not be invariant under spatial rotations if $\alpha_i$ are numbers and $\psi$ is a scaler.

30. Show that the orbital angular momentum operator does not commute with the Dirac Hamiltonian.

31. Prove that

    
    

    $$\left[ \alpha_i, \alpha_j \right] = 2i \epsilon_{ijk} \Sigma_k .$$
    
    

32. Show that $\vec{\Sigma} \cdot \hat{p}$ commutes with the Dirac Hamiltonian.

33. Show that at high energies

    
    

    $$\gamma^5 u = \left( \begin{array}{cc} \vec{\sigma}\cdot\hat{p} & 0 \\ 0 & \vec{\sigma}\cdot\hat{p} \end{array} \right) u ,$$
    
    

    where $u$ is the electron spinor. That is, show that in the extreme relativistic limit, the chirality operator ($\gamma^5$) is equal to the helicity operator; and so, for example, $\frac{1}{2}(1-\gamma^5)u=u_L$ corresponds to an electron of negative helicity.

34. For a massive fermion, show that handedness is not a good quantum number. That is, show that $\gamma^5$ does not commute with the Hamiltonian. However, verify that helicity is conserved but is frame dependent. In particular, show that the helicity is reversed by ``overtaking'' the particle concerned.

35. For an electron of momentum $\vec{p}=(p\sin\theta,0,p\cos\theta)$, calculate the $\lambda=+1/2$ helicity eigenspinor.