Chapter 5
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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.22 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, that remains constant for motion in a stationary potential U( x), so that we may rewrite the expression under this integral as23
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 the 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 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. At 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 consisting of straight-line segments. To have another look at 22 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.
23 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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that assumption, let us generalize the path integral to multi-dimensional geometries. Fortunately, the simple structure of Eq. (51b) makes such generalization virtually evident:
3D
t
t
2
i
dr
m dr
propagator
G(r, t; r t ) exp S r D r
S L r
d
U r d (5.55) as a path
0, 0
( ) [ ( )],
,
( )
.
d
d
integral
t
t 2
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 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/ – as it 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. 24
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
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
24 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 minor importance.) Chapter 5
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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 that 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 the two surfaces and minimizes the functional I. This is of course a well-known problem of the calculus of variations,25 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 ' ) r
d ' k (r ' ) r
d ,
(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 the tunneling path (in the WKB limit) corresponds to the classical (so-called instanton 26) trajectory of the same particle moving in the inverted potential U inv(r). If the initial point r0 is fixed, this trajectory may be readily found by the 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 25 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.
26 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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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.27
5.4. Revisiting harmonic oscillator
Let us return to the 1D harmonic oscillator, now 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 its key characteristics.
Fortunately, the bra-ket formalism helps to make such calculations.
First, introducing normalized (dimensionless) operators of coordinates and momentum:28
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
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: ˆ
ˆ
i
m
ˆ p
Creation
0 1/ 2
ˆ†
a
ˆ x i
.
(5.65b) operator:
2
2
m 0
definition
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.
27 For more on this interesting issue see, e.g., M. Buttiker and R. Landauer, Phys. Rev. Lett. 49, 1739 (1982), and references therein.
28 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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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 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 that Eqs. (65) yield
Creation-
annihilation
†
1
i
operators:
a ˆ, a ˆ
ˆ i ˆˆ
, 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
a ˆ 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 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:
Chapter 5
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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: N ˆ a ˆ n N 1 ˆ
.
(5.79)
n
a n
It is time to stop calculations for a minute, and translate these results into plain English: if n is an eigenket of the operator N ˆ with the 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 on 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
...
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,29 while the latter operator kills (“annihilates”) such excitation.30 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:
29 For electromagnetic field oscillators, such excitations are called photons; for mechanical wave oscillators, phonons, etc.
30 This is exactly why a ˆ † is called the creation operator, and a ˆ , the annihilation operator.
Chapter 5
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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 that 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)
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.31 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.
31 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).
Chapter 5
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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 n a ˆ† a ˆ n a ˆ† A n 1 A A'
n .
(5.88)
n
n
n 1
Comparing the first and the last form of this equality, we see that A’n-1 = n/ An = n 1/2, so that 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) as32
ˆ†
a n n
1 1/ 2 n 1 ,
ˆ
1/ 2
a n n
n 1 .
(5.89) Fock state
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
1/ 2
x
Coordinate’s
0
n' ˆ x n
1/2
1/ 2
n
( n )
1
n
( n )
1
. (5.92) matrix
n' , n 1
n' , n 1
1/2
1/ 2
n' , n 1
n' , n 1
2
2
m
elements
0
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
32 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.
Chapter 5
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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
,
that
so
,
2
0
2
2 m
2
2
i.e. both partial energies are equal to En/2, just as in a classical oscillator.33
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)
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.
33 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 that the Fock states n are not their eigenstates. This fact maps on the well-known oscillations of these partial energies (with the frequency 20) in a classical oscillator, at the full energy staying constant.
Chapter 5
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Essential Graduate Physics
QM: Quantum Mechanics
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): 34
P
U
X
,
2
P
m X.
