Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 10.1

Spinors — Schwartz M.D. z-lib.org

Physics — QFT · dirac-equation-nonrelativistic-limit

Problem

We saw that the Dirac equation predicted that there is interaction between the electron spin and the magnetic field,

\vec{S}\vec{B}

, with strength

\mu_B = \frac{\hbar c}{2m_e}

. When the electron has angular momentum

\vec{L}

, such as in an atomic orbital, there is also a

\vec{B}\vec{L}

interaction and a spin-orbit coupling

\vec{S}\vec{L}

. The Dirac equation (along with symmetry arguments) predicts the strength of all three interactions, as well as other corrections. To see the effect of these terms on the hydrogen atom, we have to take the non-relativistic limit. (a) For the Schr\"odinger equation, we need the Hamiltonian, not the Lagrangian. Find the Dirac Hamiltonian by writing the Dirac equation as

i\partial_t \psi = H_D \psi

. Write the Hamiltonian in terms of momenta

p_i

rather than derivatives

\partial_i

. (b) Calculate

(H_D + eA_0)^2

in the Weyl representation for

\psi = (\psi_L \psi_R)

. Leave in terms of

\sigma_i

p_i

and

A_i

. Put back in the factors of

c

and

\hbar

, keeping the charge

e

dimensionless. (c) Now take the square root of this result and expand in

\frac{1}{c}

, subtracting off the zero-point energy

mc^2

, i.e. compute

H = H_D - mc^2

to order

c^0

. Looking at the

\sigma_i

term, how big are the electron's electric and magnetic dipole moments? (d) The size of the terms in this Hamiltonian are only meaningful because the spin and angular momentum operators have the same normalization. Check the normalization of the angular momentum operators

L_i = \varepsilon_{ijk} x_j p_k

and the spin operators

S_i = \frac{1}{2}\sigma_i

by showing that they both satisfy the rotation algebra:

[J_i, J_j] = i\varepsilon_{ijk} J_k

. (e) The gyromagnetic ratio,

g_e

(sometimes called the

g

-factor), is the relative size of the

\vec{S}\vec{B}

and

\vec{L}\vec{B}

interactions. Choose a constant magnetic field in the

z

direction, then isolate the

B_z L_z

coupling in

H

. Extract the electron gyromagnetic ratio

g_e

by writing the entire coupling to the magnetic field in the Hamiltonian as

\mu_B B_z(L_z + g_e S_z) = B_z \mu_z

, with

\vec{\mu} \equiv \mu_B(\vec{L} + g_e \vec{S})

. How could you experimentally measure

g_e

(e.g. with spectroscopy of the hydrogen atom)? (f) In spherical coordinates, the Schr\"odinger equation has a

\vec{L}^2

term. With spin, you might expect that this becomes

\vec{L}\vec{\mu} = \mu_B(\vec{L}^2 + g_e \vec{L}\vec{S})

, making the

\vec{L}\vec{S}

term proportional to the

g

-factor. This is wrong. It misses an important relativistic effect, Thomas precession. It is very hard to calculate directly, but easy to calculate using symmetries. With no magnetic field, the atom, with spin included, is still rotationally invariant. Which of

\vec{J} = \vec{L} + \vec{S}

\vec{\mu} = \vec{L} + g_e \vec{S}

is conserved (i.e. commutes with

H

)? Using this result, how does the spin-orbit coupling depend on

g_e

? (g) There are additional relativistic effects coming from the Dirac equation. Expand the Dirac equation to next order in

\frac{1}{c^2}

, producing a term that scales as

\vec{p}^4

. (h) Now let us do some dimensional analysis -- there is only one scale

m_e

. Show that the electron's Compton wavelength, the classical electron radius,

r_e

, the Bohr radius,

a_0

, and the inverse-Rydberg constant,

\mathrm{Ry}^{-1}

, are all

m_e

times powers of

\alpha_e

. Are the splittings due to the

p^4

term fine structure (

\Delta E \sim \alpha_e^2 E

), hyperfine structure (

\Delta E \sim \alpha_e^4 E

) or something else? [Hint: write out a formula for the energy shift using time-dependent perturbation theory, then see which of the above length scales appears.]

Related theory

Limit Superior And Limit Inferior

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