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z2 2
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(219)
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22 Four Representations of Quantum States
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Viewed through the transform (218), the eigenstate n (x) is simply proportional to the nth power of z: f n (z) = K(x, z) n (x) dx z
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= z|n
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(220)
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The notation z| will be explained soon The inverse transformation for (218) reads as follows: (x) =
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K(x, z) f (z) s (dz)
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(221)
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Here s (dz) is the Gaussian measure on the plane: s (dz) = 1 |z|2 i |z|2 dx d y = dz dz , e e 2 (222)
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with z = x + i y The transform (218) maps the Hilbert space L2 (R) onto the space F B of entire analytical functions that are square-integrable with respect to s (dz):
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n z n converges absolutely for all z C ,
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that is, its convergence radius is in nite, and f
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| f (z)|2 s (dz) <
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The Hilbert space F B is known as a Fock Bargmann Hilbert space It is equipped with the scalar product
f 1| f 2 =
f 1 (z) f 2 (z) s (dz) =
n! 1n 2n
(224)
From (220) a natural orthonormal basis of F B is immediately found to be 1 f n (z) = n! z
(225)
226 Operators in Fock Bargmann Representation
The annihilation operator a is represented as a derivation, whereas its adjoint is a multiplication operator: a f (z) = d f (z) , dz z a f (z) = f (z) (226)
2 The Standard Coherent States: the Basics
d In consequence, the number operator N realizes as a dilatation (Euler), N = z dz , and the Hamiltonian becomes a rst order differential operator:
H= z
d 1 + dz 2
(227)
Position and momentum then assume a quasi-symmetric form: Q = lc d 1 + z dz , P = i p c d 1 z dz (228)
23 Schr dinger Coherent States
Equipped with the basic quantum mechanical material presented in the previous section, we are now in the position to describe the coherent states appearing in (21) We rst note that in position (like in momentum) representation, the ground state, 0 (x) =
4 x 1 4l 2 e c , 2 2 l c 2
(229)
is a Gaussian centered at the origin Then, let us ask the question: what quantum states could keep this kind of Gaussian localization in other points of the real line | (q) (x)|2 e const(x q) ,
q R
(230)
In our Fock Hilbertian framework, the question amounts to nding the expansion coef cients b n such that
| (q) =
b n |n
(231)
The answer is immediate after having a look at the Bergman kernel K(x, z)
231 Bergman Kernel as a Coherent State
Let us rst simplify our notation by putting from now on = 1, m = 1, = 1 1 l c = 2 = p c Consider again the expansion (219) of the kernel K(x, z): 1 K(x, z) = exp 4 z2 2 1 x z 2
2 +
n=0 1 (q + i 2
zn n (x) , n!
(232)
where we have noted that n = n Let us put z =
phase
p) and adopt the notation
qp 1 1 |z|2 2 K(x, z) = e 2 e ix p e i 2 e 2 (x q) = x |z = x |q, p 4 s s
(233)
23 Schr dinger Coherent States
These are the Schr dinger or nonnormalized coherent states in position representation (the index s is for Schr dinger ) In Fock representation they read as
|z = |q, p =
s s n=0
zn |n n!
(234)
We have obtained a continuous family of states, labeled by all points of the complex plane, and elements of the Hilbert space H with as orthonormal basis the set of kets |n , n N
232 A First Fundamental Property
For a given value of the labeling parameter z, the coherent state |z is an eigenvector
of the annihilation operator a, with eigenvalue z,
a|z =
s n=0
zn a|n = n!
zn n|n 1 = z|z n! s
(235)
It thus follows from this equation that (i) f (a)|z = f (z)|z for an analytical function of z (with appropriate condis s
tions), (ii) and z|a = z z|
233 Schr dinger Coherent States in the Two Other Representations
In momentum representation with variable k, coherent states |z = |q, p are ess s
sentially Gaussians centered at k:
phase
qp 1 2 1 |z|2 k |z = e 2 e ixk e i 2 e 2 (k p) 4 s
(236) =
1 (x 2
In Fock Bargmann representation with variable
+ ik) C, one gets (237)
|z =
s n=0
zn n!
|n =
1 i zn = e z = e 2 (xq+k p) e 2 (x p qk) n! n!
One thus obtains a Gaussian depending on z multiplied by a phase factor involving the form Iz = 1 (x p qk) = z After multiplication by Gaussian factors 2 present in the measures s (dz) and s (d ), one gets a Gaussian localization in the complex plane: e