8.514: Many-body phenomena in condensed matter and atomic physics Last modi ed: September 24, 2003
1 Lecture 1. Coherent States.
We start the course with the discussion of coherent states. These states are of interest
because they provide
a method to describe on equal terms both particles and photons
a connection to classical physics (mechanics and electrodynamics)
tools for the construction of path integral, to be discussed later
The coherent states also provide a natural entry point into the method of second
quantization that will be introduced in the next lecture.
1.1 Harmonic oscillator the creation and annihilation operators
Particle in a parabolic potential:
p2 + m!2 q2 p = ;ih@ q p] = ih
H=
(1)
q
2m
2
The ground state width can be found by minimizing energy:
h2 + m!22 ! min
hHi =
(2)
2m2
2
which gives = (h=m!)1=2 .
It will be convenient to use nondimensionalized variables q = q~, p = (h=)p~, so that
the classical phase volume is rescaled by h. Thus we obtain
h! p~2 + q~2 p~ = ;i@ q~ p~] = i
H=
(3)
q~
2
We shall study the Hamiltonian (3) below having in mind the quantum-mechanical particle
problem. However, later we shall nd that the quantized electromagnetic eld is also
described by a set of harmonic oscillators of the form (3).
The canonical creation and annihilation operators are dened as
a = p1 (q~ + ip~) a+ = p1 (q~ ; ip~)
(4)
2
2
They can be used to express q, p and H as follows:
q = p a + a+ p = i ph a+ ; a
(5)
2
2
h! a+a + aa+ = h! a+a + 1 H=
(6)
2
2
1
The operators a and a+ obey the commutation relation
Proof:
a a+] = 1
(7)
aa+ ; a+a = 12 ((q~ + ip~)(q~ ; ip~) ; (q~ ; ip~)(q~ + ip~)) = i (p~q~ ; q~p~) = 1
(8)
q0 (q) + 0 (q) = 0 0 =0 = ;q ln 0 = ;q2 =2
(10)
As a simple application of the operators a and a+, let us reconstruct the main facts
of the harmonic oscillator quantum mechanics.
1. The ground state j0i, also called vacuum state, provides the lowest possible energy
expectation value
1
1
+
h0 jHj0 i = h! h0 ja aj0 i +
= h! ha0 ja0 i +
(9)
2
2
which gives the condition a0 = 0, i.e., (q + ip)0 = 0.
Let us nd the ground (vacuum) state in the q-representation. Using the units with
the length = 1, i.e., q~, p~ instead of q, p, we write
0
0
This leads to a Gaussian wavefunction
0 (q) = 1=4
;
exp
;q 2 =2
E0 = h!=2
(11)
2. The higher energy states can be obtained from the ground state. Starting with the
commutation relations,
a+H = (H ; h!) a+ aH = (H + h!) a
(12)
one can show that the states n (q) = (a+)n0 (q) are the eigenstates. Indeed, consider
1 = a+0 and apply the rst relation (12):
(H ; h!) 1 = a+H0 = E0a+ 0 = E01
(13)
which gives E1 = E0 + h! = 3h!=2 and
1 (q) = p1 (q ; @q ) 0 = p2 q exp ;q2 =2
(14)
2h
2h
Subsequently, from 1 one obtains the eigenstate 2 (q) / (2q2 ; 1) exp (;q2 =2) with
the energy E2 = E1 + h! = 5h!=2, and so on. The recursion relation n = a+n 1,
En = En 1 + h!, gives
n = An (a+)n0 En = h! n + 12
(15)
;
;
2
where we inserted the normalization factors An.
The factors An can be determined from
1 = A2n h(a+)n0 j(a+)n0 i = A2nh0ja:::aa+ :::a+ j0i
(16)
H
= A2nh0 jan 1 h!
+ 12 (a+ )n 1j0i
(17)
= A2nnh0 jan 1(a+)n 1j0 i = ::: = Ann!h0 j0 i = Ann!
(18)
which gives An = (n!) 1=2 . The normalized oscillator eigenstates
1
jni = p (a+ )n j0i j0i = 0
(19)
n!
form an orthonormal complete set of functions, providing a basis in the oscillator Hilbert
space. The ground state j0i is also known as the vacuum state.
3. The operators a and a+ written as matrices in the basis of states (19) have nonzero
matrix elements only between the states jni and jn 1i:
p
p
hnja+ jmi = nnm+1 hmjajni = nnm+1
(20)
while all other matrix elements are zero.
