3. A system of two uncoupled quantum harmonic oscillators is described by the following Hamil- tonian: 武,1 成,! 2m2 2m, (a) Re-express this Hamiltonian in the form H (â, â), thus in terms of the raising/lowering operators, carefully defining all quantities. This two QHO system is described by the basis of states In)n2) = In, n) where a m) = Vmn - 1). It experiences a coupling potential V(à, 9).
3. A system of two uncoupled quantum harmonic oscillators is described by the following Hamil- tonian: 武,1 成,! 2m2 2m, (a) Re-express this Hamiltonian in the form H (â, â), thus in terms of the raising/lowering operators, carefully defining all quantities. This two QHO system is described by the basis of states In)n2) = In, n) where a m) = Vmn - 1). It experiences a coupling potential V(à, 9).
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Transcribed Image Text:3. A system of two uncoupled quantum harmonic oscillators is described by the following Hamil-
tonian:
民,1
2m,
2m2
(a) Re-express this Hailtonian in the form Ĥ (â, ây), thus in terms of the raising/lowering
operators, carefully defining all quantities.
This two QHO system is described by the basis of states |m,)|n2) = In1, n2) where
a na) = Vn – 1). It experiences a coupling potential V(â, ŷ).
(b) Considering only the lowest four states, with zero or single quanta occupancies i.e n1, n2 =
0,1 construct the 4 x 4 Hamiltonian matrix for the case where:
Vay (3, 6) = 9(0, + â;)(@, + â;).
( for neatness we wrote ât rather than â, for the raising operator in the question above).
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