221A Lecture Notes Supplemental Material on Harmonic Oscillator 1 Number-Phase Uncertainty To discuss the harmonic oscillator with the Hamiltonian p2 1 H = + mω2x2, (1) 2m 2 we have defined the annihilation operator rmω ip a = x + , (2) 2¯h mω the creation operator a†, and the number operator N = a†a. In some discussions, it is useful to define the “phase” operator Θ by √ √ a = eiΘ N, a† = Ne−iΘ. (3) Obviously the phase is ill-defined when N = 0, but apart from that, it is a useful notion. It is particularly useful when we discuss the classical limit N 1. One can define the “phase eigenstate” ∞ |θi = X einθ|ni. (4) n=1 By acting the phase operator eiΘ = a √1 , N 1 ∞ ∞ eiΘ|θi = a√ X einθ|ni = X einθ|n − 1i N n=1 n=1 ∞ = X ei(m+1)θ|mi = |0i + eiθ|θi. (5) m=0 It is almost an eigenstate of the phase operator, the failure due to the obvious problem with n = 0 state as anticipated from its definition. We can also calculate the inner products ∞ ∞ ∞ hθ0|θi = X X hm|e−imθ0 einθ|ni = X ein(θ−θ0). (6) n=1 m=1 n=1 1 This is almost the delta function +∞ 1 0 δ(θ − θ0) = X ein(θ−θ ). (7) 2π n=−∞ The number eigenstate is expressed correspondingly as Z 2π |ni = dθe−inθ|θi, (8) 0 which works for all n except for n = 0. Again ignoring the subtlety with the n = 0 state, we can derive the number-phase uncertainty principle. Study the commutator 1 1 1 [N, eiΘ] = [N, a√ ] = [N, a]√ = −a√ = −eiΘ. (9) N N N Therefore, roughly speaking, ∂ N = i . (10) ∂Θ Indeed, this makes sense on the “phase eigenstate,” ∞ ∂ hθ|N = X e−inθhn|n = i hθ|. (11) n=1 ∂θ Therefore, it leads to the “canonical commutation relation” [N, Θ] = i, (12) leading to the uncertainty principle 1 ∆N∆Θ ≥ . (13) 2 2 Coherent State of Harmonic Oscillator I’ve expanded discussions on the coherent state beyond Sakurai. Here is my lecture note on this subject. We saw that the uncertainty of the state |ki is actually larger than the minimum uncertainty h¯ ∆x∆p = (2k + 1). (14) 2 2 It appears odd that states with larger k, which we expect to behave more classicaly, are more uncertain. Moreover, expectation values of x and p vanish for energy eigenstates hk|x|ki = 0, hk|p|ki = 0. (15) Therefore even for large k, the energy eigenstates do not share characteristics we expect for classical oscillators. But how do we make a classical oscillator actually oscillate? Let’s say we are talking about a pendulum. To make it oscillate, what we do is to exert a force on it, pull the pendulum up, make sure the pendulum is settled in your hand, and release it. Namely, pull, hold, and release. Why not do the same in quantum mechanics? To pull a pendulum, we have to add an additional term to the potential 1 V = mω2x2 − F x, (16) 2 where F is the force we exert on the pendulum. Because the added term is linear in x, we can complete the square 1 1 V = mω2(x − x )2 − mω2x2, (17) 2 0 2 0 so that the pendulum settles to the position x0 6= 0. The force for this 2 purpose is given by F = mω x0. Because the pulled pendulum still has a quadratic potential, it is a modified harmonic oscillator. It settles to a ground state |0i0, which is annihilated by the modified annihilation operator rmω ip rmω a0 = (x − x ) + = a − x . (18) 2¯h 0 mω 2¯h 0 Therefore, the new ground state satisfies the equation rmω 0 = a0|0i0 = a − x |0i0. (19) 2¯h 0 In other words, rmω a|0i0 = x |0i0. (20) 2¯h 0 This is an eigenequation for the annihilation operator a. 3 In general, the eigenstates for the annilation operator can be found as fol- lows. Note that the annihilation operator is not Hermitian, and its eigenvalue does not have to be real. Define ∞ n ∞ n † f f efa |0i = X (a†)n|0i = X √ |ni, (21) n=0 n! n=0 n! for a complex number f. If you act the annihilation operator on this state, ∞ n ∞ n ∞ n fa† X f X f √ X f a e |0i = √ a|ni = √ n|n−1i = q |n−1i. (22) n=0 n! n=1 n! n=1 (n − 1)! We used the fact that n = 0 state does not contribute because it cannot be lowered by the annihilation operator. Changing the dummy index n to n+1, ∞ n+1 ∞ n f