The Klein-Gordon equation
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• The Klein–Gordon equation, ( + m2 )φ(x) = 0,
(2.A)
is an equation for a free relativistic particle with zero spin. The transformation law of a scalar field φ(x) under Lorentz transformations is given by φ (Λx) = φ(x). • The equation for the spinless particle in an electromagnetic field, Aμ is obtained by changing ∂μ → ∂μ + iqAμ in equation (2.A), where q is the charge of the particle.
2.1. Solve the Klein–Gordon equation. 2.2. If φ is a solution of the Klein–Gordon equation calculate the quantity
∂φ∗ 3 ∗ ∂φ −φ Q = iq d x φ . ∂t ∂t 2.3. The Hamiltonian for a free real scalar field is 1 d3 x[(∂0 φ)2 + (∇φ)2 + m2 φ2 ] . H= 2 Calculate the Hamiltonian H for a general solution of the Klein–Gordon equation. 2.4. The momentum for a real scalar field is given by P = − d3 x∂0 φ∇φ . Calculate the momentum P for a general solution of the Klein–Gordon equation.
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Problems
2.5. Show that the current1 i jμ = − (φ∂μ φ∗ − φ∗ ∂μ φ) 2 satisfies the continuity equation, ∂ μ jμ = 0. 2.6. Show that the continuity equation ∂μ j μ = 0 is satisfied for the current i jμ = − (φ∂μ φ∗ − φ∗ ∂μ φ) − qAμ φ∗ φ , 2 where φ is a solution of Klein–Gordon equation in external electromagnetic potential Aμ . 2.7. A scalar particle in the s–state is moving in the potential −V, r < a 0 qA = , 0, r>a where V is a positive constant. Find the dispersion relation, i.e. the relation between energy and momentum, for discrete particle states. Which condition has to be satisfied so that there is only one bound state in the case V < 2m? 2.8. Find the energy spectrum and the eigenfunctions for a scalar particle in a constant magnetic field, B = Bez . 2.9. Calculate the reflection and the transmission coefficients of a Klein– Gordon particle with energy E, at the potential 0, z 0 where U0 is a positive constant. 2.10. A particle of charge q and mass m is incident on a potential barrier 0, z < 0, z > a A0 = , U0 , 0 < z < a where U0 is a positive constant. Find the transmission coefficient. Also, find the energy of particle for which the transmission coefficient is equal to one. 2.11. A scalar particle of mass m and charge −e moves in the Coulomb field of a nucleus. Find the energy spectrum of the bounded states for this system if the charge of the nucleus is Ze.
θ 2.12. Using the two-component wave function , where θ = 12 (φ + mi ∂φ ∂t ) χ and χ = 12 (φ − mi ∂φ ∂t ), instead of φ rewrite the Klein–Gordon equation in the Schr¨ odinger form. 1
Actually this is current density.
Chapter 2. The Klein–Gordon equation
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2.13. Find the eigenvalues of the Hamiltonian from the previous problem. Find the nonrelativistic limit of this Hamiltonian. 2.14. Determine the velocity operator v = i[H, x], where H is the Hamiltonian obtained in Problem 2.12. Solve the eigenvalue problem for v. 2.15. In the space of two–component wave functions the scalar product is defined by 1 ψ1 |ψ2 = d3 xψ1† σ3 ψ2 . 2 (a) Show that the Hamiltonian H obtained in Problem 2.12 is Hermitian. (b) Find expectation values of the Hamiltonian H, and the velocity v in 1 the state e−ip·x . 0
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