4. It is convenient to dene the so-called number operator n^ = a+ a which counts
the number of energy quanta in the QM particle problem, or the number of photons for
quantized E&M eld. In the energy basis jni, the number operator is diagonal:
1
+
n^jni = a ajni = njni H = h! n^ + 2
(21)
;
;
;
;
;
1.2 Denition of coherent states.
The coherent states are dened as eigenstates of the operator a:
ajvi = vjvi
(22)
where v is a complex parameter. Expanded in the energy basis (19), jvi = Pn=0 cnjni,
the coherent state can be reconstructed from
1
ajvi =
X
1
n=0
p
cn njn ; 1i =
X
1
n=0
vcnjni
(23)
p
Comparing the coecients, obtain a recursion relation cn = (v= n)cn 1, leading to
n
v
p
cn =
c
(24)
n! 0
The coecient c0 is determined from normalization
X
X 2n
1 = jcnj2 = jvnj! jc0j2 = e v 2 jc0j2
(25)
n=0
n=0
;
1
1
j j
3
Finally,
X vn
2
2
+
p jni = e v =2 eva j0i
jv i = e v =2
n=0 n!
1
;j j
(26)
;j j
As an example, consider the distribution of the number of quanta n^ = a+a in a
coherent state. Since n^ jni = njni, the distribution is given by
2n
pn = jcnj2 = e v 2 jvnj!
(27)
This is a Poisson distribution with the mean n = jvj2.
;j j
1.3 The quasiclassical interpretation of coherent states
As we shall see below, the coherent states represent the points of the classical phase space
(q p). This can be conjectured most easily from their time dependence. Applying the
Schrodinger equation i@t = H to the number states, we have
1
jni(t) = e i(n+ 2 )!t jni
(28)
;
for the number states. Combined with (26), this gives
n
2 =2 X v
v
p
jv i(t) = e
(29)
e i(n+ 12 )!t jni = e i!t=2 jv(t)i
n=0 n!
with
(30)
v(t) = e i!t v
This denes a circular trajectory in the complex v plane, suggesting the correspondence
with classical coordinate and momentum,
1
;j j
;
;
;
q = c v p = c v v = v + iv
(31)
where c is a scaling factor. The relation of coherent states with the points in a classical
phase space will be claried below.
Let us nd the form of a coherent state in the q-representation, v (q) = hqjvi. As
before, we use the units in which the length = 1, and write
vv (q) = hqjajvi = p1 hqj (q + @q ) jvi = p1 (q + @q ) v (q)
(32)
2
2
p
Solving the equation q + = 2v, obtain
p
ln = ;q2 =2 + v~q + const: v~ = 2v
(33)
and, nally,
v (q) = A exp ; 21 (q ; v~)2 jAj = 1=4 e (v~ )2 =2 = 1=4 e (v )2
(34)
0
00
0
00
0
;
4
;
00
;
;
00
p
with v~ = 2v . The probability jv (q)j2 has a form of a gaussian centered at q = Re(~v),
which agrees with the above interpretation
of v as a point in the phase space (with the
p
scaling factor taking value c = 2).
A more detailed picture of the phase-space density is provided by the Wigner distribution function
Z
W (q p) = 21h hq + 12 xj ^jq ; 12 xieixp=hdx
(35)
where ^ is the density matrix. For a pure state (q), the density matrix in position space
is just ^qq = (q)(q ), and the matrix element in (35) is
1
1
h:::i = (q ; x) (q + x)
(36)
2
2
The interpretation of the Wigner function as a phase-space density is supported by the
following observations. One can check that the function (35) is real and normalized to
unity. Also, the coordinate and momentum distributions, obtained by integrating over
the conjugate variable, are reproduced correctly. The distribution in q is
00
00
1
;1
0
0
Z
W (q p)dp = hqj ^jqi
(37)
which is equal to j(q)j2 for a pure state, while the distribution in p is
Z
W (q p)dq = ::: = 21h j(p)j2
R
where (p) = (q)e iqpdq.
For a coherent state jvi, the Wigner function is given by
(38)
;
2 Z
W (q p) = j2Ajh
(39)
e 12 (q+ 12 x v~)2 e 12 (q 12 x v~)2 eixpdx
jAj2 Z
= 2h
e (q v~ )2 eixp 21 ( 12 x+iv~ )2 dx = 21h e (q v )2 (p v~ )2
(40)
p
p
with v~ = 2v, v~ = 2v. The gaussian distribution, centered at q = v~ , p = v~ , evolves
in time as if carried by the classical harmonic oscillator phase ow. Since v(t) = e i!t v,
the center of the gaussian packet is circling around the phase space origin:
W (q p t) = 21h e (q v~ cos !t)2 (p v~ sin !t)2
(41)
For any jvi, the width of the Wigner distribution is the same as for the vacuum state
j0i. Thus one can conclude that a coherent state can be thought of as a displaced vacuum
state. This interpretation will be substantiated in Problem 2, PS#1.
1
;
;
;
;
;
;1
1
;
;
0
00
;
;
;
0
;
;
00
;1
0
00
0
00
;
;
;j j
5
;
;j j
1.4 Coherent states vector algebra
Here we discuss the the vector space propeties of coeherent states. Normally, the states
appearing in quantum mechanics are orthogonal, or can be made orthogonal in some
natural way, which provides an orthonormal basis in Hilbert space. The situation with
coherent states is quite dierent.
Let us start with evaluating the overlap:
n
1
1 2 X (uv )
2
hujv i = e 2 u e 2 v
(42)
= e 21 u 2 21 v 2 +uv
n=0 n!
which shows that the coherent states are not orthogonal. On the other hand, Eq.(42)
gives overlap decreasing exponentially as a function of the distance between u and v in
the complex plane:
2
jhujv ij2 = e u v
(43)
For generic classical states, juj jvj 1, the overlap is very small, which is consistent with
the intuition that dierent classical states are orthogonal in the quantum mechanical sense.
Recalling the interpretation of the complex v plane as a phase space, q~ = v , p~ = v , we
see that the overlap falls to zero at the length scale of the order of the
p wavepacket width
p
set by Planck's constant, i.e. by the uncertainty relation, q / h, p h= / h.
Another property of coherent states is completeness in the vector algebra sense. (A set
of vectors is called complete if linear combinations of these vectors span the entire vector
space.) The property is seen most readily from the formula know as unity decomposition:
Z
d2v = ^1
jv ihv j
(44)
Proof can be obtained by evaluating the matrix elements of the operator on the left hand
side of Eq. (44) between the number states
Z
Z Z
2 E
m n 2
m+n i(n m) rdrd
D Z
m jvihvj dv n = e v 2 pv v dv =
e r 2 r pe
(45)
0
m!n!
m!n!
Z
2n
2r
r
2
= mn e
dr
(46)
n! = mn
0
(we used polar coordinates v = rei ).
Using the formula (44), one can express any operator in terms of coherent states:
ZZ
2 2
u
M^ = ^1M^ ^1 = juihvj M (u v) d vd
(47)
2
with the matrix elements M (u v) = hujM^ jvi. This formula can be useful in calculations,
as well as in formal manipulations (we shall use it later to derive Feynman path integral).
As another application of Eq. (44), let show that the coherent states form an overcomplete set, i.e. they are not linearly independent. Indeed, by writing
Z
Z
Z
2
2
2
1 u 2 1 v 2 +uv d u
1 u v2d u
d
u
uv
v
u
^
2
2
2
jv i = 1jv i = juihujv i
e
= jui e
= jui e
(48)
1
;
j
j
;
j j
;
;j
j
j ;
j j
; j
0
;
1
;j j
;
;
1
;
;
j
j ;
6
j j
;
;
j
; j
00
we express the state jvi as a superposition of the states jui with ju ; vj 1.
The overcompleteness (48) should not come as a surprise. The coherent states, parameterized by complex numbers, form a continuum, and thus there are way too many
of them to form an a set of independent vectors. In contrast, the number states, which
provide a basis of the oscillator Hilbert space, are a countable set.
To summarize, the coherent states are non-orthogonal and form an over-complete set.
There have been many attempts to reduce the number of these states to a `neccessary
minimum,' by identifying a good subset that could serve as a basis. Even though some of
the proposals are very interesting (e.g. Perelomov lattices1) it is probably more natural to
use the entire space of coherent states, coping with the overcompleteness and not favoring
some of the states to the others.
1.5 Coordinate and momentum uncertainty
We already mentioned, while discussing the Wigner function, that the coherent states
form wavepackets in the phase space of width corresponding to the absolute minimum
required by the uncertainty relation. Let us estimate coordinate uncertainty of a state
jui:
2
2
2 (u + u)2 + 1
2
hujq^2 jui = huj(a + a+ )2 jui = huja2 + a+ + 2a+ a + 1jui =
2
2
2
2
2
2
(hujq^jui)2 =
huja + a+ jui = (u + u)2
2
2
2 = h
hujq^2 jui = hujq^2 jui ; (hujq^jui)2 =
(49)
2 2m!
The uncertainty does not depend on u, which is consistent with the observations made
using Wigner function. Similarly, for momentum uncertainty,
(ih)2 huj(a+ ; a)2jui = (ih)2 huja+2 + a2 ; 2a+a ; 1jui = h2 1 ; (u ; u)2
hujp^2jui =
22
22
22
2
2
2
(hujp^jui)2 = (2ih)2 huja+ ; ajui = (2ih)2 (u ; u)2
h2 hm!
hujp^2jui = hujp^2 jui ; (hujp^jui)2 = 2 =
(50)
2
2
which is also independent of u. The uncertainty product hp^2i1=2 hq^2i1=2 equals 12 h, which
is the lower bound required by the uncertainty relation. Below we shall see that coherent
states can be naturally generalized to a broader class of states that minimize uncertainty
product.
One can consider lattices in the complex plane, vm n = mu1 + nu2, m n 2 Z . Perelomov shown that
the lattice fvm ng generates an undercomplete set of coherent states fjvm n ig if the area of the lattice
unit cell is greater than 2h , and an overcomplete set if the area is less than 2h. The borderline lattices,
having the unit cell area equal to 2h , are overcomplete just by one vector. After any single vector is
removed from such a lattice, it becomes a complete set.
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