a
t ˆ a t
ˆ
e(r) ini
fin
d t n ini
,
2
e
a
where n e e(r)/ e(r) is the local direction of the vector e(r), symbols “ini” and “fin” denote the initial and final states of the corresponding system (the electromagnetic field in the first long bracket, and the atom in the second bracket), and the density a of the continuous atomic states should be calculated at its final energy E fin = E ini + .
As a reminder, in the Heisenberg picture of quantum dynamics, the initial and final states are time-independent, while the creation-annihilation operators are functions of time. In the Golden Rule formula (25), as in any perturbative result, this time dependence has to be calculated ignoring the perturbation – in this case, the field-atom interaction. For the field’s creation-annihilation operators, this dependence coincides with that of the usual 1D oscillator – see Eq. (5.141), in which 0 should be, in our current notation, replaced with :
ˆ a( t) ˆ a( )
0 i t
e , ˆ†
a ( t) ˆ†
a ( )
0 i t
e
.
(9.26)
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Hence Eq. (25) becomes
2
2
fin
ˆ†
a (0) i
e t ˆ
i t
ˆ
a(0) e
e(r) ini
fin
d( t) n ini
.
(9.27a)
e
a
Now let us multiply the first long bracket by exp{ i t}, and the second one by exp{- i t}: 2
2
fin
ˆ†
a (0) 2 i t
e
ˆ
ˆ
a(0) e(r) ini
fin
d( t)
i t
n e
ini
.
(9.27b)
e
a
This, mathematically equivalent form of the previous relation shows more clearly that at resonant photon absorption, only the annihilation operator gives a significant time-averaged contribution to the first bracket matrix element. (As a reminder, the quantum-mechanical Golden Rule for time-dependent perturbations is a result of averaging over a time interval much larger than 1/ – see Sec. 6.6.) Similarly, according to Eq. (4.199), the Heisenberg operator of the dipole moment, corresponding to the increase of atom’s energy by , has the Fourier components that differ in frequency from only by ~ << , so that its time dependence virtually compensates the additional factor in the second bracket of Eq.
(27b), and this bracket also may have a substantial time average. Hence, in the first bracket we may neglect the fast-oscillating term, whose average over time interval ~1/ is very close to zero.16
Now let us assume, first, that we use the same detector, characterized by the same matrix element of the quantum transition, i.e. the same second bracket in Eq. (27), and the same final state density a, for measurement of various electromagnetic fields – or just of the same field at different points r. Then we are only interested in the behavior of the first, field-related bracket, and may write
fin
ˆ ae(r) ini 2 fin
ˆ ae(r) ini
fin
ˆ ae(r) ini
* ini
ˆ† *
a e (r) fin
fin
ˆ ae(r) ini
, (9.28)
where the creation-annihilation operators are implied to be taken at t = 0, i.e. in the Schrödinger picture, and the initial and final states are those of the field alone. Second, let us now calculate the total rate of transitions to all available final states of the given mode e(r). If such states formed a full and orthonormal set, we could use the closure relation (4.44), applied to the final states, to write Photon
† *
†
2
Γ ini
ˆ a e (r) fin
fin
ˆ ae(r) ini
ini
ˆ a ˆ a ini
*
e (r) e(r) n
e(r) ,
(9.29) counting
ini
rate
fin
where, for a given field mode, n
†
ini is the expectation value of the operator n ˆ a ˆ a ˆ for the initial state of the electromagnetic field. In the more realistic case of fields in relatively large volumes, V >> 3, with their virtually continuous spectrum of final states, the middle equality in this relation is not strictly valid, but it is correct to a constant multiplier,17 which we are currently not interested in. Note, however, that Eq. (29) may be substantially wrong for high- Q electromagnetic resonators (“cavities”), which may make just one (or a few) modes available for transitions. (Quantum electrodynamics of such cavities will be briefly discussed in Sec. 4 below.)
Let us apply Eq. (29) to several possible quantum states of the mode.
16 This is essentially the same rotating wave approximation (RWA), which was already used in Sec. 6.5 and beyond – see, e.g., the transition from Eq. (6.90) to the first of Eqs. (6.94).
17 As the Golden Rule shows, this multiplier is proportional to the density f of the final states of the field.
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(i) First, as a sanity check, the ground initial state, n = 0, gives no photon absorption at all. The interpretation is easy: the ground state field, cannot emit a photon that would ionize an atom in the counter. Again, this does not mean that the ground-state “motion” is not observable (if you still think so, please review the Casimir effect discussion in Sec. 1), just that it cannot ionize the trigger atom –
because it does not have any spare energy for doing that.
(ii) All other coherent states (Fock, Glauber, squeezed, etc.) of the field oscillator give the same counting rate, provided that their nini is the same. This result may be less evident if we apply Eq. (29) to the interference of two light beams from the same source – say, in the double-slit or the Bragg-scattering configurations. In this case, we may represent the spatial distribution of the field as a sum e(r) e (r) e (r) .
(9.30)
1
2
Here each term describes one possible wave path, so that the operator product in Eq. (29) may be a rapidly changing function of the detector position. For this configuration, our result (29) means that the interference pattern (and its contrast) are independent of the particular state of the electromagnetic field’s mode.
(iii) Surprisingly, the last statement is also valid for a classical mixture of the different eigenstates of the same field mode, for example for its thermal-equilibrium state. Indeed, in this case we need to average Eq. (29) over the corresponding classical ensemble, but it would only result in a different meaning of averaging n in that equation; the field part describing the interference pattern is not affected.
The last result may look a bit counter-intuitive because common sense tells us that the stochasticity associated with thermal equilibrium has to suppress the interference pattern contrast. These expectations are (partly :-) justified because a typical thermal source of radiation produces many field modes j, rather than one mode we have analyzed. These modes may have different wave numbers kj and hence different field distribution functions e j(r), resulting in shifted interference patterns. Their summation would indeed smear the interference, suppressing its contrast.
So the use of one photon detector is not the best way to distinguish different quantum states of an electromagnetic field mode. This task, however, may be achieved using the photon counting correlation technique shown in Fig. 2.18
controllable
semi-transparent
delay
mirror
light
source
detector 2
detector 1
count
statistics
Fig. 9.2. Photon count
calculation
correlation measurement.
18 It was pioneered as early as the mid-1950s (i.e. before the advent of lasers), by Robert Hanbury Brown and Richard Twiss. Their second experiment was also remarkable for the rather unusual light source – the star Sirius!
(Their work was an effort to improve astrophysics interferometry techniques.)
Chapter 9
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In this experiment, the counter rate correlation may be characterized by the so-called second-order correlation function of the counting rates,
( t) ( t )
Second-
(2)
g ( )
1
2
,
(9.31) order
( t) ( t)
correlation
1
2
function
where the averaging may be carried out either over many similar experiments, or over a relatively long time interval t >> , with usual field sources – due to their ergodicity. Using the normalized correlation function (31) is very convenient because the characteristics of both detectors and the beam splitter (e.g., a semi-transparent mirror, see Fig. 2) drop out from this fraction.
