Quantum Mechanics by Konstantin K. Likharev - HTML preview
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QM: Quantum Mechanics
1 k
U
k
k
1
1
U
ˆ p , U ( ˆ x, t)
ik x
i
x
(5.32b)
x
ˆ
ˆ k 1
k
.
k
k x
k
k
k
1
! ˆ
1
(
)!
1
ˆ x
But the last sum is just the Taylor expansion of the derivative U/ x. Indeed,
U
1
k'
k
U
U
U
k
k
k
'
1
' 1
'
1
ˆ x
ˆ x
x
(5.33)
k'
ˆ k 1
k
,
ˆ
x
k
k' x x
k
k x
k
k
k
'
0
! ˆ
ˆ
'0
'! ˆ ' 1
1
(
)!
1
ˆ x
where at the last step, the summation index was changed from k’ to k – 1. As a result, we may recast Eq.
(5.32b) as
ˆ p , U( ˆ x, t)
,
(5.34)
x
i U(ˆ x, t)
ˆ x
so Eq. (30) yields:
Heisenberg
ˆ p
d
equation
x
U ( ˆ x, t).
(5.35)
for
dt
ˆ x
momentum
This equation also coincides with the classical equation of motion! Moreover, averaging Eqs. (29) and (35) over the initial state (as Eq. (4.191) prescribes), we get similar results for the expectation values:14
Ehrenfest
d x
p
d p
x
x
U
theorem
,
.
(5.36)
dt
m
dt
x
However, it is important to remember that the similarity of these quantum-mechanical equations and their classical mechanics analogs is superficial, and the degree of difference between the two mechanics very much depends on the problem. As one extreme, let us consider the case when a particle’s state, at any moment between t 0 and t, may be accurately represented by one, relatively px-
narrow wave packet. Then we may interpret Eqs. (36) as the equations of the essentially classical motion of the wave packet’s center, and consider this fact as a manifestation of the correspondence principle.
However, even in this case, it is important to remember the purely quantum mechanical effects of nonzero wave packet broadening and its spread in time, which were discussed in Sec. 2.2.
As an opposite extreme, let us revisit the “leaky” potential well discussed in Sec. 2.5 – see Fig.
2.15. Since both the potential U( x) and the initial wavefunction of that system are symmetric relative to point x = 0 at all times, the right-hand sides of both Eqs. (36) identically equal zero, and hence they predict that the average values of the coordinate and the momentum stay equal to zero at all times. Of course, this prediction is correct, but it does not tell us much about the rich dynamics of the system: the finite lifetime of the metastable state, the formation of two wave packets, their waveform and propagation speed (see Fig. 2.17), and about the insights the full solution gives for the quantum measurement theory and the system’s irreversibility. Another similar example is the energy band theory (Sec. 2.7), with its purely quantum effect of the allowed energy bands and forbidden energy gaps, of which Eqs. (36) give no clue.
To summarize, the Ehrenfest theorem is useful as an illustration of the correspondence principle and as the sanity check of quantum-mechanical calculation results, but its predictive power should not be exaggerated.
14 The equation set (36) constitutes the Ehrenfest theorem, named after its author, Paul Ehrenfest.
Chapter 5
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QM: Quantum Mechanics
5.3. The Feynman path integral
As has been already mentioned, even within the realm of wave mechanics, the bra-ket language may simplify some calculations that would be very bulky using the notation used in Chapters 1-3.
Probably the best example is the famous alternative, path-integral formulation of quantum mechanics.15
I will review this important concept, cutting one math corner for the sake of brevity.16 (This shortcut will be clearly marked below.)
Let us inner-multiply both parts of Eq. (4.157a), which is essentially the definition of the time-evolution operator, by the bra-vector of state x,
x ( t) x ˆ u ( t, t ) ( t ) , (5.37)
0
0
insert the identity operator before the ket-vector on the right-hand side, and then use the closure condition in the form of Eq. (4.252), with x’ replaced with x 0: x ( t) dx x ˆ u ( t, t ) x x ( t ) .
(5.38)
0
0
0
0
0
According to Eq. (4.233), this equality may be represented as
( x, t)
dx x ˆ u( t, t ) x ( x , t ).
(5.39)
0
0
0
0
0
Comparing this expression with Eq. (2.44), we see that the long bracket in this relation is nothing other than the 1D propagator that was discussed in Sec. 2.2, i.e.
G( x, t; x , t ) x ˆ u ( t, t ) x .
(5.40)
0
0
0
0
Let me hope that the reader sees that this equality corresponds to the physical sense of the propagator.
Now let us break the time segment [ t 0, t] into N (for the time being, not necessarily equal) parts by inserting ( N – 1) intermediate points (Fig. 4) with
t t t t
t ,
(5.41)
0
1
k
N 1
and use the definition (4.157) of the time evolution operator to write
ˆ u ( t, t ) ˆ u ( t, t
) ˆ u ( t
, t
)
.
(5.42)
ˆ u ( t , t ) ˆ u ( t , t )
0
N 1
N 1
N 2
2
1
1
0
x
x
x
x
x
x
0
1
k
N 2
N 1
Fig. 5.4. Time partition and coordinate
notation at the initial stage of the
Feynman path integral’s derivation.
t
t
...
t
...
t
t
t
0
1
k
N 2
N 1
15 This formulation was developed in 1948 by Richard P. Feynman. (According to his memories, this work was motivated by a “mysterious” remark by P. Dirac in his pioneering 1930 textbook on quantum mechanics.) 16 A more thorough discussion of the path-integral approach may be found in the famous text by R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, first published in 1965. (For its latest edition by Dover in 2010, the book was emended by D. Styler.) For a more recent monograph, which reviews more applications, see L. Schulman, Techniques and Applications of Path Integration, Wiley, 1981.
Chapter 5
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After plugging Eq. (42) into Eq. (40), let us insert the identity operator, again in the closure form (4.252), but written for xk rather than x’, between each two partial evolution operators including the time argument tk. The result is
G( x, t; x t ) dx
dx
(5.43)
dx
x ˆ u ( t, t
) x
x
ˆ u ( t
, t
) x
x ˆ u ( t , t ) x .
0, 0
N 1
N 2
1
N 1
N 1
N 1
N 1
N 2
N 2
1
1
0
0
The physical sense of each integration variable xk is the wavefunction’s argument at time tk – see Fig. 4.
The key Feynman’s step was the realization that if all intervals are taken similar and sufficiently small, tk – tk- 1 = d → 0, all the partial brackets participating in Eq. (43) may be expressed via the free-particle’s propagator given by Eq. (2.49), even if the particle is not free, but moves in a stationary potential profile U( x). To show that, let us use either Eq. (4.175) or Eq. (4.181), which, for a small time interval d, give the same result:
i
i ˆ 2
p
ˆ
ˆ
u (
d , ) exp
d
H
exp
d U ˆ x
d
.
(5.44)
2
m
Generally, an exponent of a sum of two operators may be treated as that of c-number arguments, and in particular factored into a product of two exponents, only if the operators commute. (In that case, we can use all the standard algebra for the exponents of c-number arguments.) In our case, this is not so because the operator p ˆ2 / 2 m does not commute with x ˆ , and hence with U( x ˆ ). However, it may be shown17 that for an infinitesimal time interval d, the non-zero commutator
ˆ 2
p
d , U ( ˆ x) d ,
0
(5.45)
2
m
proportional to ( d)2, may be ignored in the first, linear approximation in d. As a result, we may factor the right-hand side in Eq. (44) by writing
i p ˆ2
i
u ˆ
( d , )
exp
exp
( ˆ
d 0
d
U x) d .
(5.46)
2 m
(This approximation is very much similar in spirit to the trapezoidal-rule approximation in the usual 1D
integration,18 which is also asymptotically impeachable.)
Since the second exponential function on the right-hand side of Eq. (46) commutes with the coordinate operator, we may move it out of each partial bracket participating in Eq. (43), with U( x) turning into a c-number function:
i ˆ 2
p
i
x
ˆ u(
d , ) x x
exp
(5.47)
d
d
d
x
U x d
exp
( )
.
2
m
But the remaining bracket is just the propagator of a free particle, so for it, we may use Eq. (2.49): 17 This is exactly the corner I am going to cut because a strict mathematical proof of this (intuitively evident) statement would take more time/space than I can afford.
