Coherent States of the Harmonic Oscillator

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 |x0i etc. 1. What is a coherent state ? Remember that the ground state |0i, being a gaussian, is a minimum uncertainty wavepacket: Proof: h¯ x2 = (a + a†)2 2mω mωh¯ p2 = − (a − a†)2 2 Since h0|(a + a†)(a + a†)|0i = h0|aa†|0i = 1 (1) h0|(a − a†)(a − a†)|0i = −h0|aa†|0i = −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 |ni is also a minimum uncertainty wave packet. Corresponding to (1) and (2) we have hn|(a + a†)(a + a†)|ni = hn|aa† + a†a|ni = hn|2a†a + [a, a†]|ni = 2n + 1 and similarly hn|(a − a†)(a − a†)|ni = −(2n + 1) which implies h¯2 h(∆x)2i h(∆p)2i = (2n + 1)2 (4) n n 4

1 so |ni is not minimal ! Clearly a crucial part in |0i being a minimal wave packet was

a|0i = 0 ⇒ h0|a†a|0i = 0 this corresponds to a sharp eigenvalue for the non-Hermitian operator mωx+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.

a|αi = α|αi (5) which implies hα|a†a|αi = |α|2. It is trivial to check that this indeed deﬁnes a minimal wave-packet

hα|(a + a†)|αi = (α + α?) hα|(a − a†)|αi = (α − α?) hα|(a + a†)(a + a†)|αi = (α + α?)2 + 1 hα|(a − a†)(a − a†)|αi = (α − α?)2 − 1 from which follows h¯ h(∆x)2i = hx2i − hxi2 = α α α 2mω hmω¯ 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 |αi, deﬁned by (6), satisﬁes the minimum uncertainty relation. They are called coherent states and we shall now proceed to study them in detail.

∞ 2m |α| 2 1 = Xhα|nhin|αi = |h0|αi|2 X = |h0|αi|2e|α| n m=0 m! solving for h0|αi we get:

− 1 |α|2 h0|αi = e 2 (11) up to a phase factor. Substituting into (10) we obtain the ﬁnal form:

∞ n − 1 |α|2 X α |αi = e 2 √ |ni (12) n=0 n! Obviously |αi 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 |ni:

∞ n ∞ n α α † X √ |ni = X (a†)n|0i = eαa |0i n=0 n! n=0 n! which implies

− 1 |α|2+αa† αa†−α?a |αi = e 2 |0i = e |0i (13)

3 3. Orthogonality and completeness relations We proceed to calculate the overlap between the coherent states using (12).

? n X − 1 |α|2− 1 |β|2 X (α β) hα|βi = hα|ni|βi = e 2 2 n n n! 1 1 = exp (− |α|2 − |β|2 + α?β) (14) 2 2 and similarly 1 1 hβ|αi = exp (− |α|2 − |β|2 + β?α) (15) 2 2 so |hα|βi|2 = hα|βihβ|αi = exp (−|α|2 − |β|2 + α?β + αβ?) or |hα|βi|2 = e−|α−β|2 . (16) Since hα|βi= 6 0 for α 6= β, we say that the set {|αi} is overcomplete. There is still, however, a closure relation: ? n m Z Z 2 (α ) α d2α |αihα| = d2α e−|α| X √ |mihn| (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 ∞ Z 2π d2α e−|α|2 (α?)nαm = dr re−r2 rm+n dφ ei(m−n)φ 0 0 Z ∞ 1 2 2 m −r2 = 2πδm,n dr (r ) e = πm!δm,n 2 0 Using this we ﬁnally get: Z d2α |αihα| = π X |nihn| = π n or equivalently Z d2α |αihα| = 1 (19) π

4 4. Coherent states in the x-representation Remember that hx0|0i is a minimal gaussian wave packet with hxi = hpi = 0. Since hx0|αi is also a minimal wave packet and

a + a† rmω <(α) = hα| |αi = hα|x|αi (20) 2 2¯h a − a† 1 =(α) = hα| |αi = √ hα|p|αi (21) 2i 2mωh¯ or  q √  hxi = ¯h 2<(α) α √ mω √ (22)  hpiα = mωh¯ 2=(α) it is natural to expect that hx0|αi 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 √ 0 0 − 1 |α|2+αa† − 1 |α|2 0 α mω (x− ip ) hx |αi = hx |e 2 |0i = e 2 hx |e 2¯h mω |0i

 q ¯h  x0 =  mω (24) q  4 mω  N = π¯h

0 0 For notational simplicity we also put y = x /x0. With these substitutions equation (23) will look like:

0 − 1 |α|2 √α (y0− d ) − 1 y02 hx |αi = 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

|α|2 √α (y0− d ) − 1 y02 − + √α y0 − √α 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 √ 0 − 1 |α|2− 1 y02− 1 α2+ 2αy0 hx |αi = Ne 2 2 2 1 √ √ = 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 − mω (x0−hxi )2+ i hpi x0− i hpi hxi hx |αi = 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 mω 1/4 i hpi x0− mω (x0−hxi )2 ψ (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:

But the |ni:s are eigenstates of the hamiltonian so:

n † n − 1 |α(0)|2 X (α(0)) − i ω¯h(n+ 1 )t (a ) |α, ti = e 2 √ e h¯ 2 √ |0i (32) n n! n! which is the same as

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−iωtα(0) while the second term only will contribute with a phase factor. Thus we have:

− i ωt −iωt |α, ti = e 2 |e α(0)i = |α(t)i (34)

So the coherent state remains coherent under time evolution. Furthermore, d α(t) = e−iωtα(0) ⇒ α(t) = −iωα(t) (35) dt or in components

( d dt <(α) = ω=(α) d (36) dt =(α) = −ω<(α) Deﬁning the expectation values,

( x(t) = hα(t)|x|α, ti (37) p(t) = hα(t)|p|α, ti we get

 q q p(t)  d x(t) = ¯h 2 d <(α) = ¯h 2ω=(α) = dt 2mω dt 2mω m (38) d q m¯hω d q m¯hω 2  dt p(t) = i 2 (−2i) dt =(α) = − 2 2ω<(α) = −mω x(t) or in a more familiar form

( d p(t) = m dt x(t) = mv(t) d 2 (39) dt p(t) = −mω x(t) i.e. x(t) and p(t) satisﬁes the classical equations of motion, as expected from Ehrenfest’s theorem. In summary, we have seen that the coherent states are minimal uncertainty wavepackets which remains minimal under time evolution. Furthermore, the time dependant expectation values of x and p satiﬁes the classical equations of motion. From this point of view, the coherent states are very natural for studying the classical limit of quantum mechanics. This will be explored in the next part.