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Unformatted text preview: 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 x etc. 1. What is a coherent state ? Remember that the ground state 0, being a gaussian, is a minimum uncertainty wavepacket:
Proof:
h
¯
(a + a† )2
2mω
mω h
¯
=−
(a − a† )2
2 x2 =
p2
Since 0(a + a† )(a + a† )0 = 0aa† 0 = 1
0(a − a† )(a − a† )0 = −0aa† 0 = −1 (1)
(2) it follows that
x2 0 p2 0 = − h2
¯
h2
¯
1(−1) =
4
4 and ﬁnally since x0 = p0 = 0, it follows that
(∆x)2 0 (∆p)2 0 = h2
¯
4 (3) We can now ask whether n is also a minimum uncertainty wave packet.
Corresponding to (1) and (2) we have
n(a + a† )(a + a† )n = naa† + a† an = n2a† a + [a, a† ]n = 2n + 1
and similarly
n(a − a† )(a − a† )n = −(2n + 1)
which implies
(∆x)2 n (∆p)2 n =
1 h2
¯
(2n + 1)2
4 (4) so n is not minimal !
Clearly a crucial part in 0 being a minimal wave packet was
a0 = 0 ⇒ 0a† a0 = 0
this corresponds to a sharp eigenvalue for the nonHermitian operator mω x+ip
even though, as we saw, there was (minimal) dispersion in both x and p. It
is natural to expect other minimal wavepackets with non zero expectation
values for x and p but still eigenfunctons of a, i.e.
aα = αα (5) which implies αa† aα = α2 . It is trivial to check that this indeed deﬁnes a
minimal wavepacket
α(a + a† )α = (α + α )
α(a − a† )α = (α − α )
α(a + a† )(a + a† )α = (α + α )2 + 1
α(a − a† )(a − a† )α = (α − α )2 − 1
from which follows
h
¯
2mω
hmω
¯
2
2
2
(∆p) α = p α − pα =
2 (∆x)2 α = x2 α − x2 =
α and accordingly
(∆x)2 α (∆p)2 α = h
¯
4 (6) So the states α, 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 nrepresentation
In the n base the coherent state look like:
α =
n cn n =
n nnα (7) Since
(a† )n
n = √ 0
n! (8) αn
nα = √ 0α
n! (9) we have and thus
α = 0α ∞
αn
√ n
n!
n=0 (10) The constant 0α is determined by normalization as follows:
1=
n αnnα = 0α2 ∞
α2m
2
= 0α2 eα
m=0 m! solving for 0α we get:
1 0α = e− 2 α 2 (11) up to a phase factor. Substituting into (10) we obtain the ﬁnal form:
1 α = e− 2 α 2 ∞
αn
√ n
n!
n=0 (12) Obviously α 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 n:
∞
∞
αn
αn
†
√ n =
(a† )n 0 = eαa 0
n!
n=0
n=0 n! which implies
1 α = e− 2 α 2 +αa† 0 = eαa
3 † −α a 0 (13) 3. Orthogonality and completeness relations
We proceed to calculate the overlap between the coherent states using (12).
αβ = n 1 αnβ = e− 2 α 2 − 1 β 2
2
n (α β )n
n! 1
1
= exp (− α2 − β 2 + α β )
2
2 (14) and similarly
1
1
β α = exp (− α2 − β 2 + β α)
2
2 (15) so
αβ 2 = αβ β α = exp (−α2 − β 2 + α β + αβ )
or
αβ 2 = e−α−β  2 . (16) Since αβ = 0 for α = β , we say that the set {α} is overcomplete. There is
still, however, a closure relation:
2 d α αα = 2 (α )n αm
√
mn
n! m!
m,n
−  α 2 d αe (17) where the measure d2 α means ”summing” over all complex values of α, i.e.
