Coherent states of the harmonic oscillator In these notes I will assume knowledge about the operator method for the harmonic oscillator corresponding to sect. 2.3 i "Modern Quantum Mechanics" by J.J. Sakurai. At a couple of places I refefer to this book, and I also use the same notation, notably x and p are operators, while the correspondig eigenkets are jx0i etc. 1. What is a coherent state ? Remember that the ground state j0i, being a gaussian, is a minimum un- certainty wavepacket: Proof: h¯ x2 = (a + ay)2 2m! m!h¯ p2 = − (a − ay)2 2 Since h0j(a + ay)(a + ay)j0i = h0jaayj0i = 1 (1) h0j(a − ay)(a − ay)j0i = −h0jaayj0i = −1 (2) it follows that h¯2 h¯2 hx2i hp2i = − 1(−1) = 0 0 4 4 and finally since hxi0 = hpi0 = 0, it follows that h¯2 h(∆x)2i h(∆p)2i = (3) 0 0 4 We can now ask whether jni is also a minimum uncertainty wave packet. Corresponding to (1) and (2) we have hnj(a + ay)(a + ay)jni = hnjaay + ayajni = hnj2aya + [a; ay]jni = 2n + 1 and similarly hnj(a − ay)(a − ay)jni = −(2n + 1) which implies h¯2 h(∆x)2i h(∆p)2i = (2n + 1)2 (4) n n 4 1 so jni is not minimal ! Clearly a crucial part in j0i being a minimal wave packet was aj0i = 0 ) h0jayaj0i = 0 this corresponds to a sharp eigenvalue for the non-Hermitian operator m!x+ip even though, as we saw, there was (minimal) dispersion in both x and p. It is natural to expect other minimal wave-packets with non zero expectation values for x and p but still eigenfunctons of a, i.e. ajαi = αjαi (5) which implies hαjayajαi = jαj2. It is trivial to check that this indeed defines a minimal wave-packet hαj(a + ay)jαi = (α + α?) hαj(a − ay)jαi = (α − α?) hαj(a + ay)(a + ay)jαi = (α + α?)2 + 1 hαj(a − ay)(a − ay)jαi = (α − α?)2 − 1 from which follows h¯ h(∆x)2i = hx2i − hxi2 = α α α 2m! hm!¯ h(∆p)2i = hp2i − hpi2 = α α α 2 and accordingly h¯ h(∆x)2i h(∆p)2i = (6) α α 4 So the states jαi, defined by (6), satisfies the minimum uncertainty relation. They are called coherent states and we shall now proceed to study them in detail. 2 2. Coherent states in the n-representation In the jni base the coherent state look like: X X jαi = cnjni = jnihnjαi (7) n n Since (ay)n jni = p j0i (8) n! we have αn hnjαi = p h0jαi (9) n! and thus 1 αn jαi = h0jαi X p jni (10) n=0 n! The constant h0jαi is determined by normalization as follows: 1 2m jαj 2 1 = Xhαjnhinjαi = jh0jαij2 X = jh0jαij2ejαj n m=0 m! solving for h0jαi we get: − 1 jαj2 h0jαi = e 2 (11) up to a phase factor. Substituting into (10) we obtain the final form: 1 n − 1 jαj2 X α jαi = e 2 p jni (12) n=0 n! Obviously jαi are not stationary states of the harmonic oscillator, but we shall see that they are the appropriate states for taking the classical limit. A very convenient expression can be derived by using the explicit expression (8) for jni: 1 n 1 n α α y X p jni = X (ay)nj0i = eαa j0i n=0 n! n=0 n! which implies − 1 jαj2+αay αay−α?a jαi = e 2 j0i = e j0i (13) 3 3. Orthogonality and completeness relations We proceed to calculate the overlap between the coherent states using (12). ? n X − 1 jαj2− 1 jβj2 X (α β) hαjβi = hαjnijβi = e 2 2 n n n! 1 1 = exp (− jαj2 − jβj2 + α?β) (14) 2 2 and similarly 1 1 hβjαi = exp (− jαj2 − jβj2 + β?α) (15) 2 2 so jhαjβij2 = hαjβihβjαi = exp (−|αj2 − jβj2 + α?β + αβ?) or jhαjβij2 = e−|α−βj2 : (16) Since hαjβi 6= 0 for α 6= β, we say that the set fjαig is overcomplete. There is still, however, a closure relation: ? n m Z Z 2 (α ) α d2α jαihαj = d2α e−|αj X p jmihnj (17) m;n n! m! where the measure d2α means "summing" over all complex values of α, i.e. integrating over the whole complex plane. Now, writing α in polar form: α = reiφ ) d2α = dφ dr r (18) we get Z Z 1 Z 2π d2α e−|αj2 (α?)nαm = dr re−r2 rm+n dφ ei(m−n)φ 0 0 Z 1 1 2 2 m −r2 = 2πδm;n dr (r ) e = πm!