f † = X √ |ni = f X √ |ni = f efa |0i . (23) n=0 n! n=0 n! Therefore, this state has an eigenvalue f for the annihilation operator. We could have guessed it. The commutation relation [a, a†] = 1 says that roughly speaking a = ∂/∂a†. Therefore, acting a just pulls out the exponent f. We have not normalized the state yet. Working out the norm, ∗n m ∗ n † 2 f f (f f) ∗ fa X X f f e |0i = hn|√ √ |mi = = e . (24) n,m n! m! n n! Therefore, the following state is a normalized eigenstate of the annihilation operator |fi = e−|f|2/2efa† |0i, a|fi = f|fi. (25) This type of state is called coherent state. Coming back to our problem, the pendulum just before the release is therefore given by the coherent state rmω | x i. (26) 2¯h 0 Now the interest is in its time evolution. At t = 0, we release the pendulum. In other words, we let the state evolve according to the original Hamiltonian without an additional force. We can address the time evolution in Heisenberg picture easier than in Schr¨odinger picture. 4 In Heisenberg picture, let us first study the equation of motion for the † 1 † annihilation and creation operators. Because H =hω ¯ (a a+ 2 ) and [a, a ] = 1, we find d ih¯ a = [a, H] =hωa. ¯ (27) dt Solving this equation is trivial, a(t) = a(0)e−iωt. (28) Similarly, we find a†(t) = a†(0)eiωt. (29) Solving the definition of the creation, annihilation operators backwards, we find the position and momentum operators s s h¯ mhω¯ x = (a + a†), p = −i (a − a†). (30) 2mω 2 Their time-dependence is then immediately obtained as s s h¯ mhω¯ x(t) = (ae−iωt + a†eiωt), p(t) = −i (ae−iωt − a†eiωt). (31) 2mω 2 On a coherent state, they have expectation values s h¯ hf|x(t)|fi = (fe−iωt + f ∗eiωt), (32) 2mω s mhω¯ hf|p(t)|fi = −i (fe−iωt − f ∗eiωt). (33) 2 Note that I used hf|a† = (a|fi)† = (f|fi)† = hf|f ∗. Specializing to the q mω released pendulum, we have f = 2¯h x0, and hence hf|x(t)|fi = x0 cos ωt, (34) hf|p(t)|fi = −mωx0 sin ωt. (35) This result is the same as the classical pendulum. 5 Another important property of coherent states is that they have the min- imum uncertainty. We can work it out easily in the following way. s s h¯ h¯ hf|x|fi = hf|a + a†|fi = (f + f ∗), (36) 2mω 2mω s s mhω¯ mhω¯ hf|p|fi = −i hf|a − a†|fi = −i (f − f ∗), (37) 2 2 h¯ h¯ hf|x2|fi = hf|(a + a†)2|fi = (f 2 + f ∗2 + (f ∗f + 1) + f ∗f), (38) 2mω 2mω mhω¯ mhω¯ hf|p2|fi = − hf|(a − a†)2|fi = − (f 2 + f ∗2 − (f ∗f + 1) − f ∗f). 2 2 (39) Therefore, we find h¯ (∆x)2 = hf|x2|fi − (hf|x|fi)2 = , (40) 2mω mhω¯ (∆p)2 = hf|p2|fi − (hf|p|fi)2 = . (41) 2 Finally, we obtain h¯ ∆x∆p = , (42) 2 indeed the minimum uncertainty state. To sum it up, the coherent state represents the closest approximation of a classical oscillator, with the minimum uncertainty and oscillating expectation value of the position and the momentum. We can obtain the same result in the Schr¨odingerpicture, which is a litte more technical than in the Heisenberg picture. The time evolution of the coherent state can be obtained as e−iHt/¯h|fi = e−iHt/¯hefa† |0ie−|f|2/2 = e−iHt/¯hefa† eiHt/¯he−iHt/¯h|0ie−|f|2/2 fe−iHt/h¯ a†eiHt/h¯ −i hω¯ t/¯h −|f|2/2 = e e 2 |0ie = efa†e−iωt |0ie−|f|2/2e−iωt/2 = |fe−iωtie−iωt/2. (43) 6 Therefore the expectation values of the postion and momentum operators are hf, t|x|f, ti = hfe−iωt|x|fe−iωti s h¯ = (fe−iωt + f ∗e+iωt) 2mω = x0 cos ωt, (44) hf, t|p|f, ti = hfe−iωt|p|fe−iωti s mhω¯ = −i (fe−iωt − f ∗e+iωt) 2 = −mωx0 sin ωt, (45) q mω where we used f = 2¯h x0. The results agree with those in the Heisenberg picture Eq. (32,33). In quantum treatment of electromagnetism, light is described by a collec- tion of photons. For a coherent light such as laser, the electric and magnetic field behave exactly like in the classical Maxwell theory. Laser is described in terms of a coherent state. 3 Coherent State Wave Functions Coherent state of course can be studied using the conventional wave func- tions. It takes a few tricks to workt it out, however. We use the Baker–Campbell–Hausdorff formula.