18 See, e.g., MA Eq. (5.2).
Chapter 5
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QM: Quantum Mechanics
1/ 2
i p ˆ2
m
m( dx)2
x exp
exp
.
(5.48)
d
d
x
i
2 m
2 i
d
2
d
As the result, the full propagator (43) takes the form
N / 2
N
2
m
m( dx)
U ( x)
G( x, t; x t ) lim
..
exp
. (5.49)
0, 0
d 0 dx
dx
dx
i
i
d
N 1
N 2
1
2 i d
k 1
2 d
t
k
N
At N and hence d ( t – t 0)/ N 0, the sum under the exponent in this expression may be approximated with the corresponding integral:
N
i m dx 2
t
i m dx 2
U( x) d U( x) d, (5.50)
k 1
2 d
2 d
t
t
0
k
and the expression in the square brackets is just the particle’s Lagrangian function L.19 The integral of this function over time is the classical action S calculated along a particular “path” x().20 As a result, defining the (1D) path integral as
N / 2
m
1D path
[
D x( )] lim d0
dx dx
dx
(5.51a) integral:
N 1
N 2
1
,
N 2 i d
definition
we can bring our result to the following (superficially simple) form:
1D
i
G( x, t; x t ) exp S x D
x .
(5.51b) propagator
0, 0
( ) [ ( )]
via path
integral
The name “path integral” for the mathematical construct (51a) may be readily explained if we keep the number N of time intervals large but still finite, and also approximate each of the enclosed integrals with a sum over M >> 1 discrete points along the coordinate axis – see Fig. 5a.
d
(a)
(b)
x
x
M
Fig. 5.5. Several 1D classical
x
x
paths: (a) in the discrete
0
0
approximation and (b) in the
t
t
t
continuous limit.
0
0
t
N 1
Then the path integral (51a) is the product of ( N – 1) sums corresponding to different values of time , each of them with M terms, each of those representing the function under the integral at a particular spatial point. Multiplying those ( N – 1) sums, we get a sum of ( N – 1) M terms, each 19 See, e.g., CM Sec. 2.1.
20 See, e.g., CM Sec. 10.3.
Chapter 5
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evaluating the function at a specific spatial-temporal point [ x, ]. These terms may be now grouped to represent all possible different continuous classical paths x[ ] from the initial point [ x 0, t 0] to the finite point [ x, t]. It is evident that the last interpretation remains true even in the continuous limit N, M
(see Fig. 5b).
Why does such a path representation of the sum make sense? This is because in the classical limit, the particle follows just a certain path, corresponding to the minimum of the action S cl. As a result, for all close trajectories, the difference ( S – S cl) is proportional to the square of the deviation from the classical trajectory. Hence, for a quasiclassical motion, with S cl >> , there is a bunch of close trajectories, with ( S – S cl) << , that give substantial contributions to the path integral. On the other hand, strongly non-classical trajectories, with ( S – S cl) >> , give phases S/ rapidly oscillating from one trajectory to the next one, and their contributions to the path integral are averaged out.21 As a result, for a quasi-classical motion, the propagator’s exponent may be evaluated on the classical path only: t
2
i
i m dx
G exp S exp U( x) d .
(5.52)
cl
cl
d
t 2
0
The sum of the kinetic and potential energies is the full energy E of the particle, which remains constant for motion in a stationary potential U( x), so we may rewrite the expression under the last integral as22
2
2
m dx
dx
dx
U ( x)
d m
E
d m
dx
Ed .
(5.53)
2
d
d
d
With this replacement, Eq. (52) yields
i x dx
i
i x
i
G exp m dxexp E( t t ) exp
p( x)
dxexp E( t t ) , (5.54)
cl
0
0
d
x
x
0
0
where p is the classical momentum of the particle. But (at least, leaving the pre-exponential factor alone) this is the WKB approximation result that was derived and studied in detail in Chapter 2!
One may question the value of such a complicated calculation, which yields results that could be readily obtained from Schrödinger’s wave mechanics. Feynman’s approach is indeed not used too often, but it has its merits. First, it has an important philosophical (and hence heuristic) value. Indeed, Eq. (51) may be interpreted by saying that the essence of quantum mechanics is the exploration, by the system, of all possible paths x(), each of them classical-like, in the sense that the particle’s coordinate x and velocity dx/ d are exactly defined simultaneously at each point. The resulting contributions to the path integral are added up coherently to form the actual propagator G, and via it, the final probability W
G2 of the particle’s propagation from [ x 0, t 0] to [ x, t]. As the scale of the action S of the motion 21 This fact may be proved by expanding the difference ( S – S cl) in the Taylor series in the path variation (leaving only the leading quadratic terms) and working out the resulting Gaussian integrals. This integration, together with the pre-exponential coefficient in Eq. (51a), gives exactly the pre-exponential factor that we have already found refining the WKB approximation in Sec. 2.4.
22 The same trick is often used in analytical classical mechanics – say, for proving the Hamilton principle, and for the derivation of the Hamilton-Jacobi equations – see, e.g., CM Secs. 10.3-4.
Chapter 5
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QM: Quantum Mechanics
decreases and becomes comparable to , more and more paths produce substantial contributions to this sum, and hence to W, providing a larger and larger difference between the quantum and classical properties of the system.
Second, the path integral provides a justification for some simple explanations of quantum phenomena. A typical example is the quantum interference effects discussed in Sec. 3.1 – see, e.g., Fig.
3.1 and the corresponding text. In that discussion, we used the Huygens principle to argue that at the two-slit interference, the WKB approximation might be restricted to contributions from two paths that pass through different slits, but otherwise consist of straight-line segments. To have another look at that assumption, let us generalize the path integral to multi-dimensional geometries. Fortunately, the simple structure of Eq. (51b) makes such generalization virtually evident:
t
t
2
3D
i
dr
m dr
G(r, t; r t ) exp S r D r
S
propagator
L r
d
U r d (5.55)
0, 0
( ) [ ( )],
,
( )
.
as a path
d
d
t
t 2
integral
0
0
where the definition (51a) of the path integral should be also modified correspondingly. (I will not go into these technical details.) For the Young-type experiment (Fig. 3.1), where a classical particle could reach the detector only after passing through one of the slits, the classical paths dominating the contribution from each slit are the straight-line segments shown in Fig. 3.1, and if they are much longer than the de Broglie wavelength, the propagator may be well approximated by the sum of two integrals of L d = ip(r) dr/ – just as this was done in Sec. 3.1.
Last but not least, the path integral allows simple solutions to some problems that would be hard to obtain by other methods. As the simplest example, let us consider the problem of tunneling in multidimensional space, sketched in Fig. 6 for the 2D case – just for the graphics’ simplicity. Here, the potential profile U( x, y) has a saddle-like shape. (Another helpful image is a mountain path between two summits, in Fig. 6 located on the top and at the bottom of the shown region.) A particle of energy E may move classically in the left and right regions with U( x, y) < E, but if E is not sufficiently high, it can pass from one of these regions to another one only via the quantum-mechanical tunneling under the pass. Let us calculate the transparency of this potential barrier in the WKB approximation, ignoring the possible pre-exponential factor. 23
y
U E
U E
1
U E
1
2
r
Fig. 5.6. A saddle-type 2D
r
x
0
potential profile and the instanton
trajectory of a particle of energy
U E
E (schematically).
2
U
U
E
E
23 Actually, one can argue that the pre-exponential factor should be close to 1, just like in Eq. (2.117), especially if the potential is smooth, in the sense of Eq. (2.107), in all spatial directions. (Let me remind the reader that for most practical applications of quantum tunneling, the pre-exponential factor is of relatively minor importance.) Chapter 5
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According to the evident multi-dimensional generalization Eq. (54), for the classically forbidden region, where E < U( x, y), and hence p(r)/ = i(r), the contributions to the propagator (55) are proportional to
r
i
e I exp E( t t )
I
d ,
(5.56)
0 ,
where
κ(r) r
r0
where may be calculated just in the 1D case – cf. Eq. (2.97):
2
2
(r)
U (r) E .
