Mathematical Methods for Physics and Engineering - Matematica.NET

QUANTUM OPERATORSan arbitrary complete set of orthonormal base states |φ i 〉 and using equation(19.11), is as follows:〈ψ | B 2 | ψ〉 = 〈ψ | B × 1 × B | ψ 〉= ∑ i= ∑ i= ∑ i= ∑ i= ∑ i〈ψ | B |φ i 〉〈φ i | B |ψ〉〈ψ | B |φ i 〉 ( 〈φ i | B |ψ〉 ∗) ∗〈ψ | B |φ i 〉 ( 〈ψ | B † |φ i 〉 ) ∗〈ψ | B |φ i 〉〈ψ | B |φ i 〉 ∗ , since B is Hermitian,|〈ψ | B |φ i 〉| 2 ≥ 0.We note, for future reference, that the Hamiltonian H for the s.h.o. is the sum oftwo terms each of this form and therefore conclude that 〈ψ|H|ψ〉 ≥0 for all |ψ〉.The energy spectrum of the simple harmonic oscillatorLet the normalised ket vector |n〉 (or |E n 〉) denote the nth energy state of the s.h.o.with energy E n . Then it must be an eigenstate of the (Hermitian) Hamiltonian Hand satisfyH|n〉 = E n |n〉 with 〈m|n〉 = δ mn .Now consider the state A|n〉 and the effect of H upon it:HA|n〉 = AH|n〉−ωA|n〉, using (19.42),= AE n |n〉−ωA|n〉=(E n − ω)A|n〉.Thus A|n〉 is an eigenstate of H corresponding to energy E n − ω and must besome multiple of the normalised ket vector |E n − ω〉, i.e.A| E n 〉≡A|n〉 = c n |E n − ω〉,where c n is not necessarily of unit modulus. Clearly, A is an operator thatgenerates a new state that is lower in energy by ω; it can thus be compared tothe operator D, which has a similar effect in the context of the z-component ofangular momentum. Because it possesses the property of reducing the energy ofthe state by ω, which, as we will see, is one quantum of excitation energy for theoscillator, the operator A is called an annihilation operator. Repeated applicationof A, m times say, will produce a state whose energy is mω lower than that ofthe original:A m |E n 〉 = c n c n−1 ···c n−m+1 |E n − mω〉. (19.44)668