Very unexpectedly for the mid-1950s, Hanbury Brown and Twiss discovered that the correlation function depends on time delay in the way shown (schematically) with the solid line in Fig. 3. It is evident from Eq. (31) that if the counting events are completely independent, g(2)() should be equal to 1
– which is always the case in the limit . (As will be shown in the next section, the characteristic time of this approach is usually between 10-11s and 10-8s, so that for its measurement, the delay time control may be provided just by moving one of the detectors by a human-scale distance between a few millimeters to a few meters.) Hence, the observed behavior at 0 corresponds to a positive correlation of detector counts at small time delays, i.e. to a higher probability of the nearly simultaneous arrival of photons to both counters. This counter-intuitive effect is called photon bunching.
(2)
g
2
1
Fig. 9.3. Photon bunching (solid line) and
antibunching for various n (dashed lines). The
n 1
lines approach level g(2) = 1 at (on the
0
time scale depending on the light source).
Let us use our simple single-mode model to analyze this experiment. Now the elementary quantum process characterized by the numerator of Eq. (31), is the correlated, simultaneous ionization of two trigger atoms, at two spatial-temporal points {r1, t} and (r2, t – }, by the same field mode, so that we need to make the following replacement in the first of Eqs. (25):
Ê (r, t) const Ê (r , t) Ê (r , t ) .
(9.32)
1
2
Repeating all the manipulations done above for the single-counter case, we get
Γ
†
†
*
*
t
t
a t a t a t a t
e r e r e r e r
(9.33)
1 Γ 2
ini
ˆ( ) ˆ(
) ˆ(
) ˆ( ) ini
( ) ( ) ( ) ( ).
1
2
1
2
Plugging this expression, as well as Eq. (29) for single-counter rates, into Eq. (31), we see that the field distribution factors (as well as the detector-specific brackets and the density of states a) cancel, giving a very simple final expression:
ˆ†
a ( t) ˆ†
a ( t ) ˆ a( t ) ˆ a( t)
(2)
g ( )
,
(9.34)
2
ˆ†
a ( t) ˆ a( t)
Chapter 9
Page 10 of 36
QM: Quantum Mechanics
where the averaging should be carried out, as before, over the initial state of the field.
Still, the calculation of this expression for arbitrary may be quite complex, because in many cases the relaxation of the correlation function to the asymptotic value g(2)() is due to the interaction of the light source with the environment, and hence requires the open-system techniques that were discussed in Chapter 7. However, the zero-delay value g(2)(0) may be calculated straightforwardly, because the time arguments of all operators are equal, so that we may write
ˆ†
a ˆ†
a ˆ ˆ a
a
Zero-delay
correlation
(2)
g ( )
0
.
(9.35)
2
ˆ†
a ˆ a
Let us evaluate this ratio for the simplest states of the field.
(i) The n th Fock state. In this case, it is convenient to act with the annihilation operators upon the ket-vectors, and by the creation operators, upon the bra-vectors, using Eqs. (19):
Photon
n ˆ†
a ˆ†
a ˆ ˆ a
a n
n 2 n( n )
1 1/2 n( n )
1 1/ 2 n 2
n n
(2)
(
)
1
1
anti-
g ( )
0
1 .
(9.36)
bunching
n ˆ†
a ˆ
2
a n
n 1 1/ 2 1/ 2
n n
n 1 2
2
n
n
We see that the correlation function at small delays is suppressed rather than enhanced – see the dashed lines in Fig. 3. This photon antibunching effect has a very simple handwaving explanation: a single photon emitted by the wave source may be absorbed by just one of the detectors. For the initial state n =
1, this is the only option, and it is very natural that Eq. (36) predicts no simultaneous counts at = 0.
Despite this theoretical simplicity, reliable observations of the antibunching have not been carried out until 1977,19 due to the experimental difficulty of driving electromagnetic field oscillators into their Fock states – see Sec. 4 below.
(ii) The Glauber state . A similar procedure, but now using Eq. (5.124) and its Hermitian conjugate,
†
*
ˆ a , yields
ˆ†
a ˆ†
a ˆ a ˆ
* *
a
Glauber
(2)
g (0)
,
1
(9.37)
field
statistics
ˆ†
a ˆ
2
a
( *
)2
for any parameter . We see that the result is different from that for the Fock states, unless in the latter case n . (We know that the Fock and Glauber properties should also coincide for the ground state, but at that state the correlation function’s value is uncertain, because there are no photon counts at all.) (iii) Classical mixture. From Chapter 7, we know that such statistical ensembles cannot be described by single state vectors, and require the density matrix w for their description. Here, we may combine Eqs. (35) and (7.5) to write
Tr ˆ w ˆ†
a ˆ†
a ˆ a ˆ a
(2)
g (0)
.
(9.38)
Trˆ w ˆ†
a ˆ a 2
19 By H. J. Kimble et al. , Phys. Rev. Lett. 39, 691 (1977). For a detailed review of phonon antibunching, see, e.g., H. Paul, Rev. Mod. Phys. 54, 1061 (1982).
Chapter 9
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Spelling out this expression is easy for the field in thermal equilibrium at some temperature T, because its density matrix is diagonal in the basis of Fock states n – see Eqs. (7.24):
E
w W , W exp
(9.39)
nn
n nn
n
n
Z n
n ,
where
exp
.
'
'
k T
n
k T
B
0
B
So, for the operators in the numerator and denominator of Eq. (38) we also need just the diagonal terms of the operator products, which have already been calculated – see Eq. (36). As a result, we get
W n( n )
1
n n
n
n ( )1 n
(2)
g ( )
0
n0
n0
n0
.
(9.40)
2
2
n
W n
n
n
n0
n0
One of the three series involved in this expression is just the usual geometric progression, n
1
,
(9.41)
n
0
1
and the remaining two series may be readily calculated by its differentiation over the parameter :
n
n
d
n
d
1
1
n n
,
n
n
d n
d
0
0
0
1
1
(
)2
(9.42)
2
2
n
n
d
n
d
2
2
2
2
1
2 2
n( n )1 n( n )1
,
2
2
n
n
d n
d
0
0
0
1
1
(
)3
and for the correlation function we get an extremely simple result independent of the parameter and hence of temperature:
(2
2 2
)
1
/(
)3 1 1
/(
)
g (0)
(9.43) Photon
1
/( )
2.
2
2
bunching
This is exactly the photon bunching effect first observed by Hanbury Brown and Twiss – see Fig.
3. We see that in contrast to antibunching, this is an essentially classical (statistical) effect. Indeed, Eq.
(43) allows a purely classical derivation. In the classical theory, the counting rate (of a single counter) is proportional to the wave intensity I, so that Eq. (31) with = 0 is reduced to
2
I
(2)
g ( )
0
, with
2
I E ( t)
*
E
.
