Problems

4.1 Solve the manifestly covariant form of the Klein-Gordon equation for the ground state of the hydrogen atom. Specifically, assume

V° = -~ V = 0 r and show that the ground state wave function can be written

rte-0re-iEt

Find e, ¡3, and E. Then examine the nonrelativistic limit by projecting out the 4>+ and 4>- components defined in Eq. (4.36). Interpret your results and compare with the Schrodinger theory.

4.2 Calculate the fine structure splitting of the energy levels for a pion bound in an atom with charge Ze. Draw an energy level diagram showing all the levels up to n = 3. (You may use the nonrelativistic form of the Klein-Gordon equation and calculate the splitting in perturbation theory using suitably modified hydrogen atom wave functions.) What are the Bohr radii of these orbits and what is v/cl Estimate the probability that a pion in the S-state will be inside the nucleus.

4.3 Calculate the Zeeman splitting of the levels up to n = 3 for a pionic atom.

4.4 Suppose a pion is bound by a scalar potential of the form

Solve the KG equation for the special case when the solution is static (i.e., independent of time). Discuss the significance of your result.

4.5 A pion of mass ¡j. is bound by a scalar one-dimensional square well potential V(x) defined to be:

(This could be a very rough model for a pion inside of a nucleus of radius R.)

(a) Solve the KG equation (4.7) in one space dimension for the positive energy ground state. (Take U(x) = V{x).)

(b) Find the value of R such that the positive energy ground state has energy

Estimate the size of the pion cloud.

(c) Find the positive and negative energy parts, as defined in Eq. (4.36), of the solution found in part (a). Discuss your result and explain how the negative energy part should be interpreted.

4.6 New two-component form for the KG equation. One of the features of the two-component form introduced in Eq. (4.36) is that it does not completely decouple positive and negative energy solutions, even if the potentials are zero. In particular, for the free particle solutions

For conceptual purposes, it might be convenient to further diagonalize H so that the non-diagonal terms come from interactions only. This can be done by defining new components:

where Ey is defined by the power series given in Eq. (4.3). Show that:

(a) The conserved norm is identical to (4.40) with r3 the "metric tensor."

(b) The equations assume the form (4.35) with the free Hamiltonian being region I region II region III

R < x V{x) = 0 0 <x < R V(x) = -M2 V0 x < 0 V{x) = oo

This completely diagonalizes the (±) states for free particles.

(c) The charge conjugation operation and nonrelativistic limits are as before,

(d) The "old" form can be transformed into the "new" form using the following transformation:

In particular, show that this transformation preserves the norm by proving that

U*taU = t3 • Also, using the explicit forms (4.47), show that

Hence U transforms into states with only an upper (or lower) component, Finally, show by direct computation that which shows explicitly that U transforms the "old" two-component form into the "new" two-component form.

Relativistic Quantum Mechanics and Field Theory

FRANZ GROSS Copyright© 2004 WILEY-VCH Verlag GmbH

CHAPTER 5

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