- The Magnetic Field Produced At A Position I By A Magnetic Dipole E G A Bar Magnet With Magnetic Dipole Moment U Is Gi 1 (142.32 KiB) Viewed 17 times
- The Magnetic Field Produced At A Position I By A Magnetic Dipole E G A Bar Magnet With Magnetic Dipole Moment U Is Gi 2 (77.56 KiB) Viewed 17 times
- The Magnetic Field Produced At A Position I By A Magnetic Dipole E G A Bar Magnet With Magnetic Dipole Moment U Is Gi 3 (32.51 KiB) Viewed 17 times
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The magnetic field produced at a position ï by a magnetic dipole (e.g. a bar magnet) with magnetic dipole moment ù is given by [3(•P)p – ] + 240, 783 () 2μο B Mo В ů 4nr 3 3 [83(™) is the Dirac delta function in 3D] In a hydrogen atom, the proton behaves like a magnetic dipole with a magnetic dipole Ope Sp, where Ip is the proton g-factor, my is the mass of the proton and Sp 2mp is the spin vector of the proton. a) Write down the dipole moment of an electron (lle) in terms of its mass (me) and its spin vector (Se Me = moment jp = . b) Starting with the Hamiltonian H' = -de B for the interaction of the electron dipole moment with the proton ma tic field, show: Hogpez (3 (S» -f) (S. .4) – so se H'E (Sp. Se) 8? () 8tmp me 3mme Mogpe? = p3
c) Treating H' as a perturbation, show that the first-order correction E(1) to the hydroger electron ground state (4 100) energy is given by: E(1) = Mogpe? 31m,m.a3 (s, se) Hint: Look up the hydrogen electron ground state wavefunction; 3(Sp_*)(Se *)-5, -5 Use the result = 0 for the ground state in your derivation. = 73
d) The total spin of the of the proton and electron is $ = Š> + Še. Express Sp. Ŝ, in terms of 52, Sž, and S.
The electron and the proton are both spin-1/2 fermions and their composite state consists of a triplet and a singlet. e) Find the expectation values (Se ?) and ($p?). ($,%) = ($)%) = f) Find ($2) for the triplet and the singlet state of the electron-proton composite spin: Triplet: ($2) = = Singlet: ($2) = g) Using the results above, calculate E(1) for the triplet and the singlet states. [The energy difference between the two states is known as the hyperfine splitting]