integrating over the whole complex plane. Now, writing α in polar form:
α = reiφ ⇒ d2 α = dφ dr r (18) we get
2 −  α 2 d αe n m (α ) α = ∞
0 −r 2 m+n dr re r 2π
0 dφ ei(m−n)φ 1 ∞ 2 2 m −r 2
= 2πδm,n
dr (r ) e
= π m!δm,n
20 Using this we ﬁnally get:
d2 α αα = π
n nn = π or equivalently
d2 α
αα = 1
π
4 (19) 4. Coherent states in the xrepresentation
Remember that x 0 is a minimal gaussian wave packet with x = p =
0. Since x α is also a minimal wave packet and
a + a†
mω
(α) = α
α =
αxα
2
2¯
h
a − a†
1
(α) = α
α = √
αpα
2i
2mωh
¯ (20)
(21) or
√
h
¯
xα = mω 2(α)
√
√ p = mω h 2(α)
¯
α (22) it is natural to expect that x α 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
√ mω
ip
1
1
2
†
2
h
x α = x e− 2 α +αa 0 = e− 2 α x eα 2¯ (x− mω ) 0
Acting from the left with x  gives us 1 :
√ mω i
1
2
hd
h
x α = e− 2 α eα 2¯ (x − mω (−i¯ dx )) x 0
1 α
√ ( x − x2 d )
0 dx
2 2 = N e− 2 α e x0 x
− 1 ( x )2
2 e 0 (23) where we have used the explicit form of x 0 and introduced the constants: h
¯
mω 4 x0 = mω
π¯
h N= (24) For notational simplicity we also put y = x /x0 . With these substitutions
equation (23) will look like:
1 2 x α = N e− 2 α e α
√ (y − d )
dy
2 1 2 e− 2 y (25) Using the commutator relation
1 eA+B = e− 2 [A,B ] eA eB
1 c.f. Sakurai p.93 5 (26) which is valid if both A and B commute with [A, B ], we get
e α
√ (y − d )
dy
2 e d
a dy and thus, noting that e
1 x α = N e− 2 α − 1 y 2
2 − =e α2
α
+ √ y
4
2 α
−√ e d
2 dy 1 2 e− 2 y (27) , is a translation operator 2 − 1 y 2 − 1 α2 +
2
2 √ 2α y √
√
1
= N exp (− (y − 2(α))2 + i 2(α)y − i(α)(α)) (28)
2 Using (22) the resulting expression for the wavefunction of the coherent state
is
mω h
x α = N e− 2¯ (x −xα ) 2 + i p x − i p x
α
α
α
h
¯
2¯
h (29) and since the last term is a constant phase it can be ignored and we ﬁnally get
ψα (x ) = ( mω 1/4 i pα x − mω (x −xα )2
¯
2¯
h
) eh
πh
¯ , (30) 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:
i i 1 ¯
¯
α, t = U (t, 0)α(0) = e− h Ht α(0) = e− h Ht e− 2 α(0) 2
n (α(0))n
√
n
n! (31) But the n:s are eigenstates of the hamiltonian so:
1 α, t = e− 2 α(0) 2
n (α(0))n − i ω¯ (n+ 1 )t (a† )n
¯
√
e h h 2 √ 0
n!
n! (32) which is the same as
i
− 1 α(0)2 − 2 ω t
2 α, t = e e
n (α(0)e−iωt a† )n
0 =
n! 1
i
exp (− α(0)2 − ω t + α(0)e−iωt a† )0
2
2
2 c.f. Sakurai chapter 2.1 6 (33) 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−iωt α(0) while the second term only
will contribute with a phase factor. Thus we have:
i α, t = e 2 ωt e−iωt α(0) = α(t) (34) So the coherent state remains coherent under time evolution. Furthermore,
d
α(t) = −iωα(t)
dt α(t) = e−iωt α(0) ⇒ (35) or in components
d
(α)
dt
d
(α)
dt = ω (α)
= −ω (α) (36) x(t) = α(t)xα, t
p(t) = α(t)pα, t (37) Deﬁning the expectation values,
we get d x(t)
dt d p(t)
dt =
= (t
h
¯
h
¯
2 d (α) = 2mω 2ω (α) = pm)
2
mω dt
h
¯
d
h
¯
i m2 ω (−2i) dt (α) = − m2 ω 2ω (α) = −mω 2 x(t) (38) or in a more familiar form
d
p(t) = m dt x(t) = mv (t)
d
p(t) = −mω 2 x(t)
dt (39) 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. 7 ...
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This note was uploaded on 03/07/2012 for the course ECON 111 taught by Professor Blank during the Spring '12 term at Alabama A&M University.
 Spring '12
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