Problem%20Set%206%202008.1 - and eigenstates of this...

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Problem Set # 6 Due. May 27 12:00 pm 1. Consider an atom, which has two non-degenerate energy levels (ground state |g> and excited |e>). The energy difference between the two levels is ħω . (a) Show that the Hamiltonian can be expressed as H= ħω /2 σ z , taking the zero of energy to be the halfway between the two levels. (b) Two 2-level atoms interacting through a dipole-dipole interaction can be described by the Hamiltonian, H=H 1 +H 2 +H d = ħω /2 σ (1) z + ħω /2 σ (2) z +V d ( σ + (1) σ - (2) + σ - (1) σ + (2) ) where σ (j) acts on atom-j, and V d is a coupling constant. Find the energy eigenvalues
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Unformatted text preview: and eigenstates of this Hamiltonian expressed in terms of the uncoupled basis set {|e> ⓧ |e>, |e> ⓧ |g>, |g> ⓧ |e>, |g> ⓧ |g>}. 2. In the state |jm>, J 2 |jm>= ħ 2 j(j+1)|jm> and J z |jm>= ħ m|jm>. (a) Show that in this state, <J x >=<J y >=0 (b) Show that in this state <J x 2 >=<J y 2 >=1/2 ħ 2 [j(j +1) - m 2 ] (c) Show that Δ J x and Δ J y are consistent with Heisenberg's uncertainty relation for angular momentum. (d) Show that the states |j, ±j> are minimum uncertainty wavepackets. 3. Sakurai Chap 3. 18 4. Sakurai Chap 3. 20...
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