m U x E
dx
a
transparency
x
x
c
c
with the pre-exponential factor equal to 1 – the fact which might be readily expected from the mnemonic rule (i) of the connection formulas.
This formula is broadly used in applied quantum mechanics, despite the approximate character of its pre-exponential coefficient for insufficiently soft barriers that do not satisfy Eq. (107). For example, Eq. (80) shows that for a rectangular barrier with thickness d >> , the WKB approximation (117) with d WKB = d underestimates T by a factor of [4 k/( k 2 + 2)]2 – equal, for example, 4, if k = , i.e.
if U 0 = 2 E. However, on the appropriate logarithmic scale (see Fig. 7b), such a factor, smaller than an order of magnitude, is just a small correction.
Note also that when E approaches the barrier’s top U max (Fig. 11), the points x c and x c ’ merge, so that according to Eq. (117), TWKB 1, i.e. the particle reflection vanishes at E = U max. So, the WKB
approximation does not describe the effect of the over-barrier reflection at E > U max. (This fact could be noticed already from Eq. (95): in the absence of the classical turning points, the WKB probability current is constant for any barrier profile.) This conclusion is incorrect even for apparently smooth barriers where one could naively expect the WKB approximation to work perfectly. Indeed, near the point x = x m where the potential reaches maximum (i.e. U( x m) = U max), we may always approximate any smooth function U( x) with the quadratic term of the Taylor expansion, i.e. with an inverted parabola: 2
m x x
0
m 2
U ( x) U
.
(2.118)
max
2
25 For the most important case TWKB << 1, Eq. (117) may be simply derived from Eqs. (105)-(106) – the exercise left for the reader.
Chapter 2
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Calculating derivatives dU/dx and d 2 U/dx 2 of this function and plugging them into the condition (107), we may see that the WKB approximation is only valid if U max – E >> 0. Just for the reader’s reference, an exact analysis of tunneling through the barrier (118) gives the following Kemble formula:26
1
T
,
(2.119) Kemble
1
exp 2 ( E U
) /
formula
max
0
valid for any sign of the difference ( E – U max). This formula describes a gradual approach of T to 1, i.e.
a gradual reduction of reflection, at the particle energy’s increase, with T = ½ at E = U max.
The last remark of this section: the WKB approximation opens a straight way toward an alternative formulation of quantum mechanics, based on the Feynman path integral. However, I will postpone its discussion until a more compact notation has been introduced in Chapter 4.
2.5. Resonant tunneling, and metastable states
Now let us move to other, conceptually different quantum effects, taking place in more elaborate potential profiles. Neither piecewise-constant nor smooth-potential models of U( x) are convenient for their quantitative description because they both require “stitching” partial de Broglie waves at each classical turning point, which may lead to cumbersome calculations. However, we may get a very good insight of the physics of quantum effects that may take place in such profiles, using their approximation by sets of Dirac’s delta functions.
Additional help in studying such effects is provided by the notions of the scattering and transfer matrices, very useful for other cases as well. Consider an arbitrary but finite-length potential “bump”
(formally called a scatterer), localized somewhere between points x 1 and x 2, on the flat potential background, say U = 0 (Fig. 12).
U ( x)
A
A
1
2
E
B
B
1
2
Fig. 2.12. De Broglie wave amplitudes
near a single 1D scatterer.
x
x
x
1
0
2
From Sec. 2, we know that the general solutions of the stationary Schrödinger equation, with a certain energy E, outside the interval [ x 1, x 2] are sets of two sinusoidal waves, traveling in the opposite directions. Let us represent them in the form
ik( x x )
ik( x x )
j
j
A e
B e
,
(2.120)
j
j
j
26 This formula was derived (in a more general form, valid for an arbitrary soft potential barrier) by E. Kemble in 1935. In some communities, it is known as the “Hill-Wheeler formula”, after D. Hill and J. Wheeler’s 1953 paper in that the Kemble formula was spelled out for the quadratic profile (118). Note that mathematically Eq. (119) is similar to the Fermi distribution in statistical physics, with an effective temperature T ef = 0/2 k B. This coincidence has some curious implications for the Fermi particle tunneling statistics.
Chapter 2
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where the index j (for now) is equal to either 1 or 2, and ( k)2/2 m = E. Note that each of the two wave pairs (129) has, in this notation, its own reference point xj, because this is very convenient for what follows. As we have already discussed, if the de Broigle wave/particle is incident from the left (i.e. B 2 =
0), the solution of the linear Schrödinger equation within the scatterer range ( x 1 < x < x 2) can provide only linear expressions for the transmitted ( A 2) and reflected ( B 1) wave amplitudes via the incident wave amplitude A 1:
A S A , B S A ,
(2.121)
2
21 1
1
11 1
where S 11 and S 21 are certain (generally, complex) coefficients. Alternatively, if a wave, with amplitude B 2, is incident on the scatterer from the right (i.e. if A 1 = 0), it can induce a transmitted wave ( B 1) and a reflected wave ( A 2), with amplitudes
B S B , A S B ,
(2.122)
1
12
2
2
22
2
where the coefficients S 22 and S 12 are generally different from S 11 and S 21. Now we can use the linear superposition principle to argue that if the waves A 1 and B 2 are simultaneously incident on the scatterer (say, because the wave B 2 has been partly reflected back by some other scatterer located at x > x 2), the resulting scattered wave amplitudes A 2 and B 1 are just the sums of their values for separate incident waves:
B S A S B ,
1
11 1
12
2
(2.123)
A S A S B .
2
21 1
22
2
These linear relations may be conveniently represented using the so-called scattering matrix S: Scattering
B
A
S
S
1
1
11
12
matrix:
S
,
S
with
definition
.
(2.124)
A
B
S
S
2
2
21
22
Scattering matrices, duly generalized, are an important tool for the analysis of wave scattering in more dimensions than one; for 1D problems, however, another matrix is often more convenient to represent the same linear relations (123). Indeed, let us solve this system for A 2 and B 2. The result is Transfer
A T A T B ,
A
A
matrix:
2
11 1
12
1 i.e.
2
T 1 ,
(2.125)
definition
B T A T B ,
B
B
2
21 1
22
1
2
1
where T is the transfer matrix, with the following elements:
S S
S
S
1
11 22
T S
,
22
T
,
11
T
, T
.
(2.126)
11
21
12
21
22
S
S
S
S
12
12
21
12
The matrices S and T have some universal properties, valid for an arbitrary (but time-independent) scatterer; they may be readily found from the probability current conservation and the time-reversal symmetry of the Schrödinger equation. Let me leave finding these relations for the reader’s exercise. The results show, in particular, that the scattering matrix may be rewritten in the following form:
i
re
t
S i
e
i ,
(2.127a)
t
re
where four real parameters r, t, , and satisfy the following universal relation: Chapter 2
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2
2
r t 1,
(2.127b)
so that only 3 of these parameters are independent. As a result of this symmetry, T 11 may be also represented in a simpler form, similar to T
*
*
22: T 11 = exp{ i}/ t = 1/ S 12 = 1/ S 21 . The last form allows a ready expression of the scatterer’s transparency via just one coefficient of the transfer matrix: 2
A
2
2
2
T
S
T
.
(2.128)
21
11
A 1 B 0
2
In our current context, the most important property of the 1D transfer matrices is that to find the total transfer matrix T of a system consisting of several (say, N) sequential arbitrary scatterers (Fig. 13), it is sufficient to multiply their matrices.
A
A
A
A
1
2
3
N 1
B
B
B
B
1
2
3
N 1
Fig. 2.13. A sequence of several 1D
scatterers.
x
x
x
x
x
1
2
3
N 1
Indeed, extending the definition (125) to other points xj ( j = 1, 2, …, N + 1), we can write
A
A
A
A
A
2
T
1 ,
3
T
2
T T
1 , etc.
(2.129)
1
2
2 1
B
B
B
B
B
2
1
3
2
1
(where the matrix indices correspond to the scatterers’ order on the x-axis), so that
AN
A
1
T T
T
...
1 .
(2.130)
N
N 1
1
B
B
N 1
1
But we can also define the total transfer matrix similarly to Eq. (125), i.e. as
AN
A
1
T 1 ,
(2.131)
B
B
N 1
1
so that comparing Eqs. (130) and (131) we get
Transfer
matrix:
T T T
T
... .
(2.132)
N
N 1
1
composite
scatterer
This formula is valid even if the flat-potential gaps between component scatterers are shrunk to zero, so that it may be applied to a scatterer with an arbitrary profile U( x), by fragmenting its length into many small segments x = xj+1 – xj, and treating each fragment as a rectangular barrier of the average height ( Uj)ef = [ U( xj+1) – U( xj)]/2 – see Fig. 14. Since very efficient numerical algorithms are readily available for fast multiplication of matrices (especially as small as 2×2 in our case), this approach is broadly used in practice for the computation of transparency of potential barriers with complicated profiles U( x). (Computationally, this procedure is much more efficient than the direct numerical solution of the stationary Schrödinger equation.)
Chapter 2
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U ( x)
A
( U )
j ef
A
1
N 1
B
B
1
N 1
Fig. 2.14. The transfer matrix approach
to a potential barrier with an arbitrary
profile.
x
x
x x
2
x
x
1
j
j 1
N 1
In order to apply this approach to several particular, conceptually important systems, let us calculate the transfer matrices for a few elementary scatterers, starting from the delta-functional barrier located at x = 0 – see Fig. 8. Taking x 1, x 2 0, we can merely change the notation of the wave amplitudes in Eq. (78) to get
i
1
S
,
S
.
(2.133)
11
1
21
i
1 i
An absolutely similar analysis of the wave incidence from the left yields
i
1
S
,
S
,
(2.134)
22
1
12
i
1 i
and using Eqs. (126), we get
Transfer
1 i
i
matrix:
T
short
.
(2.135)
scatterer
i
1 i
As a sanity check, Eq. (128) applied to this result, immediately brings us back to Eq. (79).
The next example may seem strange at the first glance: what if there is no scatterer at all between the points x 1 and x 2? If the points coincide, the answer is indeed trivial and can be obtained, e.g., from Eq. (135) by taking W = 0, i.e. = 0:
1 0
Identity
T
I
(2.136)
0
matrix
0 1
- the so-called identity matrix. However, we are free to choose the reference points x 1,2 participating in Eq. (120) as we wish. For example, what if x 2 – x 1 = a? Let us first take the forward-propagating wave alone: B 2 = 0 (and hence B 1 = 0); then
ik( x x )
ik( x x ) ik( x x )
1
2
1
2
A e
A e
e
.
(2.137)
2
1
1
1
The comparison of this expression with the definition (120) for j = 2 shows that A 2 = A 1 exp{ ik( x 2 – x 1)}
= A 1 exp{ ika}, i.e. T 11 = exp{ ika}. Repeating the calculation for the back-propagating wave, we see that T 22 = exp{- ika}, and since the space interval provides no particle reflection, we finally get Transfer
matrix:
ika
e
0
spatial
T
,
(2.138)
a
interval
ika
0
e
independently of a common shift of points x 1 and x 2. At a = 0, we naturally recover the special case (136).
Chapter 2
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Now let us use these simple results to analyze the double-barrier system shown in Fig. 15. We could of course calculate its properties as before, writing down explicit expressions for all five traveling waves shown by arrows in Fig. 15, then using the boundary conditions (124) and (125) at each of points x 1,2 to get a system of four linear equations, and finally, solving it for four amplitude ratios.
a
W x x
W x x 2
1
E
Fig. 2.15. The double-barrier system. The
dashed lines show (schematically) the quasi-
levels of the metastable-state energies.
x
x
x
1
2
However, the transfer matrix approach simplifies the calculations, because we may immediately use Eqs. (132), (135), and (138) to write
1 i
i eika
0 1 i
i
T T T T
.
(2.139)
a
ika
i
1 i 0
e
i
1 i
Let me hope that the reader remembers the “row by column” rule of the multiplication of square matrices;27 using it for the last two matrices, we may reduce Eq. (139) to
1 i
i 1
(
ika
i ) e
ika
i e
T
.
(2.140)
ika
ika
i
1 i i e
1
( i ) e
Now there is no need to calculate all elements of the full product T, because, according to Eq. (128), for the calculation of barrier’s transparency T we need only one its element, T 11: 1
1
Double
T
.
(2.141) barrier:
2
2
2
T
ika
ika
transparency
11
e
1
( i )2 e
This result is somewhat similar to that following from Eq. (71) for E > U 0: the transparency is a
-periodic function of the product ka, reaching its maximum (T = 1) at some point of each period – see Fig. 16a. However, Eq. (141) is different in that for >> 1, the resonance peaks of the transparency are very narrow, reaching their maxima at ka kna n, with n = 1, 2, …
The physics of this resonant tunneling effect28 is the so-called constructive interference, absolutely similar to that of electromagnetic waves (for example, light) in a Fabry-Perot resonator formed by two parallel semi-transparent mirrors.29 Namely, the incident de Broglie wave may either N
27 In an analytical form: AB
A B , where N is the matrix rank (in our current case, N = 2).
jj'
jj" j"j'
j" 1
28 In older literature, it is sometimes called the Townsend (or “Ramsauer-Townsend”) effect. However, it is more common to use that term only for a similar effect at 3D scattering – to be discussed in Chapter 3.
29 See, e.g., EM Sec. 7.9.
Chapter 2
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tunnel through the two barriers or undertake, on its way, several sequential reflections from these semi-transparent walls. At k = kn, i.e. at 2 ka = 2 kna = 2 n, the phase differences between all these partial waves are multiples of 2, so that they add up in phase – “constructively”. Note that the same constructive interference of numerous reflections from the walls may be used to interpret the standing-wave eigenfunctions (1.84), so that the resonant tunneling at >> 1 may be also considered as a result of the incident wave’s resonance induction of such a standing wave, with a very large amplitude, in the space between the barriers, with the transmitted wave’s amplitude proportionately increased.
(a)
(b)
2
3
.
0
Im
1
0.8
1
2
1
0.6
T
k
k
0
Re
0.4
0
.
1
Fig. 2.16. Resonant tunneling through a
0.2
potential well with delta-functional walls:
0
.
3
(a) the system’s transparency as a
0
function of ka, and (b) calculating the
0
0.5
1
1.5
2
ka /
resonance’s FWHM at >> 1.
As a result of this resonance, the maximum transparency of the system is perfect (Tmax = 1) even at , i.e. in the case of very low transparency of each of the two component barriers. Indeed, the denominator in Eq. (141) may be interpreted as the squared length of the difference between two 2D
vectors, one of length 2, and another of length (1 – i)2 = 1 + 2, with the angle = 2 ka + const between them – see Fig. 16b. At the resonance, the vectors are aligned, and their difference is smallest (equal to 1) so that Tmax = 1. (This result is exact only if the two barriers are exactly equal.) The same vector diagram may be used to calculate the so-called FWHM, a common acronym for the Full Width [of the resonance curve at its] Half-Maximum. By definition, this is the difference k = k+
– k- between such two values of k, on the opposite slopes of the same resonance, at that T = Tmax/2 – see the arrows in Fig. 16a. Let the vectors in Fig. 16b, drawn for >> 1, be misaligned by a small angle
~ 1/2 << 1, so that the length of the difference vector is much smaller than the length of each vector. To double its length squared, and hence to reduce T by a factor of two in comparison with its maximum value 1, the arc between the vectors, equal to 2 , should also become equal to 1, i.e. 2(2 k a + const)
= 1. Subtracting these two equalities from each other, we get
1
k k k
k .
(2.142)
2
a
Now let us use the simple system shown in Fig. 15 to discuss an issue of large conceptual significance. For that, consider what would happen if at some initial moment (say, t = 0) we have placed a 1D quantum particle inside the double-barrier well with >> 1, and left it there alone, without any incident wave. To simplify the analysis, let us assume that the initial state of the particle coincides with one of the stationary states of the infinite-wall well of the same size – see Eq. (1.84): Chapter 2
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2 1/2
n
( x )
0
, ( x)
.
(2.143)
n
sin k ( x x )
k
n
n
1 ,
where
,
,
1 ,...
2
a
n
a
At , this is just an eigenstate of the system, and from our analysis in Sec. 1.5 we know the time evolution of its wavefunction:
2 1/2
E
k 2
( x, t) ( x) exp
sin
( ) exp ,
with
, (2.144)
n
i tn
k x x
n
1
i tn
n
n
a
n
2 m
telling us that the particle remains in the well at all times with constant probability W( t) = W(0) = 1.
However, if the parameter is large but finite, the de Broglie wave would slowly “leak out”
from the well, so that W( t) would slowly decrease. Such a state is called metastable. Let us derive the law of its time evolution, assuming that at the slow leakage, with a characteristic time >> 1/ n, does not affect the instant wave distribution inside the well, besides the gradual, slow reduction of W.30 Then we can generalize Eq. (144) as
W
2 1/ 2
( x, t)
sin k ( x x ) exp
exp
, (2.145)
n
1
i tn A i k x
t
n
n
B i k x
t
n
n
a
making the probability of finding the particle in the well equal to W 1. As the last form of Eq. (145) shows, this function is the sum of two traveling waves, with equal magnitudes of their amplitudes and equal but opposite probability currents (5):
1/ 2
W
W n
A B
,
2
I
A k
,
I I .
(2.146)
2
A
n
a
m
m 2
B
A
a a
But we already know from Eq. (79) that at >> 1, the delta-functional wall’s transparency T equals 1/2, so that the wave carrying current IA, incident on the right wall from the inside, induces an outcoming wave outside of the well (Fig. 17) with the following probability current:
1
1 n W
I T I
I
.
(2.147)
R
A
2
A
2
2
2 ma
~ v
gr
I
I
L
R
E 1
v t
0
v t
x
gr
gr
Fig. 2.17. Metastable state’s decay in the simple model of a 1D potential well
formed by two low-transparent walls – schematically.
Absolutely similarly,
1
I
I I .
(2.148)
L
2
B
R
30 This virtually evident assumption finds its formal justification in the perturbation theory to be discussed in Chapter 6.
Chapter 2
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Now we may combine the 1D version (6) of the probability conservation law for the well’s interior: dW
I I 0 ,
(2.149)
R
L
dt
with Eqs. (147)-(148) to write
dW
1 n
W .
(2.150)
dt
2
ma 2
This is just the standard differential equation,
Metastable
dW
1
state:
W ,
(2.151)
decay law
dt
of the exponential decay, W( t) = W(0)exp{- t/}, where the constant , in our case equal to 2
ma
2
,
(2.152)
n
is called the metastable state’s lifetime. Using Eq. (2.33b) for the de Broglie waves’ group velocity, for our particular wave vector giving v gr = kn/ m = n/ ma, Eq. (152) may be rewritten in a more general form,
Metastable
t
state:
a
,
(2.153)
lifetime
T
where the attempt time t a is equal to a/ v gr, and (in our particular case) T = 1/2, in which it is valid for a broad class of similar metastable systems.31 Equality may be interpreted in the following semi-classical way. The particle travels back and forth between the confining potential barriers, with the time interval t a between the sequential moments of incidence, each time attempting to leak through the wall, with the success probability equal to T, so the reduction of W per each incidence is W = – W T, in the limit >> 1 (i.e. T << 1) immediately leading to the decay equation (151) with the lifetime (153).
Another useful look at Eq. (152) may be taken by returning to the resonant tunneling problem in the same system, and expressing the resonance width (142) in terms of the incident particle’s energy: 2
2
2
2
2
k k
k
n
n
1
n
E
k
.
(2.154)
2
2
2
2 m
m
m
a
ma
Comparing Eqs. (152) and (154), we get a remarkably simple, parameter-independent formula32
Energy-time
uncertainty
E .
(2.155)
relation
31 Essentially the only requirement is to have the attempt time t A to be much longer than the effective time (the instanton time, see Sec. 5.3 below) of tunneling through the barrier. In the delta-functional approximation for the barrier, the latter time is equal to zero, so that this requirement is always fulfilled.
32 Note that the metastable state’s decay (2.151) may be formally obtained from the basic Schrödinger equation (1.61) by adding an imaginary part, equal to (- E/2), to its eigenenergy En. Indeed, in this case Eq. (1.62) becomes an( t) = constexp{ -i( En – i E/2} t/} constexp{ -iEnt/}exp{ - Et/2} = constexp{ -iEnt/}exp{ -
t/2}, so that W( t) an( t)2 exp{ -t/}. Such formalism, which hides the physical origin of the state’s decay, may be convenient for some calculations, but misleading in other cases, and I will not use it in this course.
Chapter 2
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This energy-time uncertainty relation is certainly more general than our simple model; for example, it is valid for the lifetime and resonance tunneling width of any metastable state in the potential profile of any shape. This seems very natural, since because of the energy identification with frequency, E = , typical for quantum mechanics, Eq. (155) may be rewritten as = 1 and seems to follow directly from the Fourier transform in time, just as the Heisenberg’s uncertainty relation (1.35) follows from the Fourier transform in space. In some cases, these two relations are indeed interchangeable; for example, Eq. (24) for the Gaussian wave packet width may be rewritten as E t =
, where E = ( d/ dk) k = v gr k is the r.m.s. spread of energies of monochromatic components of the packet, while t x/ v gr is the time scale of packet’s passage through a fixed observation point x.
However, Eq. (155) it is much less general than Heisenberg’s uncertainty relation (1.35). Indeed, in the non-relativistic quantum mechanics we are studying now, the Cartesian coordinates of a particle, the Cartesian components of its momentum, and the energy E are regular observables, represented by operators. In contrast, time is treated as a c-number argument, and is not represented by an operator, so that Eq. (155) cannot be derived in such general assumptions as Eq. (1.35). Thus the time-energy uncertainty relation should be used with caution. Unfortunately, not everybody is so careful. One can find, for example, wrong claims that due to this relation, the energy dissipated by any system performing an elementary (single-bit) calculation during a time interval t has to be larger than / t.33 Another incorrect statement is that the energy of a system cannot be measured, during a time interval t, with an accuracy better than / t.34
Now that we have a quantitative mathematical description of the metastable state’s decay (valid, again, only if >> 1, i.e. if >> t a), we may use it for discussion of two important conceptual issues of quantum mechanics. First, this is one of the simplest examples of systems that may be considered, from two different points of view, as either Hamiltonian (and hence time- reversible), or open (and hence irreversible). Indeed, from the former point of view, our particular system is certainly described by a time-independent Hamiltonian of the type (1.41), with the potential energy
U x W x x x x
(2.156)
1
2
- see Fig. 15 again. In this point of view, the total probability of finding the particle somewhere on the axis x remains equal to 1, and the full system’s energy, calculated from Eq. (1.23),
E
*
x t H ˆ
,
x, t d 3 x
,
(2.157)
remains constant and completely definite ( E = 0). On the other hand, since the “emitted” wave packets would never return to the potential well,35 it makes sense to look at the well’s region alone. For such a 33 On this issue, I dare to refer the reader to my own old work K. Likharev , Int. J. Theor. Phys. 21, 311 (1982), which provided a constructive proof (for a particular system) that at reversible computation, whose idea had been put forward in 1973 by C. Bennett (see, e.g., SM Sec. 2.3), energy dissipation may be lower than this apparent
“quantum limit”.
34 See, e.g., a discussion of this issue in the monograph by V. Braginsky and F. Khalili, Quantum Measurement, Cambridge U. Press, 1992.
35 For more realistic 2D and 3D systems, this statement is true even if the system as a whole is confined inside some closed volume, much larger than the potential well housing the metastable states. Indeed, if the walls Chapter 2
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truncated, open system (for which the space beyond the interval [ x 1, x 2] serves as its environment), the probability W of finding the particle inside this interval, and hence its energy E = WEn, decay exponentially per Eq. (151) – the decay equation typical for irreversible systems. We will return to the discussion of the dynamics of such open quantum systems in Chapter 7.
Second, the same model enables a preliminary discussion of one important aspect of quantum measurements. As Eq. (151) and Fig. 17 show, at t >> , the well becomes virtually empty ( W 0), and the whole probability is localized in two clearly separated wave packets with equal amplitudes, moving from each other with the speed v gr, each “carrying the particle away” with a probability of 50%. Now assume that an experiment has detected the particle on the left side of the well. Though the formalisms suitable for quantitative analysis of the detection process will not be discussed until Chapter 7, due to the wide separation x = 2 v gr t >> 2 v gr of the packets, we may safely assume that such detection may be done without any actual physical effect on the counterpart wave packet.36 But if we know that the particle has been found on the left side, there is no chance to find it on the right side. If we attributed the full wavefunction to all stages of this particular experiment, this situation might be rather confusing.
Indeed, that would mean that the wavefunction at the right packet’s location should instantly turn into zero – the so-called wave packet reduction (or “collapse”) – a hypothetical, irreversible process that cannot be described by the Schrödinger equation for this system, even including the particle detectors.
However, if (as was already discussed in Sec. 1.3) we attribute the wavefunction to a certain statistical ensemble of similar experiments, there is no need to involve such an artificial notion. The two-packet picture we have calculated (Fig. 17) describes the full ensemble of experiments with all systems prepared in the initial state (143), i.e. does not depend on the particle detection results. On the other hand, the “reduced packet” picture (with no wave packet on the right of the well) describes only a sub-ensemble of such experiments, in which the particles have been detected on the left side. As was discussed on classical examples in Sec. 1.3, for such redefined ensemble the probability distribution is rather different. So, the “wave packet reduction” is just a result of a purely accounting decision of the observer.37 I will return to this important discussion in Sec. 10.1 – on the basis of the forthcoming discussion of open systems in Chapters 7 and 8.
2.6. Localized state coupling, and quantum oscillations
Now let us discuss one more effect specific to quantum mechanics. Its mathematical description may be simplified using a model potential consisting of two very short and deep potential wells. For that, let us first analyze the properties of a single well of this type (Fig. 18), which may be modeled similarly to the short and high potential barrier – see Eq. (74), but with a negative “weight”: U x W x,
with W 0 .
(2.158)
providing such confinement are even slightly uneven, the emitted plane-wave packets will be reflected from them, but would never return to the well intact. (See SM Sec. 2.1 for a more detailed discussion of this issue.) 36 This argument is especially convincing if the particle’s detection time is much shorter than the time tc = 2 v gr t/ c, where c is the speed of light in vacuum, i.e. the maximum velocity of any information transfer.
37 “The collapse of the wavefunction after measurement represents nothing more than the updating of that scientist’s expectations.” N. D. Mermin, Phys. Today, 72, 53 (Jan. 2013).
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In contrast to its tunnel-barrier counterpart (74), such potential sustains a stationary state with a negative eigenenergy E < 0, and a localized eigenfunction , with 0 at x .
U ( x)
0
W
( x)
x
E 0
Fig. 2.18. Delta-functional
potential well and its localized
1/ 1/
eigenstate (schematically).
Indeed, at x 0, U( x) = 0, so the 1D Schrödinger equation is reduced to the Helmholtz equation (1.83), whose localized solutions with E < 0 are single exponents, vanishing at large distances:38
Ae x for
,
x ,
0
2
2
( x) x
E .
(2.159)
0
with
,
0
Ae x for
,
x ,
0
2 m
(The coefficients before the exponents have been selected equal to satisfy the boundary condition (76) of the wavefunction’s continuity at x = 0.) Plugging Eq. (159) into the second boundary condition, given by Eq. (75), but now with the negative sign before W, we get
2 m
A
A
W
A ,
(2.160)
2
in which the common factor A 0 may be canceled. This equation39 has one solution for any W > 0: m
W
,
(2.161)
0
2
and hence the system has only one (ground) localized state, with the following eigenenergy:40
2
2
2
0
W
m
E E
.
(2.162)
0
2 m
2 2
Now we are ready to analyze localized states of the two-well potential shown in Fig. 19:
a
a
U ( x) W x x ,
with W 0
.
(2.163)
2
2
Here we may still use the single-exponent solutions, similar to Eq. (159), for the wavefunction outside the interval [- a/2, + a/2], but inside the interval, we need to take into account both possible exponents: x
x
a
a
C e
C e
C sinh x
C cosh x
,
for
x ,
(2.164)
A
S
2
2
38 See Eqs. (56)-(58), with U 0 = 0.
39 Such algebraic equations for linear differential equations are frequently called characteristic.
40 Note that this E 0 is equal, by magnitude, to the constant E 0 that participates in Eq. (79). Note also that this result was actually already obtained, “backward”, in the solution of Problem 1.12(ii), but that solution did not address the issue of whether the calculated potential (158) could sustain any other localized eigenstates.
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with the parameter defined as in Eq. (159). The last of these equivalent expressions is more convenient because due to the symmetry of the potential (163) to the central point x = 0, the system’s eigenfunctions should be either symmetric (even) or antisymmetric (odd) functions of x (see Fig. 19), so that they may be analyzed separately, only for one half of the system, say x 0, and using just one of the hyperbolic function (164) in each case.
a / 2 U x
a / 2
0
x
S
E A
Fig. 2.19. A system of two coupled
E
potential wells, and its localized
S
A
eigenstates (schematically).
For the antisymmetric eigenfunction, Eqs. (159) and (164) yield
a
sinh x
,
0
for
x ,
C
2
(2.165)
A
A
a
a
a
sinh
exp x ,
for x,
2
2
2
where the front coefficient in the lower line has been selected to satisfy the condition (76) of the wavefunction’s continuity at x = + a/2 – and hence at x = – a/2. What remains is to satisfy the condition (75), with a negative sign before W, for the derivative’s jump at that point. This condition yields the following characteristic equation:
a
a
2 mW
a
a
a
0
sinh
cosh
sinh
,
1
i.e. coth
2
,
(2.166)
2
2
2
2
2
a
where 0, given by Eq. (161), is the value of for a single well, i.e. the reciprocal spatial width of its localized eigenfunction – see Fig. 18.
Figure 20a shows both sides of Eq. (166) as functions of the dimensionless product a, for several values of the parameter 0 a, i.e. of the normalized distance between the two wells. The plots show, first of all, that as the parameter 0 a is decreased, the LHS and RHS plots cross (i.e. Eq. (166) has a solution) at lower and lower values of a. At a << 1, the left-hand side of the last form of this equation may be approximated as 2/ a. Comparing this expression with the right-hand side of the characteristic equation, we see that this transcendental equation has a solution (i.e. the system has an antisymmetric localized state) only if 0 a > 1, i.e. if the distance a between the two narrow potential wells is larger than the following value,
2
1
a
,
(2.167)
min
0
W
m
which is equal to the characteristic spread of the wavefunction in a single well – see Fig. 18. (At a
a min, a 0, meaning that the state’s localization becomes weaker and weaker.) Chapter 2
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5
5
4
4
A
S
3
3
LHS (166)
(172)
LHS
2
5
.
1
2
0
.
1
5
.
1
1 a 5
.
0
1
0
RHS
0
.
1
a 5
.
0
0
RHS
0
0
0
1
2
3
a
0
1
2
3
a
Fig. 2.20. Graphical solutions of the characteristic equations of the two-well system, for: (a) the antisymmetric eigenstate (165), and (b) the symmetric eigenstate (171).
In the opposite limit of large distances between the potential wells, i.e. 0 a >> 1, Eq. (166) shows that a >> 1 as well, so that its left-hand side may be approximated as 2(1 + exp{ – a}), and the equation yields
1 exp a .
(2.168)
0
0 0
This result means that the eigenfunction is an antisymmetric superposition of two virtually unperturbed wavefunctions (159) of each partial potential well:
1
a
a
x
x x
x x
x x
,
(2.169)
A
R L ,
where
R
0
,
L
0
2
2
2
and the front coefficient is selected in such a way that if the eigenfunction 0 of each well is normalized, so is A. Plugging the middle (more exact) form of Eq. (168) into the last of Eqs. (159), we can see that in this limit the antisymmetric state’s energy is only slightly higher than the eigenenergy E 0 of a single well, given by Eq. (162):
2
2
m
E E
a E
W
a .
(2.170)
A
0 1
2
exp
0
,
where
0
exp
2
0
0
The s ymmetric eigenfunction has a form reminding Eq. (165), but still different from it:
a
cosh x
,
0
for
x ,
C
2
(2.171)
S
S
a
a
a
cosh
exp x ,
for
x,
2
2
2
giving a characteristic equation similar in structure to Eq. (166), but with a different left-hand side: a
a
0
1 tanh
2
.
(2.172)
2
a
Figure 20b shows both sides of this equation for several values of the parameter 0 a. It is evident that in contrast to Eq. (166), Eq. (172) has a unique solution (and hence the system has a localized symmetric Chapter 2
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eigenstate) for any value of the parameter 0 a, i.e. for any distance between the partial wells. In the limit of very close wells (i.e. their strong coupling), 0 a << 1, we get a << 1, tanh( a/2) 0, and Eq. (172) yields 20, leading to a four-fold increase of the eigenenergy’s magnitude in comparison with that of the single well:
m(2 W )2
E 4 E
,
for a 1.
(2.173)
S
0
2 2
0
The physical meaning of this result is very simple: two very close potential wells act (on the symmetric eigenfunction only!) together, so that their “weights” W U( x) dx just add up.
In the opposite, weak coupling limit, i.e. 0 a >> 1, Eq. (172) shows that a >> 1 as well, so that its left-hand side may be approximated as 2(1 – exp{ – a}), and the equation yields
1 exp a .
(2.174)
0
0 0
In this limit, the eigenfunction is a symmetric superposition of two virtually unperturbed wavefunctions (159) of each partial potential well:
1
,
(2.175)
S x
R x L x
2
and the eigenenergy is also close to the energy E 0 of a partial well, but is slightly lower: E E
a E
E E
,
(2.176)
S
0 1
2
exp 0
,
that
so
2
0
A
S
where is again given by the last of Eqs. (170).
So, the eigenenergy of the symmetric state is always lower than that of the antisymmetric state.
The physics of this effect (which remains qualitatively the same in more complex two-component systems, most importantly in diatomic molecules such as H2) is evident from the sketch of the wavefunctions A and S, given by Eqs. (165) and (171), in Fig. 19. In the antisymmetric mode, the wavefunction has to vanish at the center of the system, so that each its half is squeezed to one half of the system’s spatial extension. Such a squeeze increases the function’s gradient, and hence its kinetic energy (1.27), and hence its total energy. On the contrary, in the symmetric mode, the wavefunction effectively spreads into the counterpart well. As a result, it changes in space slower, and hence its kinetic energy is also lower.
Even more importantly, the symmetric state’s energy decreases as the distance a is decreased, corresponding to the effective attraction of the partial wells. This is a good toy model of the strongest (and most important) type of atomic cohesion – the covalent (or “chemical”) bonding.41 In the simplest case of the H2 molecule, each of two electrons of the system, in its ground state,42 reduces its kinetic energy by spreading its wavefunction around both hydrogen nuclei (protons), rather than being confined near one of them – as it had to be in a single atom. The resulting bonding is very strong: in chemical units, 429 kJ/mol, i.e. 18.6 eV per molecule. Perhaps counter-intuitively, this quantum-mechanical 41 Historically, the development of the quantum theory of such bonding in the H2 molecule (by Walter Heinrich Heitler and Fritz Wolfgang London in 1927) was the breakthrough decisive for the acceptance of the then-emerging quantum mechanics by the community of chemists.
42 Due to the opposite spins of these electrons, the Pauli principle allows them to be in the same orbital ground state – see Chapter 8.
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covalent bonding is even stronger than the strongest classical ( ionic) bonding due to electron transfer between atoms, leading to the Coulomb attraction of the resulting ions. (For example, the atomic cohesion in the NaCl molecule is just 3.28 eV.)
Now let us analyze the dynamic properties of our model system (Fig. 19) because such a pair of weakly coupled potential wells is our first example of the very important class of two-level systems.43 It is easiest to do in the weak-coupling limit 0 a >> 1, when the simple results (168)-(170) and (174)-(176) are quantitatively valid. In particular, Eqs. (169) and (175) enable us to represent the quasi-localized states of the particle in each partial well as linear combinations of its two eigenstates: 1
1
,
.
(2.177)
R x
S x A x
L x
S x A x
2
2
Let us perform the following thought (“gedanken”) experiment: place a particle, at t = 0, into one of these quasi-localized states, say R( x), and leave the system alone to evolve, so that 1
( x )
0
, ( x)
x x .
(2.178)
R
( )
( )
S
A
2
According to the general solution (1.69) of the time-independent Schrödinger equation, the time dynamics of this wavefunction may be obtained simply by multiplying each eigenfunction by the corresponding complex-exponential time factor:
1
E S
E A
( x, t)
( x)exp
( )exp
.
(2.179)
S
i
t
x
A
i
t
2
From here, using Eqs. (170) and (176), and then Eqs. (169) and (175) again, we get
1
i t
i t
iE t
( x, t)
( x) exp ( x)exp
exp
0
S
A
2
(2.180)
t
t
E t
( x) cos
i ( x) sin
exp
0
i
.
R
L
This result implies, in particular, that the probabilities W R and W L to find the particle, respectively, in the right and left wells change with time as
t
t
W cos2
,
W sin2
,
(2.181) Quantum
R
L
oscillations
mercifully leaving the total probability constant: W R + W L = 1. (If our calculation had not passed this sanity check, we would be in big trouble.)
This is the famous effect of quantum oscillations 44 of the particle’s wavefunction between two similar, coupled subsystems, with the frequency
43 As we will see later in Chapter 4, these properties are similar to those of spin-½ particles; hence two-level systems are frequently called the spin-½-like systems.
44 Sometimes they are called the Bloch oscillations, but more commonly the last term is reserved for a related but different effect in spatially-periodic systems – to be discussed in Sec. 8 below.
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2
E E
A
S
.
(2.182)
In its last form, this result does not depend on the assumption of weak coupling, though the simple form (181) of the oscillations, with its 100% probability variations, does. (Indeed, at a strong coupling of two subsystems, the very notion of the quasi-localized states R and L is ambiguous.) Qualitatively, this effect may be interpreted as follows: the particle, placed into one of the potential wells, tries to escape from it via tunneling through the potential barrier separating the wells. (In our particular system, shown in Fig. 17, the barrier is formed by the spatial segment of length a, which has the potential energy, U =
0, higher than the eigenstate energy – E 0.) However, in the two-well system, the particle can only escape into the adjacent well. After the tunneling into that counterpart well, the particle tries to escape from it, and hence comes back, etc. – very much as a classical 1D oscillator, initially deflected from its equilibrium position, at negligible damping.
Some care is required at using such interpretation for quantitative conclusions. In particular, let us compare the period T 2/ of the oscillations (181) with the metastable state’s lifetime discussed in the previous section. For our particular model, we may use the second of Eqs. (170) to write 4 E
t
0
exp a
T
a
a
a , (2.183)
0 ,
exp
a
0
exp 0 ,
for
1
2 E
2
0
0
where t a 2/0 2/ E 0 is the effective attempt time. On the other hand, according to Eq. (80), the transparency T of our potential barrier, in this limit, scales as exp{-20 a},45 so that according to the general relation (153), the lifetime is of the order of t aexp{20 a} >> T. This is a rather counterintuitive result: the speed of particle tunneling into a similar adjacent well is much higher than that, through a similar barrier, to the free space!
In order to show that this important result is not an artifact of our delta-functional model of the potential barrier, and also compare T and more directly, let us analyze the quantum oscillations between two weakly coupled wells, now assuming that the (symmetric) potential profile U( x) is sufficiently soft (Fig. 21), so that all its eigenfunctions S and A are at least differentiable at all points.46 If the barrier’s transparency is low, the quasi-localized wavefunctions R( x) and L( x) = R(- x) and their eigenenergies may be found approximately by solving the Schrödinger equations in one of the wells, neglecting the tunneling through the barrier, but the calculation of requires a little bit more care.
Let us write the stationary Schrödinger equations for the symmetric and antisymmetric solutions in the form
2
2
2
2
d
d
E U ( x)
E U x
(2.184)
A
A ,
A
( )
2
S
S ,
2
S
m dx
2
2
m dx
45 It is hard to use Eq. (80) for a more exact evaluation of T in our current system, with its infinitely deep potential wells, because the meaning of the wave number k is not quite clear. However, this is not too important, because in the limit 0 a >> 1, the tunneling exponent makes the dominant contribution into the transparency –
see, again, Fig. 2.7b.
46 Such a smooth well may have more than one quasi-localized eigenstate, so that the proper state (and energy) index n is implied in all remaining formulas of this section.
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multiply the former equation by S and the latter one by A, subtract them from each other, and then integrate the result from 0 to . The result is
2
2
2
d
d
( E E )
S
A
dx
.
dx
(2.185)
A
S S A
2
2
A
2
S
m
dx
dx
0
0
If U( x), and hence d 2A,S/ dx 2, are finite for all x, we may integrate the right-hand side by parts to get
2
d
d
S
A
( E E ) dx
.
(2.186)
A
S
S
A
A
S
2 m
dx
dx
0
0
U( x)
( x)
( x)
L
R
E
Fig. 2.21. Weak coupling between two
a
0
a
x
similar, soft potential wells.
2
x
2
x
'
c
c
So far, this result is exact (provided that the derivatives participating in it are finite at each point); for weakly coupled wells, it may be further simplified. Indeed, in this case, the left-hand side of Eq. (186) may be approximated as
E E
( E E ) dx A
S ,
(2.187)
A
S S A
2
0
because this integral is dominated by the vicinity of point x = a/2, where the second terms in each of Eqs. (169) and (175) are negligible, and the integral is equal to ½, assuming the proper normalization of the function R( x). On the right-hand side of Eq. (186), the substitution at x = vanishes (due to the wavefunction’s decay in the classically forbidden region), and so does the first term at x = 0, because for the antisymmetric solution, A(0) = 0. As a result, the energy half-split may be expressed in any of the following (equivalent) forms:
2
2
2
d
d
d
( )
0
A ( )
0
( )
0
R ( )
0
( )
0
L ( ).
0
(2.188)
2
S
R
L
m
dx
m
dx
m
dx
It is straightforward (and hence left for the reader’s exercise) to show that within the limits of the WKB approximation’s validity, Eq. (188) may be reduced to
x '
x '
c
t a
c
exp ( x' ) dx' ,
that
so
T
exp ( x' ) dx' ,
(2.189)
t
2
a
x
x
c
c
where t a is the time period of the classical motion of the particle, with the energy E E A E S, inside each well, the function ( x) is defined by Eq. (82), and x c and x c ’ are the classical turning points limiting the potential barrier at the level E of the particle’s eigenenergy – see Fig. 21. The result (189) is evidently a natural generalization of Eq. (183), so that the strong relationship between the times of Chapter 2
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particle tunneling into the continuum of states and into a discrete eigenstate, is indeed not specific for the delta-functional model. We will return to this fact, in its more general form, at the end of Chapter 6.
2.7. Periodic systems: Energy bands and gaps
Let us now proceed to the discussion of one of the most important issues of wave mechanics: particle motion through a periodic system. As a precursor to this discussion, let us calculate the transparency of the potential profile shown in Fig. 22 (frequently called the Dirac comb): a sequence of N similar, equidistant delta-functional potential barriers, separated by ( N – 1) potential-free intervals a.
a
a
I
T I
A
A
E
Fig. 2.22. Tunneling through a
Dirac comb: a system of N similar,
equidistant barriers, i.e. ( N – 1)
similar coupled potential wells.
x
x
x
x
1
2
N
According to Eq. (132), its transfer matrix is the following product
T T T T
T T
...
,
(2.190)
a
a
( N )
1 N terms
with the component matrices given by Eqs. (135) and (138), and the barrier height parameter defined by the last of Eqs. (78). Remarkably, this multiplication may be carried out analytically,47 giving 1
2
sin ka cos ka
2
T T
cos Nqa
sin Nqa ,
(2.191a)
11
2
sin
qa
where q is a new parameter, with the wave number dimensionality, defined by the following relation: cos qa cos ka sin .
ka
(2.191b)
For N = 1, Eqs. (191) immediately yield our old result (79), while for N = 2 they may be readily reduced to Eq. (141) – see Fig. 16a. Fig. 20 shows its predictions for two larger numbers N, and several values of the dimensionless parameter .
Let us start the discussion of the plots from the case N = 3, when three barriers limit two coupled potential wells between them. Comparison of Fig. 23a and Fig. 16a shows that the transmission patterns, and their dependence on the parameter , are very similar, besides that in the coupled-well system, each resonant tunneling peak splits into two, with the ka-difference between them scaling as 1/. From the discussion in the last section, we may now readily interpret this result: each pair of resonance peaks of transparency corresponds to the alignment of the incident particle’s energy E with the pair of energy levels E A, E S of the symmetric and antisymmetric states of the system. However, in contrast to the 47 This formula will be easier to prove after we have discussed the properties of Pauli matrices in Chapter 4.
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system shown in Fig. 19, these states are metastable, because the particle may leak out from these states just as it could in the system studied in Sec. 5 – see Fig. 15 and its discussion. As a result, each of the resonant peaks has a non-zero energy width E, obeying Eq. (155).
(a)
(b)
N 3
N 10
0.8
3
.
0
0.8
0.6
0.6
T
T
3
.
0
0.4
0
.
1
0.4
0
.
1
0.2
0
.
3
0.2
0
.
3
0
0
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
ka /
ka /
Fig. 2.23. The Dirac comb’s transparency as a function of the product ka for three values of . Since the function T( ka) is -periodic (just like it is for N = 2, see Fig. 16a), only one period is shown.
A further increase of N (see Fig. 23b) results in the increase of the number of resonant peaks per period to ( N – 1), and at N the peaks merge into the so-called allowed energy bands (frequently called just the “energy bands”) with average transparency T ~ 1, separated from similar bands in the adjacent periods of function T ( ka) by energy gaps 48 where T 0. Notice the following important features of the pattern:
(i) at N , the band/gap edges become sharp for any , and tend to fixed positions (determined by but independent of N);
(ii) the larger is well coupling (the smaller is ), the broader are the allowed energy bands and the narrower are the gaps between them.
Our previous discussion of the resonant tunneling gives us a clue for a semi-quantitative interpretation of this pattern: if ( N – 1) potential wells are weakly coupled by tunneling through the potential barriers separating them, the system’s energy spectrum consists of groups of ( N – 1) metastable energy levels, each group being close to one of the unperturbed eigenenergies of the well. (According to Eq. (1.84), for our current example shown in Fig. 22, with its rectangular potential wells, these eigenenergies correspond to kna = n.)
Now let us recall that in the case N = 2, analyzed in the previous section, the eigenfunctions (169) and (175) differed only by the phase shift between their localized components R( x) and
L( x), with = 0 for one of them (S) and = for its counterpart. Hence it is natural to expect that for other N as well, each metastable energy level corresponds to an eigenfunction that is a superposition of similar localized functions in each potential well, but with certain phase shifts between them.
Moreover, we may expect that at N , i.e. for periodic structures,49 with 48 In solid-state (especially semiconductor) physics and electronics, the term bandgaps is more common.
49 This is a reasonable 1D model, for example, for solid-state crystals, whose samples may feature up to ~109
similar atoms or molecules in each direction of the crystal lattice.
Chapter 2
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U ( x a) U ( x),
(2.192)
when the system does not have the ends that could affect its properties, the phase shifts between the localized wavefunctions in all couples of adjacent potential wells should be equal, i.e.
i
( x a) ( x) e
(2.193a)
for all x.50 This equality is the (1D version of the) much-celebrated Bloch theorem. 51 Mathematical rigor aside,52 it is a virtually evident fact because the particle’s density w( x) = *( x)( x), which has to be periodic in this a-periodic system, may be so only is constant. For what follows, it is more convenient to represent the real constant in the form qa, so that the Bloch theorem takes the form Bloch
theorem:
iqa
( x a) ( x) e
.
(2.193b)
1D version
The physical sense of the parameter q will be discussed in detail below, but we may immediately notice that according to Eq. (193b), an addition of (2/ a) to this parameter yields the same wavefunction; hence all observables have to be (2/ a)-periodic functions of q. 53
Now let us use the Bloch theorem to calculate the eigenfunctions and eigenenergies for the infinite version of the system shown in Fig. 22, i.e. for an infinite set of delta-functional potential barriers – see Fig. 24.
a
a
a
En
Fig. 2.24. The simplest periodic potential:
an infinite Dirac comb.
x
x
x
j
j 1
To start, let us consider two points separated by one period a: one of them, xj, just left of the position of one of the barriers, and another one, xj+1, just left of the following barrier – see Fig. 24 again.
50 A reasonably fair classical image of is the geometric angle between similar objects – e.g., similar paper clips – attached at equal distances to a long, uniform rubber band. If the band’s ends are twisted, the twist is equally distributed between the structure’s periods, representing the constancy of . (I have to confess that, due to the lack of time, this was the only “lecture demonstration” in my Stony Brook QM courses.) 51 Named after F. Bloch who applied this concept to the wave mechanics in 1929, i.e. very soon after its formulation. Note, however, that an equivalent statement in mathematics, called the Floquet theorem, has been known since at least 1883.
52 I will recover this rigor in two steps. Later in this section, we will see that the function obeying Eq. (193) is indeed a solution to the Schrödinger equation. However, to save time/space, it will be better for us to postpone until Chapter 4 the proof that any eigenfunction of the equation, with periodic boundary conditions, obeys the Bloch theorem. As a partial reward for this delay, that proof will be valid for an arbitrary spatial dimensionality.
53 The product q, which has the dimensionality of linear momentum, is called either the quasimomentum or (especially in solid-state physics) the “crystal momentum” of the particle. Informally, it is very convenient (and common) to use the name “quasimomentum” for the bare q as well, despite its evidently different dimensionality.
Chapter 2