(5.57)
2 m
Hence the path integral in this region is much simpler than in the classically allowed region because the spatial exponents are purely real and there is no complex interference between them. Due to the minus sign before I in the exponent (56), the largest contribution to G evidently comes from the trajectory (or a narrow bundle of close trajectories) for which the integral I has the smallest value, so the barrier transparency may be calculated as
3D
r
tunneling
2
2 I
in WKB
T G
e
exp 2 κ r
( ' )
r
d ' ,
(5.58)
limit
r
0
where r and r0 are certain points on the opposite classical turning-point surfaces: U(r) = U(r0) = E – see Fig. 6.
Thus the barrier transparency problem is reduced to finding the trajectory (including the points r and r0) that connects these two surfaces and minimizes the functional I. This is of course a well-known problem of the calculus of variations,24 but it is interesting that the path integral provides a simple alternative way of solving it. Let us consider an auxiliary problem of particle’s motion in the potential profile U inv(r) that is inverted relative to the particle’s energy E, i.e. is defined by the following equality: U (r) E E U (r).
(5.59)
inv
As was discussed above, at fixed energy E, the path integral for the WKB motion in the classically allowed region of potential U inv( x, y) (that coincides with the classically forbidden region of the original problem) is dominated by the classical trajectory corresponding to the minimum of r
r
S p (r ' ) dr ' k (r ' ) dr, (5.60)
inv
inv
inv
0
r
0
r
where kinv should be determined from the WKB relation
2
2
k (r)
inv
E U (r).
(5.61)
2
inv
m
But comparing Eqs. (57), (59), and (61), we see that kinv = κ at each point! This means that in the WKB
limit, the tunneling path corresponds to the classical (so-called instanton 25) trajectory of the same 24 For a concise introduction to the field see, e.g., I. Gelfand and S. Fomin, Calculus of Variations, Dover, 2000, or L. Elsgolc, Calculus of Variations, Dover, 2007.
25 In the quantum field theory, the instanton concept may be formulated somewhat differently and has more complex applications – see, e.g. R. Rajaraman, Solitons and Instantons, North-Holland, 1987.
Chapter 5
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particle moving in the inverted potential U inv(r). If the initial point r0 is fixed, this trajectory may be readily found by means of classical mechanics. (Note that the initial kinetic energy, and hence the initial velocity of the instanton launched from point r0 should be zero because by the classical turning point definition, U inv(r0) = U(r0) = E.) Thus the problem is further reduced to a simpler task of maximizing the transparency (58) by choosing the optimal position of r0 on the equipotential surface U(r0) = E – see Fig. 6. Moreover, for many symmetric potentials, the position of this point may be readily guessed even without calculations – as it is in Problems 6 and 7, left for the reader’s exercise.
Note that besides the calculation of the potential barrier’s transparency, the instanton trajectory has one more important implication: the so-called traversal time t of the classical motion along it, from the point r0 to the point r, in the inverted potential defined by Eq. (59), plays the role of the most important (though not the only one) time scale of the particle’s tunneling under the barrier.26
5.4. Revisiting harmonic oscillator
Now let us return to the 1D harmonic oscillator, which may be understood as any system, regardless of its physical nature, described by the Hamiltonian (4.237) with the potential energy (2.111): ˆ 2
2
p
m ˆ 2
x
Harmonic
ˆ
0
H
.
(5.62) oscillator:
2 m
2
Hamiltonian
In Sec. 2.9 we have used a “brute-force” (wave-mechanics) approach to analyze the eigenfunctions
n( x) and eigenvalues En of this Hamiltonian, and found that, unfortunately, this approach required relatively complex mathematics, which does not enable an easy calculation of the key characteristics of the system. Fortunately, the bra-ket formalism helps to make such calculations.
First, by introducing the normalized (dimensionless) operators of coordinates and momentum:27
ˆ x
ˆ p
ˆ
ˆ
,
,
(5.63)
x
m x
0
0 0
where x 0 (/ m0)1/2 is the natural coordinate scale discussed in detail in Sec. 2.9, we can represent the Hamiltonian (62) in a very simple and x p symmetric form:
0
ˆ
H
2 2
ˆ
ˆ
.
(5.64)
2
This symmetry, as well as our discussion of the very similar coordinate and momentum representations in Sec. 4.7, hints that much may be gained by treating the operators
ˆ
and
ôn equal footing. Inspired
by this clue, let us introduce a new operator
ˆ
ˆ
i
m
ˆ p
Annihilation
0 1/ 2
ˆ a
ˆ x i
.
(5.65a) operator:
2
2
m
definition
0
26 For more on this interesting issue see, e.g., M. Buttiker and R. Landauer, Phys. Rev. Lett. 49, 1739 (1982), and references therein.
27 This normalization is not really necessary, it just makes the following calculations less bulky – and thus more aesthetically appealing.
Chapter 5
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Since both operators
ˆ
and
ˆ
correspond to real observables, i.e. have real eigenvalues and hence are
Hermitian (self-adjoint), the Hermitian conjugate of the operator a îs simply its complex conjugate: Creation
ˆ
ˆ
i
m
ˆ p
0 1/ 2
operator:
ˆ†
a
definition
ˆ x i
.
(5.65b)
2
2
m 0
Because of the reason that will be clear very soon, a ˆ† and
a
ˆ (in this order!) are called the creation and
annihilation operators.
Now solving the simple system of two linear equations (65) for
ˆ
and
ˆ
, we get the following
reciprocal relations:
1/ 2
a ˆ a ˆ†
a ˆ a ˆ†
a ˆ a ˆ†
1/ 2 a
ˆ a ˆ†
ˆ
ˆ
,
,
i.e. x ˆ
, p ˆ m
.
(5.66)
0
2
2 i
m
0
2
2 i
Our Hamiltonian (64) includes only squares of these operators. Calculating them, we have to be careful to avoid swapping the new operators, because they do not commute. Indeed, for the normalized operators (63), Eq. (2.14) gives
1
ˆ
,ˆ
ˆ x, ˆ p iI,ˆ
(5.67)
2
x m
0
0
so Eqs. (65) yield
Creation-
annihilation
†
1
i
operators:
a ˆ, a ˆ
ˆ i ˆ ˆ
, i ˆ
ˆ ˆ
, ˆ ˆ
, I ˆ
.
(5.68)
commutation
2
2
relation With such due caution, Eq. (66) gives
2
1
†2
2
†
†
2
1
2
ˆ
ˆ
ˆ a ˆ a ˆ ˆ a
a
ˆ a ˆ a ,
ˆ2
a ˆ†
a
ˆ ˆ†
a
a
ˆ†
a ˆ a .
(5.69)
2
2
Plugging these expressions back into Eq. (64), we get
ˆ
0
H
a ˆ a ˆ† a ˆ† a ˆ
.
(5.70)
2
This expression is elegant enough but may be recast into an even more convenient form. For that, let us rewrite the commutation relation (68) as
ˆ a
a ˆ† a ˆ† a ˆ I ˆ
(5.71)
and plug it into Eq. (70). The result is
ˆ
0 2ˆ† ˆ ˆ
1
H
a a I
,
(5.72)
0 N
ˆ I ˆ
2
2
where, in the last form, one more (evidently, Hermitian) operator
Number
operator:
N ˆ
a ˆ†
a ˆ
(5.73)
definition
has been introduced. Since, according to Eq. (72), the operators H ând N ˆ differ only by the addition of the identity operator and multiplication by a c-number, these operators commute. Hence, according to Chapter 5
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the general arguments of Sec. 4.5, they share a set of stationary eigenstates n (they are frequently called the Fock states), and we can write the standard eigenproblem (4.68) for the new operator as N ˆ n N n ,
(5.74)
n
where Nn are some eigenvalues that, according to Eq. (72), determine also the energy spectrum of the oscillator:
1
E N
.
(5.75)
n
0
n
2
So far, we know only that all eigenvalues Nn are real; to calculate them, let us carry out the following calculation – splendid in its simplicity and efficiency. Consider the result of the action of the operator N ôn the ket-vector a ˆ † n. Using the definition (73) and then the associative rule of the bra-ket formalism, we may write
N ˆ a†
ˆ n
a†
ˆ a
ˆ a†
ˆ n
a†
ˆ
ˆ a
a †
ˆ
n .