(9.44)
2
E
I
For a sinusoidal field, the intensity is constant, and g(2)(0) = 1. (This is also evident from Eq. (37), because the classical state may be considered as a Glauber state with .) On the other hand, if the intensity fluctuates (either in time, or from one experiment to another), the averages in Eq. (44) should be calculated as
I k
(
w I ) I k dI, with
(
w I ) dI ,
1
and k ,
1 2
,
(9.45)
0
0
Chapter 9
Page 12 of 36
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where w( I) is the probability density. For classical statistics, the probability is an exponential function of the electromagnetic field energy, and hence its intensity:
w( I ) Ce I ,
where 1/ k T ,
(9.46)
B
so that Eqs. (45) yield:20
C
exp I dI C / ,
1
hence
and
C
,
0
(9.47)
k
k
k
1
1/ , for k 1,
I
(
w I ) I dI C
exp I I dI
exp k
d
k
k
0
0
0
2 / 2 for
,
2.
Plugging these results into Eq. (44), we get g(2)(0) = 0, in complete agreement with Eq. (43).
For some field states, including the squeezed ground states discussed at the end of Sec. 5.5, values g(2)(0) may be even higher than 2 – the so-called super-bunching. Analyses of two cases of such super-bunching are offered for the reader’s exercise – see the problem list in the chapter’s end.
9.3. Photon emission: spontaneous and stimulated
In our simple model of photon counting, considered in the last section, the trigger atom in the counter absorbed a photon. Now let us have a look at the opposite process of spontaneous emission of photons by an atom in an excited state, still using the same electric-dipole approximation (24) for the atom-to-field interaction. For this, we may still use the Golden Rule for the model depicted in Fig. 6.12, but now the roles have changed: we have to associate the operator A ˆ with the electric dipole moment of the atom, while the operator B ˆ , with the electric field, so that the continuous spectrum of the system b represents the plurality of the electromagnetic field modes into which the spontaneous radiation may happen. Since now the transition increases the energy of the electromagnetic field, and decreases that of the atom, after the multiplication of the field bracket in Eq. (27a) by exp{- i t}, and the second, by exp{+ i t}, we may keep only the photon creation operator whose time evolution (26) compensates this additional fast “rotation”. As a result, the Golden Rule takes the following form:
Spontaneous
2
2
photon
fin
ˆ†
ˆ
a 0
fin d e(r) ini
,
(9.48)
s
f
emission
rate
where all operators and states are time-independent (i.e. taken in the Schrödinger picture), and f is the density of final states of the electromagnetic field – which in this problem plays the role of the atom’s environment.21 Here the electromagnetic field oscillator has been assumed to be initially in the ground state – the assumption that will be changed later in this section.
This relation, together with Eq. (19), shows that for the field’s matrix element be different from zero, the final state of the field has to be the first excited Fock state, n = 1. (By the way, this is exactly 20 See, e.g., MA Eq. (6.7c) with n = 0 and n = 1.
21 Here the sum over all electromagnetic field modes j may be smuggled back. Since in the quasi-static approximation kja << 1, which is necessary for the interaction representation by Eq. (24), the matrix elements in Eq. (48) are virtually independent on the direction of the wave vectors, and their magnitudes are fixed by , the summation is reduced to the calculation of the total f for all modes, and the averaging of e 2(r) – see below.
Chapter 9
Page 13 of 36
QM: Quantum Mechanics
the most practicable way of generating an excited Fock state of a field oscillator.) With that, Eq. (48) yields
2
2
ˆ
ˆ
Γ fin
d e(r) ini
fin
e
d (r) ini
,
(9.49)
s
f
d
f
where the density f of the excited electromagnetic field states should be calculated at the energy E =
, and ed is the Cartesian component of the vector e(r) along the electric dipole’s direction. The expression for the density f was our first formula in this course – see Eq. (1.1).22 From it, we get 2
dN
V
,
(9.50)
f
2
3
dE
c
where the bounding volume V should be large enough to ensure spectrum’s virtual continuity: V >> 3 =
(2 c/)3. Because of that, in the normalization condition used to simplify Eq. (9), we may consider e 2(r) constant. Let us represent this square as a sum of squares of the three Cartesian components of the vector e(r): one of those ( ed) aligned with the dipole’s direction; due to the space isotropy we may write 2
2
2
2
e e e e 3 2
e .
(9.51)
d
1
2
d
As a result, the normalization condition yields
2
1
e
.
(9.52)
d
3 V
0
and Eq. (49) gives the famous (and very important) formula23
Free-space
1
4 3
2
1
4 3
spontaneous
*
ˆ
ˆ
ˆ
Γ
fin
d ini
fin d ini
ini
d fin
.
(9.53)
s
emission
4 3 3
c
4 3 3
c
rate
0
0
Leaving a comparison of this formula with the classical theory of radiation,24 and the exact evaluation of s for a particular transition in the hydrogen atom, for reader’s exercises, let me just estimate its order of magnitude. Assuming that d ~ er B e2/ m e( e 2/40) and ~ E H m e( e 2/40)2/2, and taking into account the definition (6.62) of the fine structure constant 1/137, we get 3
Γ
2
e
~
3
~ 3107.
(9.54)
4
c
0
This estimate shows that the emission lines at atomic transitions are typically very sharp. With the present-day availability of high-speed electronics, it also makes sense to evaluate the time scale = 1/
of the typical quantum transition: for a typical optical frequency ~ 31015 s-1, it is close to 1 ns. This is 22 If the same atom is placed into a high- Q resonant cavity (see, e.g., EM 7.9), the rate of its photon emission is strongly suppressed at frequencies between the cavity resonances (where f 0) – see, e.g., the review by S.
Haroche and D. Klepner, Phys. Today 42, 24 (Jan. 1989). On the other hand, the emission is strongly (by a factor
~ (3/ V) Q, where V is cavity’s volume) enhanced at resonance frequencies – the so-called Purcell effect, discovered by E. Purcell in the 1940s. For a brief discussion of this and other quantum electrodynamic effects in cavities, see the next section.
23 This was the breakthrough result obtained by P. Dirac in 1927, which jumpstarted the whole field of quantum electrodynamics. An equivalent expression was obtained from more formal arguments in 1930 by V. Weisskopf and E. Wigner, so that sometimes Eq. (53) is (very unfairly) called the “Weisskopf-Wigner formula”.
24 See, e.g., EM Sec. 8.2, in particular Eq. (8.29).
Chapter 9
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exactly the time constant that determines the time-delay dependence of the photon counting statistics of the spontaneously emitted radiation – see Fig. 3. Colloquially, this is the temporal scale of the photon emitted by an atom.25
Note, however, that the above estimate of is only valid for a transition with a non-zero electric-dipole matrix element. If it equals zero, i.e. the transition does not satisfy the selection rules,26 – say, due to the initial and final state symmetry – it is “forbidden”. The “forbidden” transition may still take place due to a different, smaller interaction (say, via a magnetic dipole field of the atom, or its quadrupole electric field27), but takes much longer. In some cases the increase of is rather dramatic –
sometimes to hours! Such long-lasting radiation is called the luminescence – or the fluorescence if the initial atom’s excitation was due to external radiation of a higher frequency, followed first by non-radiative transitions down the ladder of energy levels.
Now let us consider a more general case when the electromagnetic field mode of frequency is initially in an arbitrary Fock state n, and from it may either get energy from the atomic system ( photon emission) or, vice versa, give such energy back to the atom ( photon absorption). For the photon emission rate, an evident generalization of Eq. (48) gives
2
fin
ˆ†
a n
e
nfin
,
(9.55)
2
s
0 1
1 ˆ†
a 0
where both brackets should be calculated in the Schrödinger picture, and s is the spontaneous emission rate (48) of the same atomic system. According to the second of Eqs. (19), at the photon emission, the final field state has to be the Fock state with n’ = n + 1, and Eq. (55) yields Stimulated
photon
( n )
1 .
(9.56)
e
s
emission rate
Thus the initial field increases the photon emission rate; this effect is called the stimulated emission of radiation. Note that the spontaneous emission may be considered as a particular case of the stimulated emission for n = 0, and hence interpreted as the emission stimulated by the ground state of the electromagnetic field – one more manifestation of the non-trivial nature of this “vacuum” state.
On the other hand, following the arguments of Sec. 2,28 for the description of radiation absorption, the photon creation operator has to be replaced with the annihilation operator, giving the rate ratio
25 The scale c of the spatial extension of the corresponding wave packet is surprisingly macroscopic – in the range of a few millimeters. Such a “human” size of spontaneously emitted photons makes the usual optical table, with its 1-cm-scale components, the key equipment for many optical experiments – see, e.g., Fig. 2.
26 As was already discussed in Sec. 5.6, for a single spin-less particle moving in a spherically-symmetric potential (e.g., a hydrogen-like atom), the orbital selection rules are simple: the only allowed electric-dipole transitions are those with l l fin- l ini = 1 and m m fin- m ini = 0 or 1. The simplest example of the transition that does not satisfy this rule, i.e. is “forbidden”, is that between the s-states ( l = 0) with n = 2 and n = 1; because of that, the lifetime of the lowest excited s-state of a hydrogen atom is as long as ~0.15 s.
27 See, e.g., EM Sec. 8.9.
28 Note, however, a major difference between the rate discussed in Sec. 2, and a in Eq. (57). In our current case, the atomic transition is still between two discrete energy levels (see Fig. 4 below), so that the rate a is Chapter 9
Page 15 of 36
QM: Quantum Mechanics
2
fin
ˆ a n
a
.
(9.57)
2
s
1 ˆ†
a 0
According to this relation and the first of Eqs. (19), the final state of the field at the photon absorption has to be the Fock state with n’ = n – 1, and Eq. (57) yields
Photon
n .
(9.58)
a
s
absorption
rate
The results (56) and (58) are usually formulated in terms of relations between the Einstein coefficients A and B defined in the way shown in Fig. 4, where the two energy levels are those of the atom, a is the rate of energy absorption from the electromagnetic field in its n th Fock state, and e is that of energy emission into the field, initially in the same state. In this notation, Eqs. (56) and (58) yield29
Einstein
A B B ,
(9.59) coefficients’
21
21
12
relation
because each of these coefficients equals the spontaneous emission rate s.
W
2
Fig. 9.4. The Einstein coefficients
E
B n
A B n
a
12
e
21
21
on the atomic quantum transition
diagram – cf. Fig. 7.6.
W 1
I cannot resist the temptation to use this point for a small detour – an alternative derivation of the Bose-Einstein statistics for photons. Indeed, in the thermodynamic equilibrium, the average probability flows between levels 1 and 2 (see Fig. 4 again) should be equal:30
W W ,
(9.60)
2
e
1
a
where W 1 and W 2 are the probabilities for the atomic system to occupy the corresponding levels, so that Eqs. (56) and (58) yield
W
n
W 1 n W n ,
i.e.
2
,
(9.61)
2 s
1 s
W
n 1
1
where n is the average number of photons in the field causing the interstate transitions. But, on the other hand, for an atomic subsystem only weakly coupled to its electromagnetic environment, we ought to have the Gibbs distribution of these probabilities:
W
exp{ E / k T}
E
2
2
B
exp
exp
.
(9.62)
W
exp{ E / k T}
k T
k T
1
1
B
B
B
proportional to f, the density of final states of the electromagnetic field, i.e. the same density as in Eq. (48) and beyond, while the rate (27) is proportional to a, the density of final (ionized) states of the “trigger” atom – more exactly, of it’s the electron released at its ionization.
29 These relations were conjectured, from very general arguments, by Albert Einstein as early as 1916.
30 This is just a particular embodiment of the detailed balance equation (7.198).
Chapter 9
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Requiring Eqs. (61) and (62) to give the same result for the probability ratio, we get the Bose-Einstein distribution for the electromagnetic field in thermal equilibrium:
1
n
(9.63)
exp{
/ k T} 1
B
- the same result as that obtained in Sec. 7.1 by other means – see Eq. (7.26b).
Now returning to the discussion of Eqs. (56) and (58), their very important implication is the possibility to achieve the stimulated emission of coherent radiation using the level occupancy inversion.
Indeed, if the ratio W 2/ W 1 is larger than that given by Eq. (62), the net power flow from the atomic system into the electromagnetic field,
power
1
,
(9.64)
s W 2 n
W n
1
may be positive. The necessary inversion may be produced using several ways, notably by intensive quantum transitions to level 2 from an even higher energy level (which, in turn, is populated, e.g., by absorption of external radiation, usually called pumping, at a higher frequency.) A less obvious, but crucial feature of the stimulated emission is spelled out by Eq. (55): as was mentioned above, it shows that the final state of the field after the absorption of energy from the atom is a pure (coherent) Fock state ( n + 1). Colloquially, one may say that the new, ( n + 1)st photon emitted from the atom is automatically in phase with the n photons that had been in the field mode initially, i.e. joins them coherently.31 The idea of stimulated emission of coherent radiation using population inversion32 was first implemented in the early 1950s in the microwave range ( masers) and in 1960 in the optical range ( lasers). Nowadays, lasers are ubiquitous components of almost all high-tech systems and constitute one of the cornerstones of our technological civilization.
A quantitative discussion of laser operation is well beyond the framework of this course, and I have to refer the reader to special literature,33 but still would like to briefly mention two key points: (i) In a typical laser, each generated electromagnetic field mode is in its Glauber (rather than the Fock) state, so that Eqs. (56) and (58) are applicable only for the n averaged over the Fock-state decomposition of the Glauber state – see Eq. (5.134).
(ii) Since in a typical laser n >> 1, its operation may be well described using quasiclassical theories that use Eq. (64) to describe the electromagnetic energy balance (with the addition of a term describing the energy loss due to field absorption in external components of the laser, including the useful load), plus the equation describing the balance of occupancies W 1,2 due to all interlevel transitions
– similar to Eq. (60), but including also the contribution(s) from the particular population inversion mechanism used in the laser. At this approach, the role of quantum mechanics in laser science is essentially reduced to the calculation of the parameter s for the particular system.
This role becomes more prominent when one needs to describe fluctuations of the laser field.
Here two approaches are possible, following the two options discussed in Chapter 7. If the fluctuations 31 It is straightforward to show that this fact is also true if the field is initially in the Glauber state – which is more typical for modes in practical lasers.
32 This idea may be traced back at least to an obscure 1939 publication by V. Fabrikant.
33 I can recommend, for example, P. Milloni and J. Eberly, Laser Physics, 2nd ed., Wiley, 2010, and a less technical text by A. Yariv, Quantum Electronics, 3rd ed., Wiley, 1989.
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are relatively small, one can linearize the Heisenberg equations of motion of the field oscillator operators near their stationary-lasing “values”, with the Langevin “forces” (also time-dependent operators) describing the fluctuation sources, and use these Heisenberg-Langevin equations to calculate the radiation fluctuations, just as was described in Sec. 7.5. On the other hand, near the lasing threshold, the field fluctuations are relatively large, smearing the phase transition between the no-lasing and lasing states. Here the linearization is not an option, but one can use the density-matrix approach described in Sec. 7.6, for the fluctuation analysis.34 Note that while the laser fluctuations may look like a peripheral issue, pioneering research in that field has led to the development of the general theory of open quantum systems, which was discussed in Chapter 7.
9.4. Cavity QED
Now I have to visit, at least in passing, the field of cavity quantum electrodynamics (usually called cavity QED for short) – the art and science of creating and using the entanglement between quantum states of an atomic system (either an atom, or an ion, or a molecule, etc.) and the electromagnetic field in a macroscopic volume called the resonant cavity (or just “resonator”, or just
“cavity”). This field is very popular nowadays, especially in the context of the quantum computation and communication research discussed in Sec. 8.5.35
The discussion in the previous section was based on the implicit assumption that the energy spectrum of the electromagnetic field interacting with an atomic subsystem is essentially continuous, so that its final state is spread among many field modes, effectively losing its coherence with the quantum state of the atomic subsystem. This assumption has justified using the quantum-mechanical Golden Rule for the calculation of the spontaneous and stimulated transition rates. However, the assumption becomes invalid if the electromagnetic field is contained inside a relatively small volume, with its linear size comparable with the radiation wavelength. If the walls of such a cavity mostly reflect, rather than absorb, radiation, then the 0th approximation the energy dissipation may be disregarded, and the particular solutions e j(r) of the Helmholtz equation (5) correspond to discrete, well-separated mode wave numbers kj and hence well-separated frequencies j.36 Due to the energy conservation, an atomic transition corresponding to energy E = E ini – E fin may be effective only if the corresponding quantum transition frequency E/ is close to one of these resonance frequencies.37 As a result of such resonant interaction, the quantum states of the atomic system and the resonant electromagnetic mode may become entangled.
A very popular approximation for the quantitative description of this effect is the so-called Rabi model,38 in which the atom is treated as a two-level system interacting with a single electromagnetic field mode of the resonant cavity. (As was shown in Sec. 6.5, this model is justified, e.g., if transitions 34 This path has been developed (also in the mid-1960s), by several researchers, notably including M. Sully and W. Lamb – see, e.g., M. Sargent III, M. Scully, and W. Lamb, Jr., Laser Physics, Westview, 1977.
35 This popularity was demonstrated, for example, by the award of the 2012 Nobel Prize in Physics to cavity QED
experimentalists S. Haroche and D. Wineland.
36 The calculation of such modes and corresponding frequencies for several simple cavity geometries was the subject of EM Sec. 7.8 of this series.
37 On the contrary, if is far from any j, the interaction is suppressed; in particular, the spontaneous emission rate may be much lower than that given by Eq. (53) – so that this result is not as fundamental as it may look.
38 After the pioneering work by I. Rabi in 1936-37.
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between all other energy level pairs have considerably different frequencies.) As the reader knows well from Chapters 4-6 (see in particular Sec. 5.1), any two-level system may be described, just as a spin-½, by the Hamiltonian ˆ
bI c σˆ . Since we may always select the energy origin that b = 0, and the state basis in which c = cn z, the Hamiltonian of the atomic subsystem may be taken in the diagonal form Ω
H ˆ σ
c ˆ
σ ˆ ,
(9.65)
a
z
z
2
where 2 c = E is the difference between the energy levels in the absence of interaction with the field. Next, according to Eq. (17), ignoring the constant ground-state energy /2 (which may be always added to the energy at the end – if necessary), the contribution of a single field mode of frequency to the total Hamiltonian of the system is
H ˆ
a
ˆ† a ˆ .
(9.66)
f
Finally, according to Eq. (16a), the electric field of the mode may be represented as
1
1/2
ˆ
E (r, t)
e(r) ˆ a †
ˆ a
,
(9.67)
i 2
so that in the electric-dipole approximation (24), the cavity-atom interaction may be represented as a product of the field by some (say, y-) Cartesian component39 of the Pauli spin-½ operator:
1/2 1
ˆ
†
H
const ˆ
E const ˆ
a a
i
a a
,
(9.68)
int
y
y
ˆ ˆ
ˆ
ˆ †
ˆ
y
2
i
where is a coupling constant (with the dimension of frequency). The sum of these three terms, Rabi
Ω
†
ˆ
ˆ
ˆ
ˆ
H H H H
ˆ σ
Hamiltonian
ˆ a ˆ a i ˆ
ˆ a †
ˆ a
.
(9.69)
a
f
int
z
y
2
giving a very reasonable description of the system, is called the Rabi Hamiltonian. Despite its apparent simplicity, using this Hamiltonian for calculations is not that straightforward.40 Only in the case when the electromagnetic field is large and hence may be treated classically, the results following from Eq.
(69) are reduced to Eqs. (6.94) describing, in particular, the Rabi oscillations discussed in Sec. 6.3.
The situation becomes simpler in the most important case when the frequencies and are very close, enabling an effective interaction between the cavity field and the atom even if the coupling constant is relatively small. Indeed, if both the and the so-called detuning (defined similarly to the parameter used in Sec. 6.5),
Ω ,
(9.70)
39 The exact component is not important for final results, while intermediate formulas simplify if the interaction is proportional to either pure ôr pure ˆ .
x
y
40 For example, an exact quasi-analytical expression for its eigenenergies (as zeros of a Taylor series in the parameter , with coefficients determined by a recurrence relation) was found only recently – see D. Braak, Phys.
Rev. Lett. 107, 100401 (2011).
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are much smaller than , the Rabi Hamiltonian may be simplified using the rotating-wave approximation, already used several times in this course. For this, it is convenient to use the spin ladder operators, defined absolutely similarly for those of the orbital angular momentum – see Eqs. (5.153): ˆ
ˆ
ˆ ˆ iˆ , so that ˆ
.
(9.71)
x
y
y
i
2
From Eq. (4.105), it is very easy to find the matrices of these operators in the standard z-basis,
0 2
0 0
σ
,
σ
,
(9.72)
0 0
2 0
and their commutation rules – which turn out to be naturally similar to Eqs. (5.154):
ˆ ,ˆ
4 ˆ ,
ˆ ,ˆ
2 ˆ .
(9.73)
z
z
In this notation, the Rabi Hamiltonian becomes
Ω
ˆ
H
σˆ
†
ˆ a ˆ a
,
(9.74)
z
ˆ
†
ˆ
ˆ a ˆ a
2
2
and it is straightforward to use Eq. (4.199) and (73) to derive the Heisenberg-picture equations of motion for the involved operators. (Doing this, we have to remember that operators of the “spin”
subsystem, on one hand, and of the field mode, on the other hand, are defined in different Hilbert spaces and hence commute – at least at coinciding time moments.) The result (so far, exact!) is i
i
ˆ a i ˆ a
ˆ ˆ
a i a
,
ˆ†
ˆ†
ˆ
ˆ
,
2
2
(9.75)
ˆ
i Ω ˆ
2
i ˆ a ˆ†
a
ˆ
,
ˆ
i ˆ†
a ˆ a
z
z
ˆ
ˆ
.
At negligible coupling, 0, these equations have simple solutions,
ˆ a t
e i t
, ˆ†
a t ei t , ˆ
t
e i t
t
,
(9.76)
Ω
,
ˆ z const
and the small terms proportional to on the right-hand sides of Eqs. (75) cannot affect these time evolution laws dramatically even if is not exactly zero. Of those terms, ones with frequencies close to the “basic” frequency of each variable would act in resonance and hence may have a substantial impact on the system’s dynamics, while non-resonant terms may be ignored. In this rotating-wave approximation, Eqs. (75) are reduced to a much simpler system of equations:
i
i
ˆ a i ˆ a
ˆ
,
ˆ†
a i ˆ†
a
ˆ
,
2
2
(9.77)
ˆ
i Ω ˆ
2 iˆ†
a ˆ
,
ˆ
i Ω ˆ
2 iˆ a ˆ
,
ˆ
i ˆ†
a ˆ
ˆ a ˆ
z
z
z
.
Alternatively, these equations of motion may be obtained exactly from the Rabi Hamiltonian (74), if it is preliminary cleared of the terms proportional to
†
ˆ
ˆ a and ˆ
a ˆ , that oscillate fast and hence
self-average to produce virtually zero effect:
Chapter 9
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Jaynes-
Ω
Cummings
ˆ
H
σˆ
ˆ†
a ˆ
a
ˆ
ˆ a ˆ
ˆ†
a
.
(9.78)
z
,
at , ,Ω
Hamiltonian
2
2
This is the famous Jaynes-Cummings Hamiltonian,41 which is basic model used in the cavity QED and its applications.42 To find its eigenstates and eigenenergies, let us note that at negligible interaction (
0), the spectrum of the total energy E of the system, which in this limit is the sum of two independent contributions from the atomic and cavity-field subsystems,
Ω
E
n E
,
with n ,
1 ,...
2
0
,
(9.79)
2
n
2
consists43 of close level pairs (Fig. 5) centered to values
1
E
.
(9.80)
n
n
2
(At the exact resonance = , i.e. at = 0, each pair merges into one double-degenerate level En.) Since at 0 the two subsystems do not interact, the eigenstates corresponding to the sublevels of the n th pair may be represented by direct products of their independent state vectors:
n 1
and
- n ,
(9.81)
where the first ket of each product represents the state of the two-level (spin-½-like) atomic subsystem, and the second ket, that of the field oscillator.
. . .
. . .
. . .
Ω/2
E
2
E 3
/ 2
2
Ω/2 2
E
2
Ω/2 E
1
E
/ 2
1
Ω/2
E
1
E 0
Ω
E Ω/2
g
atom field
total system
Fig. 9.5. The energy spectrum (79) of the Jaynes-Cummings Hamiltonian in the limit << .
Note again that the energy is referred to the ground-state energy /2 of the cavity field.
As we know from Chapter 6, even weak interaction may lead to strong coherent mixing44 of quantum states with close energies (in this case, the two states (81) within each pair with the same n), 41 It was first proposed and analyzed in 1963 by two engineers, Edwin Jaynes and Fred Cummings, in a Proc.
IEEE publication, and it took the physics community a while to recognize and acknowledge the fundamental importance of that work.
42 For most applications, the baseline Hamiltonian (78) has to be augmented by additional term(s) describing, for example, the incoming radiation and/or the system’s coupling to the environment, for example, due to the electromagnetic energy loss in a finite- Q-factor cavity – see Eq. (7.68).
43 Only the ground state level E g = –/2 is non-degenerate – see Fig. 5.
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while their mixing with the states with farther energies is still negligible. Hence, at 0 < , << , a good approximation of the eigenstate with E En is given by a linear superposition of the states (81): Jaynes-
c c c n 1 c n ,
(9.82) Cummings
n
eigenstates
with certain c-number coefficients c. This relation describes the entanglement of the atomic eigenstates
and with the Fock states number n and n – 1 of the field mode. Let me leave the (straightforward) calculation of the coefficients ( c) for each of two entangled states (for each n) for the reader’s exercise.
(The result for the corresponding two eigenenergies ( En) may be again represented by the same anticrossing diagram as shown in Figs. 2.29 and 5.1, now with the detuning as the argument.) This calculation shows, in particular, that at = 0 (i.e. at = ), c+ = c- = 1/2 for both states of the pair.
This fact may be interpreted as a (coherent!) equal sharing of an energy quantum = by the atom and the cavity field at the exact resonance.
As a (hopefully, self-evident) by-product of the calculation of c is the fact that the dynamics of the state n described by Eq. (82), is similar to that of the generic two-level system that was repeatedly discussed in this course – the first time in Sec. 2.6 and then in Chapters 4-6. In particular, if the composite system had been initially prepared to be in one component state, for example 0 (i.e.
with the atom excited, while the cavity in its ground state), and then allowed to evolve on its own, after some time interval t ~ 1/ it may be found definitely in the counterpart state 1, including the first excited Fock state n = 1 of the field mode. If the process is allowed to continue, after the equal time interval t, the system returns to the initial state 0, etc. This most striking prediction of the Jaynes-Cummings model was directly observed, by G. Rempe et al., only in 1987, although less directly this model was repeatedly confirmed by numerous experiments carried out in the 1960s and 1970s.
This quantized version of the Rabi oscillations can only persist in time if the inevitable electromagnetic energy losses (not described by the basic Jaynes-Cummings Hamiltonian) are somehow compensated – for example, by passing a beam of particles, externally excited into the higher-energy state , though the cavity. If the losses become higher, the dissipation suppresses quantum coherence, in our case the coherence between two components of each pair (82), as was discussed in Chapter 7. As a result, the transition from the higher-energy atomic state to the lower-energy state , giving energy
to the cavity ( n – 1 n), which is then rapidly drained into the environment, becomes incoherent, so that the system’s dynamics is reduced to the Purcell effect, already mentioned in Sec. 3. A quantitative analysis of this effect is left for the reader’s exercise.
The number of interesting physics games one can play with such systems – say by adding external sources of radiation at a frequency close to and , in particular with manipulated time-dependent amplitude and/or phase, is always unlimited.45 Unfortunately, my time/space allowance for the cavity QED is over, and for further discussion, I have to refer the interested reader to special literature.46
44 In some fields, especially chemistry, such mixing is frequently called hybridization.
45 Most of them may be described by adding new terms to the basic Jaynes-Cummings Hamiltonian (78).
46 I can recommend, for example, either C. Gerry and P. Knight, Introductory Quantum Optics, Cambridge U.
Press, 2005, or G. Agarwal, Quantum Optics, Cambridge U. Press, 2012.
Chapter 9
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9.5. The Klein-Gordon and relativistic Schrödinger equations
Now let me switch gears and discuss the basics of relativistic quantum mechanics of particles with a non-zero rest mass m. In the ultra-relativistic limit pc >> mc 2 the quantization scheme of such particles may be essentially the same as for electromagnetic waves, but for the intermediate energy range, pc ~ mc 2, a more general approach is necessary. Historically, the first attempts47 to extend the non-relativistic wave mechanics into the relativistic energy range were based on performing the same transitions from classical observables to their quantum-mechanical operators as in the non-relativistic limit:
p ˆ
p i
E H ˆ
,
i .
(9.83)
t
The substitution of these operators, acting on the Schrödinger-picture wavefunction (r, t), into the classical relation (1) between the energy E and momentum p (for of a free particle) leads to the following formulas:
Table 9.1. Deriving the Klein-Gordon equation for a free relativistic particle. 48
Non-relativistic limit
Relativistic case
Classical
1
2
E
p
2
2
2
E c p mc 2
2
mechanics
2 m
2
Wave
1
i Ψ
i 2
Ψ i Ψ
2
c i
2 Ψ (
2
mc )2 Ψ
mechanics
t
2 m
t
The resulting equation for the non-relativistic limit, in the left-bottom cell of the table, is just the usual Schrödinger equation (1.28) for a free particle. Its relativistic generalization, in the right-bottom cell, usually rewritten as
Klein-
1 2
2
2
mc
Gordon
Ψ Ψ ,
0
with
,
(9.84)
equation
c 2 t 2
is called the Klein-Gordon (or sometimes “Klein-Gordon-Fock”) equation. The fundamental solutions of this equation are the same plane, monochromatic waves
Ψ r
( , t)
exp ik r t.
(9.85)
as in the non-relativistic case. Indeed, such waves are eigenstates of the operators (83), with eigenvalues, respectively,
p k,
and E
,
(9.86)
so that their substitution into Eq. (84) immediately returns us to Eq. (1) with the replacements (86): 47 This approach was suggested in 1926-1927, i.e. virtually simultaneously, by (at least) V. Fock, E. Schrödinger, O. Klein and W. Gordon, J. Kudar, T. de Donder and F.-H. van der Dungen, and L. de Broglie.
48 Note that in the left, non-relativistic column of this table, the energy is referred to the rest energy mc 2, while in its right, relativistic column, it is referred to zero – see Eq. (1).
Chapter 9
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E
2
2
2
ck mc
1/2
.
(9.87)
Though one may say that this dispersion relation is just a simple combination of the classical relation (1) and the same basic quantum-mechanical relations (86) as in non-relativistic limit, it attracts our attention to the fact that the energy as a function of the momentum k has two branches, with E–
(p) = – E+(p) – see Fig. 6a. Historically, this fact has played a very important role in spurring the fundamental idea of particle- antiparticle pairs. In this idea (very similar to the concept of electrons and holes in semiconductors, which was discussed in Sec. 2.8), what we call the “vacuum” actually corresponds to all quantum states of the lower branch, with energies E–(p) < 0, being completely filled, while the states on the upper branch, with energies E+(p) > 0, being empty. Then an externally supplied energy,
Δ E E E E
E
mc
,
(9.88)
2 2 0
may bring the system from the lower branch to the upper one (Fig. 6b). The resulting excited state is interpreted as a combination of a particle (formally, of the infinite spatial extension) with the energy E+
and the momentum p, and a “hole” (antiparticle) of the positive energy (– E-) and the momentum –p.
This idea49 has led to a search for, and discovery of the positron: the electron’s antiparticle with charge q
= + e, in 1932, and later of the antiproton and other antiparticles.
(a)
(b)
E
E
2
mc
ck
Fig. 9.6. (a) The free-particle
Δ E
dispersion relation resulting from
0
0
the Klein-Gordon and Dirac
k
2
k
mc
equations, and (b) the scheme of
E
creation of a particle-antiparticle
E
pair from the vacuum.
Free particles of a finite spatial extension may be described, in this approach, just as in the non-relativistic Schrödinger equation, by wave packets, i.e. linear superpositions of the de Broglie waves (85) with close wave vectors k, and the corresponding values of given by Eq. (87), with the positive sign for the “usual” particles, and negative sign for antiparticles – see Fig. 6a above. Note that to form, from a particle’s wave packet, a similar wave packet for the antiparticle, with the same phase and group velocities (2.33a) in each direction, we need to change the sign not only before , but also before k, i.e.
to replace all component wavefunctions (85), and hence the full wavefunction, with their complex conjugates.
Of more formal properties of Eq. (84), it is easy to prove that its solutions satisfy the same continuity equation (1.52), with the probability current density j still given by Eq. (1.47), but a different expression for the probability density w – which becomes very similar to that for j: i * Ψ
i
w
Ψ
c.c. ,
j
(9.89)
2
*
c.c.
2 mc
t
2 m
49 Due to the same P. A. M. Dirac!
Chapter 9
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– very much in the spirit of the relativity theory, treating space and time on equal footing. (In the non-relativistic limit p/ mc 0, Eq. (84) allows the reduction of this expression for w to the non-relativistic Eq. (1.22): w *.)
The Klein-Gordon equation may be readily generalized to describe a particle moving in external fields; for example, the electromagnetic field effects on a particle with charge q may be described by the same replacement as in the non-relativistic limit (see Sec. 3.1):
ˆ
ˆ
p P qAr, t
ˆ
ˆ
,
H H q(r, t) ,
(9.90)
where Pˆ i
is the canonical momentum operator (3.25), and the vector- and scalar potentials, A and , should be treated appropriately – either as c-number functions if the electromagnetic field quantization is not important for the particular problem, or as operators (see Secs. 1-4 above) if it is.
However, the practical value of the resulting relativistic Schrödinger equation is rather limited, for two main reasons. First of all, it does not give the correct description of particles with spin. For example, for the hydrogen-like atom/ion problem, i.e. the motion of an electron with the electric charge
– e, in the Coulomb central field of an immobile nucleus with charge + Ze, the equation may be readily solved exactly50 and yields the following spectrum of (doubly-degenerate) energy levels: 1
/ 2
2
2
Z
1/ 2
2
E mc 1
, with n l
Z
l
(9.91)
2
½2 2 2 ½,
where n = 1, 2,… and l = 0, 1,…, n – 1 are the same quantum numbers as in the non-relativistic theory (see Sec. 3.6), and 1/137 is the fine structure constant (6.62). The three leading terms of the Taylor expansion of this result in the small parameter Z are as follows:
2
2
4
4
2
Z
Z
n
3
E mc 1
.
(9.92)
2
4
2 n
2 n l ½ 4
The first of these terms is just the rest energy of the particle. The second term,
2
2
2 4
Z
mZ e
E
2
1
0
2
E mc
E Z E
(9.93)
n
2 2
n
4
n
n
0
, with
,
2
2 2 2
2 2
0
H
reproduces the non-relativistic Bohr’s formula (3.201). Finally, the third term,
4
4
Z
n
E
2
3
2 2 n n
3
mc
,
(9.94)
2 4
n l ½ 4
2
mc l ½ 4
is just the perturbative kinetic-relativistic contribution (6.51) to the fine structure of the Bohr levels (93).
However, as we already know from Sec. 6.3, for a spin-½ particle such as the electron, the spin-orbit interaction (6.55) gives an additional contribution to the fine structure, of the same order, so that the net result, confirmed by experiment, is given by Eq. (6.60), i.e. is different from Eq. (94). This is very natural, because the relativistic Schrödinger equation does not have the very notion of spin.
Second, even for massive spinless particles (such as the Z 0 bosons), for which this equation is believed to be valid, the most important problems are related to particle interactions at high energies of 50 This task is left for the reader’s exercise.
Chapter 9
Page 25 of 36
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the order of ~ 2 mc 2 and beyond – see Eq. (88). Due to the possibility of creation and annihilation of particle-antiparticle pairs at such energies, the number of particles participating in such interactions is typically considerable (and variable), and the adequate description of the system is given not by the relativistic Schrödinger equation (which is formulated in single-particle terms), but by the quantum field theory – to which I will devote only a few sentences in the very end of this chapter.
9.6. Dirac’s theory
The real breakthrough toward the quantum relativistic theory of electrons (and other spin-½
fermions) was achieved in 1928 by P. A. M. Dirac. For that time, the structure of his theory was highly nontrivial. Namely, while formally preserving, in the coordinate representation, the same Schrödinger-picture equation of quantum dynamics as in the non-relativistic quantum mechanics,51
i
ˆ
H ,
(9.95)
t
it postulates that the wavefunction it describes is not a scalar complex function of time and coordinates, but a four-component column-vector (sometimes called the bispinor) of such functions, its Hermitian-conjugate bispinor † being a 4-component row-vector of their complex conjugates:
Ψ (r, t)
1
Ψ (r, t)
Ψ
2
,
Ψ†
Ψ*(r, t), Ψ*(r, t), Ψ*(r, t), Ψ*(r, t) , (9.96)
1
2
3
4
Ψ (r, t)
3
Ψ (r, t)
4
and that the Hamiltonian participating in Eq. (95) is a 44 matrix defined in the Hilbert space of bispinors . For a free particle, the postulated Hamiltonian looks amazingly simple: 52
51 After the “naturally-relativistic” form of the Klein-Gordon equation (84), this apparent return to the non-relativistic Schrödinger equation may look very counter-intuitive. However, it becomes a bit less surprising taking into account the fact (whose proof is left for the reader’s exercise) that Eq. (84) may be also recast into the form (95) for a two-component column-vector (sometimes called spinor), with a Hamiltonian which may be represented by a 22 matrix – and hence expressed via the Pauli matrices (4.105) and the identity matrix I.
52 Moreover, if the time derivative participating in Eq. (95), and the three coordinate derivatives participating (via the momentum operator) in Eq. (97), are merged into one 4-vector operator / xk {, /( ct)}, the Dirac equation (95) may be rewritten in an even simpler, manifestly Lorentz-invariant 4-vector form (with the implied summation over the repeated index k = 1, ..., 4 – see, e.g., EM Sec. 9.4):
iσ
ˆ γ
Ψ ,
0
where ˆγ
k
ˆ γ , ˆ γ , ˆ γ
1
2
3
0 - ˆ , γˆ βˆ ,
x
i ˆ
k
σ
0
4
where μ mc/ – just as in Eq. (84). Note also that, very counter-intuitively, the Dirac Hamiltonian (97) is linear in the momentum, while the non-relativistic Hamiltonian of a particle, as well as the relativistic Schrödinger equation, are quadratic in p. In my humble opinion, the Dirac theory (including the concept of antiparticles it has inspired) may compete for the title of the most revolutionary theoretical idea in physics of all times, despite such strong contenders as Newton’s laws, Maxwell’s equations, Gibbs’ statistical distribution, Bohr’s theory of the hydrogen atom, and Einstein’s general relativity.
Chapter 9
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QM: Quantum Mechanics
Free-
particle’s
2
ˆ
H c ˆα ˆ
ˆ
p mc .
(9.97)
Hamiltonian
where pˆ = – i is the same 3D vector operator of momentum as in the non-relativistic case, while the operators αˆ and ˆ may be represented in the following shorthand 22 form:
Dirac
0ˆ ˆσ
I ˆ 0ˆ
ˆα
β ˆ
,
operators
.
(9.98a)
ˆ
σ 0ˆ
ˆ
0ˆ I
The operator αˆ , composed of the Pauli vector operators σˆ , is also a vector in the usual 3D
space, with each of its 3 Cartesian components being a 44 matrix. The particular form of the 22
matrices corresponding to the operators σând I în Eq. (98a) depends on the basis selected for the spin state representation; for example, in the standard z-basis, in which the Cartesian components of σâre represented by the Pauli matrices (4.105), the 44 matrix form of Eq. (98a) is
0 0 0 1
0 0 0 i
0 0 1 0
1 0 0
0
0 0 1 0
0 0
i
0
0 0 0
1
0 1 0
0
α
, α
, α
, β
. (9.98b)
x
y
z
0 1 0 0
0 i 0
0
1
0
0
0
0 0 1
0
1 0 0 0
i
0
0
0
0 1 0 0
0 0 0
1
It is straightforward to use Eqs. (98) to verify that the matrices x, y, z and satisfy the following relations:
α2 α2 α2 β2 ,
I
(9.99)
x
y
z
α α α α α α α α α α α α α β βα α β βα α β βα 0 , (9.100)
x
y
y
x
y
z
z
y
z
x
x
z
x
x
y
y
z
z
i.e. anticommute.
Using these commutation relations, and acting essentially as in Sec. 1.4, it is straightforward to show that any solution to the Dirac equation obeys the probability conservation law, i.e. the continuity equation (1.52), with the probability density:
†
w
,
(9.101)
and the probability current,
j Ψ† c ˆα Ψ ,
(9.102)
looking almost as in the non-relativistic wave mechanics – cf. Eqs. (1.22) and (1.47). Note, however, the Hermitian conjugation used in these formulas instead of the complex conjugation, to form the scalars w, jx, jy, and jz from the 4-component state vectors (96).
This close similarity is extended to the fundamental, plane-wave solutions of the Dirac equations is free space. Indeed, plugging such solution, in the form
u 1
k r
u
i
t
2 ikr
t
Ψ u e
e
,
(9.103)
u
3
u
4
Chapter 9
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