(5.100)
0
m
x
On the so-called phase plane, with the Cartesian coordinates x and p, these equations describe 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 oscillations 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
,
(5.101)
2
m
m
0
0
determined by the initial conditions – see Fig. 8.)
p /
m 0
P /
m
0
Fig. 5.8. Representations of various states of a harmonic
/ 2
oscillator on the phase plane. The bold black point
represents a classical state with the complex amplitude
, with the dashed line showing its trajectory. The (very
X
n 0
x
imperfect) classical images of the 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 .
n 1
Finally, the 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
For the forthcoming comparison with quantum states, it is convenient to describe this classical motion by the following dimensionless complex variable
1
P( t)
( t)
X ( t) i
,
(5.102)
2 x
m
0
0
which is essentially the standard complex-number representation of the representing point’s position on the 2D phase plane, with A/2 x 0. With this definition, Eqs. (100) are conveniently merged into one equation,
i
,
(5.103)
0
34 If Eqs. (100) are not evident, please consult a classical mechanics course – e.g., CM Sec. 3.2 and/or Sec. 10.1.
Chapter 5
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with an evident, very simple solution
t
( ) ( )
0
exp i t ,
(5.104)
0
where the constant (0) may be complex, and is just the (normalized) classical complex amplitude of oscillations.35 This equation describes sinusoidal oscillations of both X( t) Re[( t)] and P Im[( t)], with a phase shift of /2 between them.
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 of 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 image36 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 defined exactly, without any uncertainty.) So, the difference between a classical state of the oscillator and its Fock state n is very profound.
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
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. Let me show that such a Glauber state,37 which is 35 See, e.g., CM Chapter 5, especially Eqs. (5.4).
36 I have to confess that such geometric mapping of a quantum state on 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, 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.
37 Named after Roy Jay Glauber who studied these states in detail in the mid-1965s, though they had been discussed in brief by Ervin Schrödinger as early as in 1926. Another popular adjective, “coherent”, for the Glauber states is very misleading, because all quantum states of all systems we have studied 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.
Chapter 5
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schematically represented in Fig. 8 by a blurred red spot around the classical point { X( t), P( t)}, is indeed possible.
Conceptually the simplest way to find the corresponding coefficients n would be to calculate
x, p, x, and p for an arbitrary set of n, and then try to optimize these coefficients to reach our goal. However, this problem may be solved much easier using wave mechanics. Indeed, let us consider the following wavefunction:
Glauber
m
state:
0 1/ 2
m 0
2
P t
( ) x
( x, t)
exp
x X t() i
,
(5.107) coordinate
2
representation
Its comparison with Eqs. (2.275) shows that this is just the ground-state wavefunction, but with the center shifted from the origin into the classical point { X( t), P( t)}. A straightforward (though a bit bulky) differentiation over x and t shows that it satisfies the oscillator’s Schrödinger equation, provided that the c-number functions X( t) and P( t) obey the classical equations (100). Moreover, a similar calculation shows that the wavefunction (107) 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)} in Eq. (107) takes this force into account.38 Since for many experimental implementations of the harmonic oscillator, the ground state may be readily formed (for example, by providing a weak coupling of the oscillator 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 much discussion.
In such a discussion, there is a substantial place for the bra-ket formalism. For example, to calculate the corresponding coefficients in the expansion (105) by wave-mechanical means,
*
( ) ( , ) ,
(5.108)
n t
n t
dx n x x t
x
x t dx
n
we would need to use not only the simple Eq. (107), but also the Fock state wavefunctions n( x), which are not very appealing – see Eq. (2.284) again. Instead, this calculation may be readily done in the bra-ket formalism, giving us one important byproduct result as well.
Let us start by expressing the double shift of the ground state (by X and P), which has led us to 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-number X
distance X along the coordinate argument x. This means
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
38 For its description, it is sufficient to solve Eqs. (100), with F( t) added to the right-hand side of the second of these equations.
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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 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 that 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 – with the opposite sign under the exponent, due to the opposite sign of the exponent in the reciprocal Fourier transform, so that 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 that we may use Eqs. (66) to recast Eq. (113) as
ˆ
†
*
ˆ
T exp ˆ a
ˆ a ,
that
so
†
T
exp *
ˆ a α ˆ†
a ,
(5.114)
where (which, generally, may be a function of time) is the c-number defined by Eq. (102). Now, according to 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. Fortunately, it turns out that a much simpler representation for 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), then39
êxp ˆ
A B
êxp ˆ
A B I
.ˆ
(5.117)
39 A proof of Eq. (117) may be readily achieved 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.
Chapter 5
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Let us apply Eqs. (116)-(117) to two cases, both with
ˆ
*
A ˆ a ˆ†
a , so that
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. Using, at the due stage 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 that 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 of this equality on the ground state’s ket
0, and using the fact that a ˆ 0 is the null-state, while according to Eq. (115), ˆ
T 0
, we finally
get a very simple and elegant result:40
Glauber
a ˆ .