(5.76)
Now using the commutation relation (71), and then Eq. (74), we may continue as
ˆ†
a
ˆ ˆ†
a
a
n ˆ†
a
ˆ†
a ˆ
ˆ
a I
n ˆ†
a ˆ ˆ
N I n ˆ†
a N
(5.77)
n
1 n Nn 1
ˆ†
a n
.
For clarity, let us summarize the result of this calculation:
ˆ
N ˆ†
a n
N
(5.78)
n
1
ˆ†
a n
.
Performing a similar calculation for the operator a ˆ , we get a similar formula, but with a different sign: N ˆ a ˆ n N 1 ˆ
.
(5.79)
n
a n
It is time to stop the calculations for a minute, and translate their results into plain English: if n
is the eigenket of the operator N ˆ with an eigenvalue Nn, then a ˆ † n and a ˆ n are also eigenkets of that operator, with the eigenvalues ( Nn + 1) and ( Nn – 1), respectively. This statement may be vividly represented by the so-called ladder diagram shown in Fig. 7.
eigenket
of
eigenvalue
...
N ˆ
a ˆ† a ˆ
a†
ˆ n
N 1
n
a ˆ† a ˆ
n
Nn
a ˆ† a ˆ
Fig. 5.7. The “ladder diagram” of eigenstates of a 1D
a ˆ n
N 1
harmonic oscillator. Arrows show the actions of the
n
creation and annihilation operators on the eigenstates.
a ˆ† a ˆ
...
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The operator a ˆ † moves the system one step up this ladder, while the operator a ˆ brings it one step down. In other words, the former operator creates a new excitation of the system,28 while the latter operator kills (“annihilates”) such excitation.29 On the other hand, according to Eq. (74) inner-multiplied by the bra-vector n, the operator N ˆ does not change the state of the system, but “counts” its position on the ladder:
ˆ
n N n n N n N .
(5.80)
n
n
This is why N îs called the number operator, in our current context meaning the number of the elementary excitations of the oscillator.
This calculation still needs completion. Indeed, we still do not know whether the ladder shown in Fig. 7 shows all eigenstates of the oscillator, and what exactly the numbers Nn are. Fascinating enough, both questions may be answered by exploring just one paradox. Let us start with some state n (read: a step of the ladder), and keep going down the ladder, applying the operator a âgain and again. According to Eq. (79), at each step, the eigenvalue Nn is decreased by one, so eventually, it should become negative. However, this cannot happen because any actual eigenstate, including the states represented by kets d a ˆ n and n, should have a positive norm – see Eq. (4.16). Comparing the norms, 2
n
n n ,
2
d
n ˆ†
a a n n N n N n n ,
(5.81)
n
we see that both of them cannot be positive simultaneously if Nn is negative.
To resolve this paradox let us notice that the action of the creation and annihilation operators on the stationary states n may consist of not only their promotion to an adjacent step of the ladder diagram but also by their multiplication by some c-numbers:
ˆ a n A n 1 ,
ˆ†
a n A' n 1 .
(5.82)
n
n
(The linear relations (78)-(79) clearly allow that.) Let us calculate the coefficients An assuming, for convenience, that all eigenstates, including the states n and ( n –1), are normalized: ˆ†
a
ˆ a
1
N
n n ,
1
n 1 n 1 n
n
n N n
n
n n 1 .
(5.83)
*
A A
*
*
n
n
A A
A A
n
n
n
n
From here, we get An = ( Nn)1/2, i.e.
ˆ
1/ 2
a n N ei n n 1 ,
(5.84)
n
where n is an arbitrary real phase. Now let us consider what happens if all numbers Nn are integers.
(Because of the definition of Nn, given by Eq. (74), it is convenient to call these integers n, i.e. to use the same letter as for the corresponding eigenstate.) Then when we have come down to the state with n
= 0, an attempt to make one more step down gives
ˆ a 0 0 1 .
(5.85)
28 For electromagnetic field oscillators, such excitations are called photons; for mechanical wave oscillators, phonons, etc.
29 This is exactly why a ˆ † is called the creation operator, and a ˆ , the annihilation operator.
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But according to Eq. (4.9), the state on the right-hand side of this equation is the “null state”, i.e. does not exist.30 This gives the (only known :-) resolution of the state ladder paradox: the ladder has the lowest step with Nn = n = 0.
As a by-product of our discussion, we have obtained a very important relation Nn = n, which means, in particular, that the state ladder shown in Fig. 7 includes all eigenstates of the oscillator.
Plugging this relation into Eq. (75), we see that the full spectrum of eigenenergies of the harmonic oscillator is described by the simple formula
1
E
,
(5.86)
n
n
,
n ,
0 ,
1 2
0
2
which was already discussed in Sec. 2.9. It is rather remarkable that the bra-ket formalism has allowed us to derive it without calculating the corresponding (rather cumbersome) wavefunctions n( x) – see Eqs. (2.284).
Moreover, this formalism may be also used to calculate virtually any matrix element of the oscillator, without using n( x). However, to do that, we should first calculate the coefficient A’n participating in the second of Eqs. (82). This may be done similarly to the above calculation of An; alternatively, since we already know that An = ( Nn)1/2 = n 1/2, we may notice that according to Eqs. (73) and (82), the eigenproblem (74), which in our new notation for Nn becomes N ˆ n n n ,
(5.87)
may be rewritten as
N ˆ n a ˆ† a ˆ n a ˆ† A n 1 A A'
n .
(5.88)
n
n
n 1
Comparing the right-hand sides of Eqs. (87) and (88), we see that A’n-1 = n/ An = n 1/2, i.e. A’n = ( n +
1)1/2exp( i n’). Taking all phases n and n’ equal to zero for simplicity, we may spell out Eqs. (82) as31
Fock state
ˆ†
a n n
1 1/ 2 n 1 ,
ˆ
1/ 2
a n n
n 1 .
(5.89) ladder
Now we can use these formulas to calculate, for example, the matrix elements of the operator x în the Fock state basis: x
x
0
†
0
†
ˆ
n' ˆ x n x n' n
n' ˆ a ˆ a n
n' ˆ a n n' ˆ
a n
0
2
2
(5.90)
x 0
1/2
n
n' n 1 n
1 1/ 2 n' n 1 .
2
Taking into account the Fock state orthonormality:
n' n ,
(5.91)
n' n
this result becomes
30 Please note again the radical difference between the null state on the right-hand side of Eq. (85) and the state described by the ket-vector 0 on the left-hand side of that relation. The latter state does exist and, moreover, represents the most important, ground state of the system, with n = 0 – see Eqs. (2.274)-(2.275).
31 A useful mnemonic rule for these key relations is that the c-number coefficient in any of them is equal to the square root of the largest number of the two states it relates.
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1/ 2
Coordinate’s
x
matrix
0
n' ˆ x n
1/2
1/ 2
n
( n )
1
n
( n )
1
. (5.92)
n' , n 1
n' , n 1
1/2
1/ 2
n' , n 1
n' , n 1
elements
2
2
m0
Acting absolutely similarly, for the momentum’s matrix elements we get a similar expression: 1/ 2
m
n' ˆ
0
p n i
1/ 2
n
( n )
1 1/ 2
(5.93)
n' , n 1
n ,' n 1
.
2
Hence the matrices of both operators in the Fock-state basis have only two diagonals adjacent to the main diagonal; all other elements (including the main-diagonal ones) are zeros.
The matrix elements of higher powers of these operators, as well as their products, may be handled similarly, though the higher the power, the bulkier the result. For example, 2
n' ˆ x n n' ˆˆ x
x n
n' ˆ x n" n" ˆ x n
n" 0
2
x
0 n" 1/2
n" 1/2
1/ 2
1
n
n 1/ 2
1
(5.94)
n' , n"1
n,' n"1
n ,
" n1
n ," n1
2 n"0
2
x 0
n n 11/2
n n
n
n ,' n2
1 21/2
(2
)
1
n ,' n2
n n .
2
,'
For applications, the most important of these matrix elements are those on its main diagonal: 2
x
2
x
n ˆ2
0
x n
2 n 1.
(5.95)
2
This expression shows, in particular, that the expectation value of the oscillator’s potential energy in the n th Fock state is
2
2
2
m
m x
0
2
0
0
1
0
1
U
x
n
n
.
(5.96)
2
2
2
2
2
This is exactly one-half of the total energy (86) of the oscillator. As a sanity check, an absolutely similar calculation for the momentum squared, and hence for the kinetic energy p 2/2 m, yields 2
2
2
1
1
2
p
0
1
p
n ˆ p n m x n m n
n
(5.97)
0 0
,
so
,
2
0
2
2 m
2
2
i.e. both partial energies are equal to En/2, just as in a classical oscillator.32
Note that according to Eqs. (92) and (93), the expectation values of both x and p in any Fock state are equal to zero:
x n ˆ x n ,
0
p n ˆ p n ,
0
(5.98)
32 Still note that operators of the partial (potential and kinetic) energies do not commute with either each other or with the full-energy (Hamiltonian) operator, so the Fock states n are not their eigenstates. This fact maps onto the well-known oscillations of these partial energies (with the frequency 20) in a classical oscillator, at the full energy staying constant.
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This is why, according to the general Eqs. (1.33)-(1.34), the results (95) and (97) also give the variances of the coordinate and the momentum, i.e. the squares of their uncertainties, ( x)2 and ( p)2. In particular, for the ground state ( n = 0), these uncertainties are
1/ 2
1/ 2
x
m
x
m
0
0 0
0
x
,
p
.
(5.99)
2
2
m
2
2
0
In the theory of precise measurements (to be reviewed in brief in Chapter 10), these expressions are often called the standard quantum limit.
5.5. Glauber states and squeezed states
There is a huge difference between a quantum stationary (Fock) state of the oscillator and its classical state. Indeed, let us write the well-known classical equations of motion of the oscillator (using capital letters to distinguish classical variables from the arguments of quantum wavefunctions): 33
P
U
X
,
2
P
m X.
(5.100)
0
m
x
The simplest method to solve these equations is to introduce the dimensionless complex variable 1
P( t)
( t)
X ( t) i
,
(5.101)
2 x
m
0
0
With this definition, Eqs. (100) are conveniently merged into one equation,
i
,
(5.102)
0
with an evident, very simple solution
( t) ( )
0
exp i t
0 ,
so per Eq. (102):
(5.103)
X t
( ) 2 x Re (0) exp
,
( ) 2
Im (0) exp
,
0
i t 0
P t
m
x
0 0
i t 0
where the constant (0) is just the (normalized) classical complex amplitude of the oscillations, so their real amplitude is A = 2 x 0( t) = 2 x 0(0) .34 By the appropriate choice of the time origin, the complex amplitude may be always made real; then X cos0 t and P –sin0 t.
On the so-called phase plane, with the Cartesian coordinates x and p, this solution describes a clockwise rotation of the representation point { X( t), P( t)} along an elliptic trajectory starting from the initial point { X(0), P(0)}. The normalization of the momentum by m0, similar to the one performed by the second of Eqs. (63), makes this trajectory pleasingly circular, with a constant radius equal to the oscillation amplitude A, corresponding to the constant full energy
2
2
2
m
P( t)
P( )
0
0
2
2
E
A ,
with A X ( t)2
const
X ( )02
,
(5.104)
2
m
m
0
0
33 If Eqs. (100) are not evident, please consult a classical mechanics course – e.g., CM Sec. 3.2 and/or Sec. 10.1.
34 See, e.g., CM Chapter 5, especially Eqs. (5.4).
Chapter 5
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determined by the initial conditions – see Fig. 8.)
p /
m
0
P /
m 0
Fig. 5.8. Representations of various states of a
/ 2
harmonic oscillator on the phase plane. The bold black
point represents a classical state with displacement
amplitude A, with the dashed line showing its
X
A
x
trajectory. The (very imperfect) classical images of the
n 0
Fock states with n = 0, 1, and 2 are shown in blue. The
blurred red spot is the (equally schematic) image of
the Glauber state , with = A/2 x 0. Finally, the n 1
magenta elliptical spot is a classical image of a
squeezed ground state – see below. Arrows show the
direction of the states’ evolution in time.
n 2
On the other hand, according to the basic Eq. (4.161), the time dependence of a Fock state, as of a stationary state of the oscillator, is limited to the phase factor exp{– iEnt/}. This factor drops out at the averaging (4.125) for any observable. As a result, in this state the expectation values of x, p, or any function thereof are time-independent; moreover, as Eqs. (98) show, x = p = 0. Taking into account Eqs. (96)-(97), the closest (though very imperfect) geometric image35 of such a state on the phase plane is a static circle of the radius An = x 0(2 n + 1)1/2, along which the wavefunction is uniformly spread – see the blue rings in Fig. 8. For the ground state ( n = 0), with the wavefunction (2.275), a better image may be a blurred round spot, of a radius ~ x 0, at the origin. (It is easy to criticize such blurring, intended to represent the non-vanishing spreads (99), because it fails to reflect the fact that the total energy of the oscillator in the state, E 0 = 0/2 is definite, without any uncertainty.) So, the difference between a classical state of the oscillator and its Fock state n is very profound; it is much similar to the difference between the classical picture of a freely moving 1D particle and the traveling de Broglie wave (1.88). However, the Fock states are not the only possible quantum states of the oscillator: according to the basic Eq. (4.6), any state described by the ket-vector
n
(5.105)
n
n0
with an arbitrary set of (complex) c-numbers n, is also its legitimate state, subject only to the normalization condition = 1, giving
2
1.
(5.106)
n
n0
35 I have to confess that such geometric mapping of a quantum state onto the phase plane [ x, p] is not exactly defined; you may think about colored areas in Fig. 8 as the regions of the observable pairs { x, p} most probably obtained in measurements. A quantitative definition of such a mapping will be given in Sec. 7.3 using the Wigner function, though, as we will see there, even such imaging has certain internal contradictions. Still, such cartoons as Fig. 8 have a substantial heuristic value, provided that their limitations are kept in mind.
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It is natural to ask: could we select the coefficients n in such a special way that the state properties would be closer to the classical one; in particular the expectation values x and p of the coordinate and momentum would evolve in time as the classical values X( t) and P( t), while the uncertainties of these observables would be, just as in the ground state, given by Eqs. (99), and hence have the smallest possible uncertainty product, x p = /2. As early as 1926, E. Schrödinger showed that the answer was positive. In particular, by using special properties of the Hermite polynomials (2.281), he showed that the corresponding wavefunction, in the coordinate representation, is
1/ 4
Glauber
m
m
P t x
( x, t)
0
(5.107) state:
exp
0
x X( t)2
( )
i
i t ,
2
wavefunction
where
t
X t P t
t
0
.
const
(5.108)
2
2
2
2
m x
0
0
This solution, whose validity may be readily verified by its substitution to the full Schrödinger equation for the oscillator’s Hamiltonian (2.271) with the account of Eqs. (100), shows that such a Glauber state 36 is essentially the ground state but with its center shifted from the phase plane’s origin to the classical oscillation point { X( t), P( t)} – see the blurred red spot in Fig. 8. Moreover, it clearly shows that the coordinate’s uncertainty, which is not affected by the x-independent phase shift ( t), does not change with time, i.e. that in the harmonic oscillator, the Gaussian wave packet (107), once formed, does not spread with time. (As we have seen in Sec. 2.2, for a free particle, this is impossible.) Moreover, a similar (though bulkier) calculation shows that the wavefunction (107), with the appropriately modified phase ( t), also satisfies the Schrödinger equation of an oscillator under the effect of a pulse of a classical force F( t), provided that the oscillator initially was in its ground state and that the classical evolution law { X( t), P( t)} takes this force into account.37 Since for many experimental implementations of the harmonic oscillator, the ground state may be readily formed (for example, by letting the oscillator relax via its weak coupling to a low-temperature environment), the Glauber state is usually easier to form than any Fock state with n > 0. This is why the Glauber states are so important and deserve a thorough discussion.
However, for such a discussion, the usual methods of wave mechanics and even the expansion (105) are rather inconvenient, because of the bulky coordinate representation (2.284) of the Fock states n. Instead, the needed calculations may be more readily done in the bra-ket formalism.
Let us start by expressing the double shift of the ground state (by X and by P), which is so evident in Eq. (107), in the operator language. Forgetting about the P for a minute, let us find the translation operator T ˆ that would produce the desired shift of an arbitrary wavefunction ( x) by a c-
X
number distance X along the coordinate argument x. This means that 36 Named after R. J. Glauber who studied these states in detail in the mid-1960s using operator methods – see below. Another popular adjective, “coherent”, for the Glauber states is very misleading, because all quantum states of all systems we have studied in this course so far, including the Fock states of the harmonic oscillator, may be represented as coherent (pure) superpositions of the basis states. This is why I will not use this term for the Glauber states.
37 To find it, it is sufficient to integrate Eqs. (100) with F( t) added to the right-hand side of the second of these equations. For their solution for an arbitrary F( t), see, e.g., CM Eqs. (5.27) and (5.34) with = 0.
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ˆ
T ( x) ( x X ) .
(5.109)
X
Representing the wavefunction as the standard wave packet (4.264), we see that 1
p( x X )
1
pX
px
ˆ
T ( x)
( )exp
. (5.110)
1/ 2
( ) exp
exp
X
2
p
i
dp
2
p
1/ 2
i
i
dp
Hence, the shift may be achieved by the multiplication of each Fourier component of the packet, with the momentum p, by exp{– ipX/}. This gives us a hint that the general form of the translation operator, valid in any representation, should be
ˆ X
p
ˆ
T exp i
.
(5.111)
X
The proof of this formula is provided merely by the fact that, as we know from Chapter 4, any operator is uniquely determined by the set of its matrix elements in any full and orthogonal basis, in particular, the basis of the momentum states p. According to Eq. (110), the analog of Eq. (4.235) for the p-
representation, applied to the translation operator (which is evidently local), is
pX
ˆ
dp p T p' ( p' ) exp
i
( p) ,
(5.112)
X
so the operator (111) does exactly the job we need it to.
The operator that provides the shift of momentum by a c-number P is absolutely similar in structure – with the opposite sign under the exponent, due to the opposite sign of the exponent in the reciprocal Fourier transform, so the simultaneous shift by both X and P may be achieved by the following translation operator:
x
P ˆ ˆ X
p
Translation
ˆ
T exp
i
.
(5.113)
operator
As we already know, for a harmonic oscillator, the creation-annihilation operators are more natural, so we may use Eqs. (66) to recast Eq. (113) as
ˆ
†
*
ˆ
T exp ˆ a
ˆ a ,
so
†
T
exp *
ˆ a ˆ†
a ,
(5.114)
where (which, generally, may be a function of time) is the c-number defined by Eq. (101). Now, getting clues from Eq. (107), we may form the Glauber state’s ket-vector just as ˆ
T 0
.
(5.115)
This formula, valid in any representation, is very elegant, but using it for practical calculations (say, of the expectation values of observables) is not too easy because of the exponent-of-operators form of the translation operator (113). Fortunately, it turns out that a much simpler representation of the Glauber state is possible. To show this, let us start with the following general (and very useful) property of exponential functions of an operator argument: if
ˆ ˆ
,
A B
ˆ
I ,
(5.116)
(where A ând B âre arbitrary linear operators, and is a c-number), then Chapter 5
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êxp A ˆ B
êxp A ˆ
B I
.ˆ
(5.117)
This relation may be readily proved by expanding the operator f ˆ ()
êxp
A B ˆ
êxp
A in the
Taylor series with respect to the c-number parameter , and then evaluating the result for = 1. (This simple exercise is left for the reader.)
Let us apply Eqs. (116)-(117) to two cases, both with
ˆ
*
A ˆ a ˆ†
a , so
êxp ˆ †
A T ,
A
êxp
ˆ T .
(5.118)
First, let us take B ˆ I ˆ ; then Eq. (116) is valid with = 0, and Eq. (117) yields ˆ † ˆ
T
I ˆ
T
,
(5.119)
This equality means that the translation operator is unitary – not a big surprise, because if we shift a classical point on the phase plane by a complex number (+) and then by (–), we certainly must come back to the initial position. Eq. (119) means merely that this fact is true for any quantum state as well.
Second, let us take B ˆ a ˆ ; in order to find the corresponding parameter , we must calculate the commutator on the left-hand side of Eq. (116) for this case. By using, at the due step of the calculation, Eq. (68), we get
ˆ ˆ
,
A B *
ˆ a ˆ†
a , ˆ a
ˆ†
a , ˆ
ˆ
a I ,
(5.120)
so in this case = , and Eq. (117) yields
ˆ † ˆ
T
ˆ aT ˆ a I.ˆ
(5.121)
We have approached the summit of this beautiful calculation. Let us consider the following operator: T ˆ ˆ † ˆ
ˆ
T aT .
(5.122)
Using Eq. (119), we may reduce this product to T ˆ
ˆ a , while the application of Eq. (121) to the same
expression (122) yields T ˆ a ˆ
ˆ
T . Hence, we get the following operator equality:
aT ˆ
ˆ
T ˆ a ˆ
ˆ
T ,
(5.123)
which may be applied to any state. Now acting by both sides on the ground state’s ket 0, and using the fact that a ˆ 0 is the null state (while per Eq. (115), ˆ
T 0
), we finally get a very simple and
elegant result:38
Glauber
a ˆ .
(5.124) state as
eigenstate
38 This result is also somewhat counterintuitive. Indeed, according to Eq. (89), the annihilation operator a ˆ , acting upon a Fock state n, “beats it down” to the lower-energy state ( n – 1). However, according to Eq. (124), the action of the same operator on a Glauber state does not lead to the state change and hence to any energy change! The resolution of this paradox is given by the representation of the Glauber state as a series of Fock states – see Eq.
(134) below. The operator a îndeed transfers each Fock component of this series to a lower-energy state, but it also re-weighs each term of the series, so the complete energy of the Glauber state remains constant.
Chapter 5
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Thus any Glauber state is one of the eigenstates of the annihilation operator, namely the one with the eigenvalue equal to the c-number parameter of the state, i.e. to the complex representation (101) of the classical point which is the center of the Glauber state’s wavefunction.39 This fact makes the calculations of all Glauber state properties much simpler. As an example, let us calculate x in the Glauber state with some c-number :
x
x
0
†
0
x ˆ x
ˆ a ˆ a
ˆ a †
ˆ a
.
(5.125)
2
2
In the first term in the parentheses, we can apply Eq. (124) directly, while in the second term, we can use the bra-counterpart of that relation,
†
*
ˆ a . Now assuming that the Glauber state is
normalized, = 1, and using Eq. (101), we get
x 0
*
x
x
0 *
X ,
(5.126)
2
2
Acting absolutely similarly, we may verify that p = P, and that x and p do indeed obey Eqs. (99), i.e. do not depend on the shift . (This simple exercise is highly recommended to the reader.) As the last sanity check, let us use Eq. (124) to re-calculate the Glauber state’s wavefunction (107). Inner-multiplying both sides of that relation by the bra-vector x, and using the definition (65a) of the annihilation operator, we get
1
p ˆ
x x ˆ i
x .
(5.127)
2 x
m
0
0
Since x is the bra-vector of the eigenstate of the Hermitian operator x ˆ , they may be swapped, with the operator giving its eigenvalue x; acting on that bra-vector by the (local!) operator of momentum, we have to use it in the coordinate representation – see Eq. (4.245). As a result, we get 1
x x
x
x .
(5.128)
2 x
m
x
0
0
But x is nothing else than the Glauber state’s wavefunction , so Eq. (128) gives a first-order differential equation:
1
x
.
(5.129)
2
x
m
x
0
0
Chasing and x to the opposite sides of the equation, and using the definition (101) of the parameter
, we can bring this equation to the following form (valid at fixed t, and hence fixed X and P): d
m
P
0 x X i
dx .
(5.130)
m
0
Integrating both parts, we return to Eq. (107).
39 This fact means that the spectrum of eigenvalues in Eq. (124), viewed as an eigenproblem, is continuous – it may be any complex number.
Chapter 5
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Now we can use Eq. (124) for finding the coefficients n in the expansion (105) of the Glauber state in the series over the Fock states n. Plugging Eq. (105) into both sides of Eq. (124), using the second of Eqs. (89) on the left-hand side, and requiring the coefficients at each ket-vector n in both parts of the resulting relation to be equal, we get the following recurrence relation:
.
(5.131)
n 1
( n )
1 1/ 2 n
Applying this relation sequentially for n = 0, 1, 2, etc., we get
n
.
(5.132)
n
( n!)1/ 2 0
Now we can find 0 from the normalization requirement (106), getting
2 n
2
.
1
(5.133)
0
n0
!
n
In this sum, we may readily recognize the Taylor expansion of the function exp{2}, so the final result (besides an arbitrary common phase multiplier) is
2
n
Glauber
exp
n .
(5.134) state vs
2
1/ 2
n
0 ( !
n )
Fock states
Hence, if the oscillator is in the Glauber state , the probabilities Wn n n* of finding the system on the n th energy level (86) obey the well-known Poisson distribution (Fig. 9): n
n
n
W
e
,
(5.135) Poisson
n
n!
distribution
where n is the statistical average of n – see Eq. (1.37):
n
nW .
(5.136)
n
n0
0.8
W
n
n 3
.
0
0.6
0.4
0
.
1
Fig. 5.9. The Poisson distribution (135)
0
.
3
for several values of n. Note that W
0.2
n are
10
defined only for integer values of n, so the
0
lines are only guides for the eye.
0
5
10
15
20
n
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Note that the result of such summation is not necessarily an integer; in our particular case, 2
n .
(5.137)
For applications, perhaps the most important property of this distribution is that for any n, Glauber state:
1/ 2
r.m.s.
~2
n
n n 2
1/ 2
~2
n ,
that
so
n
n
n
.
(5.138)
uncertainty
Another important property is that at n >> 1, the Poisson distribution approaches the Gaussian one, with Wn peaking at n = n = 2, and a small relative r.m.s. uncertainty: n/ n << 1 – see Fig. 9.
Now let us discuss the Glauber state’s evolution in time. In the wave-mechanics language, it is completely described by the dynamics (100) of the c-number shifts X( t) and P( t) participating in the wavefunction (107). An alternative and equivalent way of dynamics description is to use the Heisenberg equation of motion. As Eqs. (29) and (35) tell us, such equations for the Heisenberg operators of coordinate and momentum have to be similar to the classical equations (100):
ˆ p
ˆ
H
x
,
ˆ
2
p m ˆ x .
(5.139)
H
H
0
H
m
Now using Eqs. (66), for the Heisenberg-picture creation and annihilation operators we get the equations
†
†
ˆ a i
ˆ a ,
ˆ a i ˆ a ,
(5.140)
H
0 H
H
0 H
which are completely similar to the classical equation (102) for the c-number parameter and its complex conjugate, and hence have the solutions identical to Eq. (103):
i t
†
†
i t
a ˆ t
( )
a ˆ ( )
0 e
0 ,
a ˆ t
( ) a ˆ (0) e 0 .
(5.141)
H
H
H
H
As was discussed in Sec. 4.6, such equations are very convenient because they enable simple calculation of time evolution of observables for any initial state of the oscillator (Fock, Glauber, or any other) by using Eq. (4.191). In particular, Eq. (141), without any calculations, shows that regardless of its initial state, the oscillator always returns to it exactly with the period 2/0.40
Applied to the particular case of the ground state of the oscillator, Eq. (141) confirms that the Gaussian wave packet of the special width (99) does not spread in time at all – even temporarily. At this point, I have to notice that there exist other ground-like states whose initial wave packets are still Gaussian but have different widths, say x < x 0/2. As we already know from Sec. 2.2, the momentum spread p has to be correspondingly larger, but the uncertainty product may still be the smallest: x p =
/2. Such squeezed ground states , with zero expectation values of x and p, may be generated from the Fock/Glauber ground state:
Squeezed
ground
ˆ
S 0
,
(5.142a)
state
by using the so-called squeezing operator:
40 Actually, this fact is also evident from the Schrödinger picture of the oscillator’s time evolution: due to the exactly equal distances 0 between the eigenenergies (86), the time functions an( t) in the fundamental expansion (1.69) of its wavefunction oscillate with frequencies n0, and hence they all share the basic time period 2/0.
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1
ˆ
*
† †
S exp
Squeezing
ˆ ˆ a
a ˆ a ˆ a ,
(5.142b)
2
operator
which depends on the complex c-number parameter = rei, where r and are real. The parameter’s modulus r determines the squeezing degree; if is real (i.e. = 0), then 2
x
r
m x
r
m
0
x
x
e ,
0 0
p
e ,
so
0 0
x
p
.
(5.143)
2
2
2
2
On the phase plane (Fig. 8), this state, with r > 0, may be represented by an oval spot squeezed along one of two mutually perpendicular axes (hence the state’s name), and stretched by the same factor er along the counterpart axis; the same formulas but with r < 0 describe squeezing along the other axis. On the other hand, the phase of the squeezing parameter determines the angle /2 of the squeezing/stretching axes about the phase plane origin – see the magenta ellipse in Fig. 8. If 0, Eqs.
(143) are valid for the variables { x’, p’} obtained from { x, p} via clockwise rotation by that angle. For any of such origin-centered squeezed ground states, the time evolution is reduced to an increase of the angle with the rate 0, i.e. to the clockwise rotation of the ellipse, without its deformation, with the angular velocity 0 – see the magenta arrows in Fig. 8. As a result, the uncertainties x and p oscillate in time with the double frequency 20. Such squeezed ground states may be formed, for example, by a parametric excitation of the oscillator,41 with a parameter modulation depth close to, but still below the threshold of the excitation of degenerate parametric oscillations.
By action of an additional external force (or by appropriate initial state preparation), the center of a squeezed state may be displaced from the origin to an arbitrary point { X, P}. Such a displaced squeezed state may be described by the action of the translation operator (113) upon the ground squeezed state, i.e. by the action of the operator product T ˆ ˆ
S on the usual (Fock/Glauber, i.e. non-
squeezed) ground state. Calculations similar to those that led us from Eq. (114) to Eq. (124), show that the displaced squeezed state is an eigenstate of the following mixed operator:
b ˆ a ˆ cosh r a ˆ† ei
sinh r ,
(5.144)
with the same parameters r and , with the eigenvalue
cosh r
*
ei
sinh r ,
(5.145)
thus generalizing Eq. (124), which corresponds to r = 0. For the particular case = 0, Eq. (145) yields
= 0, i.e. the action of the operator (144) on the squeezed ground state yields the null state. Just as Eq.
(124) in the case of the Glauber states, Eqs. (144)-(145) make the calculation of the basic properties of the squeezed states (for example, the proof of Eqs. (143) for the case = = 0) very straightforward.
Unfortunately, I do not have more time/space for a further discussion of the squeezed states in this chapter (besides a few problems given for the reader’s exercise), but their importance for precise quantum measurements will be discussed in Sec. 10.2 below.42
41 For a discussion and classical theory of this effect, see, e.g., CM Sec. 5.5.
42 For more on the squeezed states see, e.g., Chapter 7 in the monograph by C. Gerry and P. Knight, Introductory Quantum Optics, Cambridge U. Press, 2005. Also, note the spectacular measurements of the Glauber and squeezed states of electromagnetic (optical) oscillators by G. Breitenbach et al., Nature 387, 471 (1997), a large Chapter 5
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5.6. Orbital angular momentum
One more blank spot to fill has been left by our study, in Sec. 3.6, of wave mechanics of particle motion in spherically symmetric 3D potentials. Indeed, while the azimuthal components of the eigenfunctions (the spherical harmonics) of such systems are very simple,
,
(5.146)
m
2 1/2 eim , with m ,0 ,1 ,...
2
their polar components include the associated Legendre functions P m l (cos), which may be expressed
via elementary functions only indirectly – see Eqs. (3.165) and (3.168). This makes all the calculations less than transparent and, in particular, does not allow a clear insight into the origin of the very simple energy spectrum of such systems – see, e.g., Eq. (3.163). The bra-ket formalism, applied to the angular momentum operator, not only enables such insight and produces a very convenient tool for many calculations involving spherically symmetric potentials but also opens a clear way toward the unification of the orbital momentum with the particle’s spin – the latter task to be addressed in the next section.
Let us start by using the correspondence principle to spell out the quantum-mechanical vector operator of the orbital angular momentum L rp of a point particle: n
n
n
x
y
z
3
Angular
momentum
ˆL ˆr ˆp ˆ r
ˆ r
ˆ
ˆ
r ,
i.e. L
ˆ r ˆ p
,
(5.147)
1
2
3
j
j' j" jj'j"
operator
j' , j" 1
ˆ p
ˆ p
ˆ p
1
2
3
where jj’j” is the Levi-Civita permutation symbol, which we have already used in Sec. 4.5, and also in Sec. 1 of this chapter in similar expressions (17)-(18). From this definition, we can readily calculate the commutation relations for all Cartesian components of the vector operators of L, r, and p, for example,
3
3
3
3
ˆ L , ˆ r
r p
r
r r p
i
r
i
r
.
(5.148)
j
j' ˆ ˆ
, ˆ
k
j"
jkj"
j' ˆ k ˆ , ˆ
j'
j"
jkj"
ˆ
k
j'j"
jkj"
ˆ j" jj'j"
k, j" 1
k , j" 1
k , j" 1
j" 1
The summary of all these calculations may be represented in similar forms:
Key
3
3
3
commutation
ˆ L , ˆ r
i
r
L p
i
p
L L
i
L
;
(5.149)
j
j'
relations
ˆ j" jj'j"
ˆ
,
, ˆ
j
j'
ˆ
,
j"
jj'j"
ˆ ˆ
ˆ
,
j
j'
j" jj'j"
j" 1
j" 1
j" 1
the last of them shows that the commutator of two different Cartesian components of Lîs proportional to its complementary component.
Also introducing, in a natural way, the (scalar!) operator of the observable L 2 L2, Operator
3
of L 2
ˆ2
ˆ2
ˆ2
ˆ2
2
L L L L
L
(5.150)
x
y
z
, j
j 1
it is straightforward to check that this operator commutes with each of the Cartesian components: (ten-fold) squeezing achieved in such oscillators by H. Vahlbruch et al., Phys. Rev. Lett. 100, 033602 (2008), and the first results on the ground state squeezing in micromechanical oscillators, with resonance frequencies 0/2 as low as a few MHz, by using their parametric coupling to microwave electromagnetic oscillators – see, e.g., E.
Wollman et al., Science 349, 952 (2015) and/or J.-M. Pirkkalainen et al., Phys. Rev. Lett. 115, 243601 (2015).
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ˆ2 ˆ
L , L
(5.151)
j 0.
This result, at first sight, may seem to contradict the last of Eqs. (149). Indeed, haven’t we learned in Sec. 4.5 that commuting operators (e.g., 2
ˆ L and any of L ˆ ) share their eigenstate sets? If yes, shouldn’t j
this set has to be common for all four angular momentum operators? The resolution in this paradox may be found in the condition that was mentioned just after Eq. (4.138), but (sorry!) was not sufficiently emphasized there. According to that relation, if an operator has degenerate eigenstates (i.e. if some Aj =
Aj’ even for j j’), they should not be necessarily all shared by another compatible operator.
This is exactly the situation with the orbital angular momentum operators, which may be schematically represented by the Venn diagram 43 shown in Fig. 10: the eigenstates of the operator 2
ˆ L
are highly degenerate,44 and their set is broader than those of any component operator L ˆ (that, as will j
be shown below, are non-degenerate – until we consider the particle’s spin).
Fig. 5.10. The Venn diagram showing the partitioning of
the set of eigenstates of the operator 2
ˆ L . Each inner sector
L ˆ
corresponds to the states shared with one of the Cartesian
z
component operators L ˆ , while the outer (shaded) ring
j
L ˆ
L ˆ
x
y
represents the eigenstates of 2
ˆ L that are not shared with
either of L ˆ – for example, all linear combinations of the
j
eigenstates of different component operators.
Let us focus on just one of these three joint sets of eigenstates – by tradition, of the operators 2
ˆ L
and L ˆ . (This tradition stems from the canonical form of the spherical coordinates, in which the polar z
angle is measured from the z-axis . Indeed, in the coordinate representation, we may write
L ˆ ˆ p
x
ˆ p
y
x i
.
(5.152)
z
y
x
y i
i
y
x
Writing the standard eigenproblem for the operator in this representation, L ˆ L , we see that it z
m
z
m
is satisfied by the eigenfunctions (146), with eigenvalues Lz = m – the fact that was already conjectured in Sec. 3.5.) More specifically, let us consider a set of eigenstates { l, m} corresponding to a certain degenerate eigenvalue of the operator 2
ˆ L , and all possible eigenvalues of the operator L ˆ , i.e. all z
possible quantum numbers m. (At this point, l is just a label of the eigenvalue of the operator 2
ˆ L ; it will
43 This is just a particular example of the Venn diagrams (introduced in the 1880s by John Venn) that show possible relations (such as intersections, unions, complements, etc.) between various sets of objects, and are a very useful notion of the general set theory.
44 Note that this particular result is consistent with the classical picture of the angular momentum vector: even when its length is fixed, the vector may be oriented in various directions, corresponding to different values of its Cartesian components. However, in the classical picture, all these components may have exactly fixed values simultaneously, while in the quantum picture, this is not true.
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be defined more explicitly in a minute.) To analyze this set, it is instrumental to introduce the so-called ladder (also called, respectively, “raising” and “lowering”) operators 45
Ladder
L ˆ L ˆ iL ˆ .
(5.153)
operators
x
y
It is simple (and hence left for the reader’s exercise) to use this definition and the last of Eqs. (149) to calculate the following commutators:
Important
commutation
L ˆ L ˆ
,
relations
L ˆ
2
, and L ˆ
L ˆ
,
L ˆ ,
(5.154)
z
z
and also to use Eqs. (149)-(150) to prove two other important operator relations: L ˆ2 L ˆ2 L ˆ L ˆ L ˆ
.
(5.155)
L ˆ
,
2 L ˆ2 L ˆ L ˆ L ˆ
z
z
z
z
Now let us rewrite the last of Eqs. (154) as
L ˆ L ˆ L ˆ L ˆ L ˆ ,
(5.156)
z
z
and act by both its sides upon the ket-vector l, m of an arbitrary common eigenstate: ˆ ˆ
ˆ ˆ
ˆ
L L l, m L L l, m L l, m .
(5.157)
z
z
Since the eigenvalues of the operator L âre equal to
z
m, in the first term of the right-hand side of Eq.
(157) we may write
ˆ L l, m m l, m .
(5.158)
z
With that, Eq. (157) may be recast as
L ˆ ˆ
ˆ
,
1
,
.
(5.159)
z L
l m
m L l m
In a spectacular similarity with Eqs. (78)-(79) for the harmonic oscillator, Eq. (159) means that the states L ˆ l, m are also eigenstates of the operator L ˆ , corresponding to eigenvalues
z
( m 1). Thus
the ladder operators work exactly as the creation and annihilation operators of a harmonic oscillator, moving the system up or down a ladder of eigenstates – see Fig. 11.
eigenket
L ôf
eigenvalue
z
l, l
l
L ˆ
ˆ
L
…
L ˆ l, m
m 1
L ˆ
ˆ
L
l, m
m
L ˆ
ˆ
L
L ˆ l, m
m 1
…
Fig. 5.11. The ladder diagram of the common
L ˆ
ˆ
eigenstates of the operators 2
ˆ L and L ˆ .
L
z
l, l
l
45 Note a substantial similarity between this definition and Eqs. (65) for the creation/annihilation operators.
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The most significant difference is that now the state ladder must end in both directions, because an infinite increase of m, with whichever sign of m, would cause the expectation values of the operator 2
2
2
2
ˆ
ˆ
ˆ
ˆ
L L L L ,
(5.160)
x
y
z
which corresponds to a non-negative observable, becoming negative. Hence there have to be two states at both ends of the ladder, with such ket-vectors l, m max and l, m min that ˆ
ˆ
L l, m
,
0
L l, m
.
0
(5.161)
max
min
Due to the symmetry of the whole problem with respect to the replacement m – m, we should have m min = – m max. This m max is exactly the quantum number traditionally called l, i.e.
Relation