δm;n 2 0 Using this we finally get: Z d2α jαihαj = π X jnihnj = π n or equivalently Z d2α jαihαj = 1 (19) π 4 4. Coherent states in the x-representation Remember that hx0j0i is a minimal gaussian wave packet with hxi = hpi = 0. Since hx0jαi is also a minimal wave packet and a + ay rm! <(α) = hαj jαi = hαjxjαi (20) 2 2¯h a − ay 1 =(α) = hαj jαi = p hαjpjαi (21) 2i 2m!h¯ or 8 q p < hxi = ¯h 2<(α) α p m! p (22) : hpiα = m!h¯ 2=(α) it is natural to expect that hx0jαi is a displaced gaussian moving with velocity v = p=m. We now proceed to show this. Using the previously derived form (13) of the coherent states expressed in terms of the ground state of the harmonic oscillator we get p 0 0 − 1 jαj2+αay − 1 jαj2 0 α m! (x− ip ) hx jαi = hx je 2 j0i = e 2 hx je 2¯h m! j0i Acting from the left with hx0j gives us 1: p 0 − 1 jαj2 α m! (x0− i (−i¯h d )) 0 hx jαi = e 2 e 2¯h m! dx0 hx j0i 0 1 2 αp 0 2 d 1 x 2 − jαj (x −x0 0 ) − ( ) = Ne 2 e x0 2 dx e 2 x0 (23) where we have used the explicit form of hx0j0i and introduced the constants: 8 q ¯h > x0 = <> m! (24) q > 4 m! : N = π¯h 0 0 For notational simplicity we also put y = x =x0. With these substitutions equation (23) will look like: 0 − 1 jαj2 pα (y0− d ) − 1 y02 hx jαi = Ne 2 e 2 dy0 e 2 (25) Using the commutator relation A+B − 1 [A;B] A B e = e 2 e e (26) 1c.f. Sakurai p.93 5 which is valid if both A and B commute with [A; B], we get jαj2 pα (y0− d ) − 1 y02 − + pα y0 − pα d − 1 y02 e 2 dy0 e 2 = e 4 2 e 2 dy0 e 2 (27) a d and thus, noting that e dy0 , is a translation operator p 0 − 1 jαj2− 1 y02− 1 α2+ 2αy0 hx jαi = Ne 2 2 2 1 p p = N exp (− (y0 − 2<(α))2 + i 2=(α)y0 − i=(α)<(α)) (28) 2 Using (22) the resulting expression for the wavefunction of the coherent state is 0 − m! (x0−hxi )2+ i hpi x0− i hpi hxi hx jαi = Ne 2¯h α h¯ α 2¯h α α (29) and since the last term is a constant phase it can be ignored and we finally get 0 m! 1=4 i hpi x0− m! (x0−hxi )2 (x ) = ( ) e h¯ α 2¯h α ; (30) α πh¯ which is the promissed result. 5. Time evolution of coherent states The time evolution of a state is given by the time evolution operator 2 U(t). Using what we know about this operator and what we have learned so far about the coherent states we can write: n − i Ht − i Ht − 1 jα(0)j2 X (α(0)) jα; ti = U(t; 0)jα(0)i = e h¯ jα(0)i = e h¯ e 2 p jni (31) n n! But the jni:s are eigenstates of the hamiltonian so: n y n − 1 jα(0)j2 X (α(0)) − i !¯h(n+ 1 )t (a ) jα; ti = e 2 p e h¯ 2 p j0i (32) n n! n! which is the same as −i!t y n − 1 jα(0)j2 − i !t X (α(0)e a ) jα; ti = e 2 e 2 j0i = n n! 1 i exp (− jα(0)j2 − !t + α(0)e−i!tay)j0i (33) 2 2 2c.f. Sakurai chapter 2.1 6 Comparing this expression with (13), it is obvious that the first and the third term in the exponent, operating on the ground state, will give us a coherent state with the time dependent eigenvalue e−i!tα(0) while the second term only will contribute with a phase factor. Thus we have: − i !t −i!t jα; ti = e 2 je α(0)i = jα(t)i (34) So the coherent state remains coherent under time evolution. Furthermore, d α(t) = e−i!tα(0) ) α(t) = −i!α(t) (35) dt or in components ( d dt <(α) = !=(α) d (36) dt =(α) = −!<(α) Defining the expectation values, ( x(t) = hα(t)jxjα; ti (37) p(t) = hα(t)jpjα; ti we get 8 q q p(t) < d x(t) = ¯h 2 d <(α) = ¯h 2!=(α) = dt 2m! dt 2m! m (38) d q m¯h! d q m¯h! 2 : dt p(t) = i 2 (−2i) dt =(α) = − 2 2!<(α) = −m! x(t) or in a more familiar form ( d p(t) = m dt x(t) = mv(t) d 2 (39) dt p(t) = −m! x(t) i.e.