Mathematical Methods for Physics and Engineering - Matematica.NET

19.2 PHYSICAL EXAMPLES OF OPERATORS19.2.3 Annihilation and creation operatorsAs a final illustration of the use of operator methods in physics we consider theirapplication to the quantum mechanics of a simple harmonic oscillator (s.h.o.).Although we will start with the conventional description of a one-dimensionaloscillator, using its position and momentum, we will recast the description interms of two operators and their commutator and show that many importantconclusions can be reached from studying these alone.The Hamiltonian for a particle of mass m with momentum p moving in aone-dimensional parabolic potential V (x) = 1 2 kx2 isH = p22m + 1 2 kx2 = p22m + 1 2 mω2 x 2 ,where its classical frequency of oscillation ω is given by ω 2 = k/m. We recall thatthe corresponding operators, p and x, do not commute and that [ p, x ] = −i.In analogy with the ladder operators used when discussing angular momentum,we define two new operators:√ mω2 x + ip √2mωand A † ≡√ mω2 x − ip √2mω.A ≡(19.39)Since both x and p are Hermitian, A and A † are Hermitian conjugates, thoughneither is Hermitian and they do not represent physical quantities that can bemeasured.Now consider the two products A † A and AA † :A † A = mω 2x2 − ipx2 + ixp2 + p22mω = H ω − i 2 [ p, x ] = H ω − 2 ,AA † = mω 2x2 + ipx2 − ixp2 + p22mω = H ω + i 2 [ p, x ] = H ω + 2 .From these it follows thatand thatFurther,Similarly,H = 1 2 ω(A† A + AA † ) (19.40)[A, A† ] = . (19.41)[ H,A] = [ 12 ω(A† A + AA † ),A ]= 1 2 ω ( A † 0+ [ A † ,A ] A + A [ A † ,A ] +0A †)= 1 2ω(−A − A) =−ωA. (19.42)[H,A† ] = ωA † (19.43).Before we apply these relationships to the question of the energy spectrum ofthe s.h.o., we need to prove one further result. This is that if B is an Hermitianoperator then 〈ψ | B 2 | ψ〉 ≥0forany |ψ〉. The proof, which involves introducing667