(5.124) state as
eigenstate
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 (102) of the classical point which is the center of the Glauber state’s wavefunction.41 This fact makes the 40 This result is also rather counter-intuitive. 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, so that the complete energy of the Glauber state remains constant.
41 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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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. (102), 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).
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 that Eq. (128) gives for it 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 (102) of the parameter
, we can bring this equation to the 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).
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
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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 that 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
The result of such summation is not necessarily integer! In our particular case, Eqs. (134)-(136) yield 2
n .
(5.137)
0.8
W
n 3
.
0
n 0.6
0.4
0
.
1
Fig. 5.9. The Poisson distribution (135)
0
.
3
0.2
for several values of n. Note that Wn are
10
defined only for integer values of n; the
0
lines are only guides for the eye.
0
5
10
15
20
n
For applications, perhaps the most important mathematical property of this distribution is 1/ 2
Glauber state:
~2
n
n n 2
1/ 2
~2
n ,
that
so
n
n
n
.
(5.138) r.m.s.
uncertainty
Another important property is that at n >> 1, the Poisson distribution approaches the Gaussian (“normal”) one, with a small relative r.m.s. uncertainty: n/ n << 1 – the trend clearly visible in Fig. 9.
Chapter 5
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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). Note again that, in contrast to the spread of the wave packet of a free particle, discussed in Sec. 2.2, in the harmonic oscillator the Gaussian packet of the special width (99) does not spread at all!
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 (103) for the c-number parameter and its complex conjugate, and hence have the solutions identical to Eq. (104):
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) using Eq. (4.191). In particular, Eq. (141) shows that regardless of the initial state, the oscillator always returns to it exactly with the period 2/0.42 Applied to the Glauber state with = 0, i.e. the ground state of the oscillator, such calculation confirms that the Gaussian wave packet of the special width (99) does not spread in time at all – even temporarily.
Now let me briefly mention the 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 will be correspondingly larger, still with the smallest possible uncertainty product: x p = /2. Such squeezed ground state , with zero expectation values of x and p, may be generated from the Fock/Glauber ground state:
Squeezed
ˆ
ground
S 0
,
(5.142a)
state using the so-called squeezing operator,
1
Squeezing
ˆ
*
† †
S exp
operator
ˆ ˆ a
a ˆ a ˆ a ,
(5.142b)
2
which depends on a 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 42 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 same time period 2/0.
Chapter 5
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2
x
r
m x
r
m
0
x
x
e ,
0 0
p
e ,
that
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,43 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, the center of a squeezed state may be displaced from the origin to an arbitrary point { X, P}. Such 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 such 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 section, but their importance for precise quantum measurements will be discussed in Sec. 10.2
below.44
43 For a discussion and classical theory of this effect, see, e.g., CM Sec. 5.5.
44 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 (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, 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).
Chapter 5
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5.6. Revisiting spherically-symmetric systems
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
Angular
3
momentum
ˆL ˆr ˆp ˆ r
ˆ r
ˆ
ˆ
r ,
i.e. L
ˆ r ˆ p
,
(5.147)
1
2
3
j
operator
j' j" jj'j"
j' 1
ˆ p
ˆ p
ˆ p
1
2
3
where each of the indices j, j’, and j” may take values 1, 2, and 3 (with j” j, j’), and 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 operators Lˆ, rˆ
p
and
,
ˆ ; for example,
3
3
3
3
L ˆ , r ˆ r ˆ p ˆ , r ˆ r ˆ r ˆ , p ˆ
i
r ˆ
i r ˆ i r ˆ , (5.148)
j
j'
k
j"
jkj"
j'
k j'
j" jkj"
k
j'j"
jkj"
k
jj'k
j"
jj'j"
k 1
k 1
k 1
k 1
The summary of all these calculations may be represented in similar compact forms:
