{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

ls3_unit_5 - THE INTERACTION OF RADIATION AND MATTER...

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 42 R. Victor Jones, April 18, 2000 V. P HOTON A BSORPTION AND E MISSION "P OOR M AN ' S " S ECOND Q UANTIZATION OF M ATERIAL S YSTEM : In treating the complete quantum mechanical problem, it is useful to recast the material (atomic) Hamiltonian in terms of an appropriate set of creation and destruction operators. To that end we make the following definition H A x = h ω x x [ V-1 ] Using the ubiquitous identity operation x x x = 1 , we may write the material Hamiltonian in second quantized form -- viz. H A = x x x H A y y y = x h ω x x y y y x = h ω x x x x [ V-2 ] In general, the operator x y b x b y applied to any state z yields x y z = b x b y z = x δ yz [ V-3 ] -- i.e. the operator changes a state z to a state x if the state is y otherwise it produces zero. In other words, the operator destroys the state y and creates a state x . The second quantization viewpoint is particularly useful in treating the interaction of a two-level material system with the radiation field. This case, is most conveniently formulate in two- vector notation with the use of Pauli spin matrices -- viz. a = 1 0 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ and b = 0 1 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ [ V-4a ] a = 1 0 [ ] and b = 0 1 [ ] [ V-4b ]
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 43 R. Victor Jones, April 18, 2000 a b = 1 0 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ 0 1 [ ] = 0 1 0 0 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ = σ + b a = 0 1 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ 1 0 [ ] = 0 0 1 0 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ = σ [ V-4c ] a a = 1 0 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ 1 0 [ ] = 1 0 0 0 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ b b = 0 1 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ 0 1 [ ] = 0 0 0 1 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ [ V-4d ] Consequently, the atomic Hamiltonian may be written H A = h ω a 1 0 0 0 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ + h ω b 0 0 0 1 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ = 1 2 h ω a −ω b ( ) 1 0 0 1 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ + 1 2 h ω a + ω b ( ) 1 0 0 1 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ [ V-5a ] and if we neglect the mean energy of the states H A 1 2 h ω ab 1 0 0 1 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ = 1 2 h ω ab σ z [ V-5b ]
Image of page 2
T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 44 R. Victor Jones, April 18, 2000 and the electric dipole interaction Hamiltonian becomes H ED = e r D r E T 0 ( ) = a a + b b [ ] e r D [ ] a a + b b [ ] r E T 0 ( ) = a a e r D [ ] b b + b b e r D [ ] a a { } r E T 0 ( ) = e a r D b σ + + e b r D a σ { } r E T 0 ( ) = r σ + + r σ { } r E T 0 ( ) [ V-6 ] From Equation [ II-24a ] in this lecture set we can write r E T 0 ( ) ˆ e l { } σ E l { } i a l { } σ t ( ) exp i r k l { } r r A [ ] i a l { } σ t ( ) exp i r k l { } r r A [ ] Χ੿ Ψએ Ωટ Ϋિ ά૏ έ૟ σ = 1 2 l { } [ V-7 ] where E l {} = h ω l { } 2 ε 0 V is the so called the electric field per photon and r r A is the location of the center of the atom under consideration. Thus Equation [ V-6 ] may be written quite generally for a two level atom as H ED = e a r D b σ + + e b r D a σ { } × ˆ e l { } s E l {} i a l { } s t ( ) exp i r k l {} r r A [ ] i a l { } s t ( ) exp i r k l { } r r A [ ] { } s = 1 2 l { } ] = h g l { } s σ + + g l { } s
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern