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The eigenfunctions at each of the points may be represented as linear superpositions of two simple waves exp{ ikx}, and the amplitudes of their components should be related by a 22 transfer matrix T
of the potential fragment separating them. According to Eq. (132), this matrix may be found as the product of the matrix (135) of one delta-functional barrier by the matrix (138) of one zero-potential interval a:
A
A
ika
1
1 i
0
i
A
j
j e
j
T T
.
(2.194)
a
B
B
ika
1
i
0
1 i B
j
j
e
j
However, according to the Bloch theorem (193b), the component amplitudes should be also related as
A
A
iqa
A
j1
iqa
j
e
0
j
e
.
(2.195)
B
B
iqa
B
j1
j 0
e
j
The condition of self-consistency of these two equations gives the following characteristic equation:
ika
e
1 i i
iqa
0
e
0
0
.
(2.196)
ika
0
e
i
1 i
iqa
0
e
In Sec. 5, we have already calculated the matrix product participating in this equation – see the second operand in Eq. (140). Using it, we see that Eq. (196) is reduced to the same simple Eq. (191b) that has jumped at us from the solution of the somewhat different (resonant tunneling) problem. Let us explore that simple result in detail. First of all, the left-hand side of Eq. (191b) is a sinusoidal function of the product qa with unit amplitude, while its right-hand side is a sinusoidal function of the product ka, with amplitude (1 + 2)1/2 > 1 – see Fig. 25,
gap gap …
2
band band …
1
1
cos qa 0
Fig. 2.25. The graphical representation of the
characteristic equation (191b) for a fixed value of the
1
parameter . The ranges of ka that yield cos qa < 1,
correspond to allowed energy bands, while those with
cos qa > 1, correspond to energy gaps between them.
20
1
2
3
4
ka /
As a result, within each half-period ( ka) = of the right-hand side, there is an interval where the magnitude of the right-hand side is larger than 1, so that the characteristic equation does not have a real solution for q. These intervals correspond to the energy gaps (see Fig. 23 again), while the complementary intervals of ka, where a real solution for q exists, correspond to the allowed energy bands. In contrast, the parameter q can take any real values, so it is more convenient to plot the eigenenergy E = 2 k 2/2 m as the function of the quasimomentum q (or, even more conveniently, of the Chapter 2
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dimensionless parameter qa) rather than ka.54 Before doing that, we need to recall that the parameter , defined by the last of Eqs. (78), depends on the wave vector k as well, so that if we vary q (and hence k), it is better to characterize the structure by another, k-independent dimensionless parameter, for example
( ka)
W
,
(2.197)
2
/ ma
so that our characteristic equation (191b) becomes
sin ka
Dirac comb:
cos qa cos ka
.
(2.198)
q vs k
ka
Fig. 26 shows the plots of k and E, following from Eq. (198), as functions of qa, for a particular, moderate value of the parameter . The first evident feature of the pattern is the 2-periodicity of the pattern in the argument qa, which we have already predicted from the general Bloch theorem arguments.
(Due to this periodicity, the complete band/gap pattern may be studied, for example, on just one interval
– qa + , called the 1st Brillouin zone – the so-called reduced zone picture. For some applications, however, it is more convenient to use the extended zone picture with – qa + – see, e.g., the next section.)
1st Brillouin zone
(a)
1st
Brillouin zone
(b)
100
4
ka
E
E 0
50
2
1
E 1
0
0
-2
–1 0 1 2
-2
–1 0 1 2
qa /
qa /
Fig. 2.26. (a) The “genuine” momentum k of a particle in an infinite Dirac comb (Fig. 24), and (b) its energy E = 2 k 2/2 m (in the units of E 0 2/2 ma 2), as functions of normalized quasimomentum, for a particular value ( = 3) of the dimensionless parameter defined by Eq. (197). Arrows in the lower right corner of panel (b) illustrate the definition of energy band ( En) and energy gap ( n) widths.
54 A more important reason for taking q as the argument is that for a general periodic potential U( x), the particle’s momentum k is not uniquely related to E, while (according to the Bloch theorem) the quasimomentum q is.
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However, maybe the most important fact, clearly visible in Fig. 26, is that there is an infinite number of energy bands, with different energies En( q) for the same value of q. Mathematically, it is evident from Eq. (198) – or alternatively from Fig. 25. Indeed, for each value of qa, there is a solution ka to this equation on each half-period ( ka) = . Each of such solutions (see Fig. 26a) gives a specific value of particle’s energy E = 2 k 2/2 m. A continuous set of similar solutions for various qa forms a particular energy band .
Since the energy band picture is one of the most practically important results of quantum mechanics, it is imperative to understand its physics. It is natural to describe this physics in two different ways in two opposite potential strength limits. In parallel, we will use this discussion to obtain simpler expressions for the energy band/gap structure in each limit. An important advantage of this approach is that both analyses may be carried out for an arbitrary periodic potential U( x) rather than for the particular model shown in Fig. 24.
(i) Tight-binding approximation. This approximation works well when the eigenenergy En of the states quasi-localized at the energy profile minima is much lower than the height of the potential barriers separating them – see Fig. 27. As should be clear from our discussion in Sec. 6, essentially the only role of coupling between these states (via tunneling through the potential barriers separating the minima) is to establish a certain phase shift qa between the adjacent quasi-localized wavefunctions un( x – xj) and un( x – xj+1).
a
a
U( x)
u ( x x )
n
j 1
u x
u ( x x )
(
x )
n
j
n
j 1
n
n
En
x
a x
Fig. 2. 27. The tight-binding
0
0
approximation (schematically).
0
x
x
x
x
j 1
j
j 1
To describe this effect quantitatively, let us first return to the problem of two coupled wells considered in Sec. 6, and recast the result (180), with restored eigenstate index n, as
En
( x, t)
(2.199)
n
a ( t) ( x) a ( t) ( x)
i
t
R
R
L
L
exp
,
where the probability amplitudes a R and a L oscillate sinusoidally in time:
a ( t) cos n t,
a ( t) i sin n t .
(2.200)
R
L
This evolution satisfies the following system of two equations whose structure is similar to Eq. (1.61a): i a
a ,
i a
a .
(2.201)
R
n L
L
n R
Eq. (199) may be readily generalized to the case of many similar coupled wells:
En
( x, t)
( ) (
) exp
,
(2.202)
n
a t u x x
j
n
j
i
t
j
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where En are the eigenenergies and un the eigenfunctions of each well. In the tight-binding limit, only the adjacent wells are coupled, so that instead of Eq. (201) we should write an infinite system of similar equations
i a
a
a ,
(2.203)
j
n
j 1
n
j 1
for each well number j, where parameters n describe the coupling between two adjacent potential wells.
Repeating the calculation outlined at the end of the last section for our new situation, for a smooth potential we may get an expression essentially similar to the last form of Eq. (188):
Tight-
2
binding
du
limit:
u ( x )
n ( a x ) ,
(2.204)
n
n
0
0
coupling
m
dx
energy
where x 0 is the distance between the well bottom and the middle of the potential barrier on the right of it
– see Fig. 27. The only substantial new feature of this expression in comparison with Eq. (188) is that the sign of n alternates with the level number n: 1 > 0, 2 < 0, 3 > 0, etc. Indeed, the number of zeros (and hence, “wiggles”) of the eigenfunctions un( x) of any potential well increases as n – see, e.g., Fig.
1.8,55 so that the difference of the exponential tails of the functions, sneaking under the left and right barriers limiting the well also alternates with n.
The infinite system of ordinary differential equations (203) enables solutions of many important problems (such as the spread of the wavefunction that was initially localized in one well, etc.), but our task right now is just to find its stationary states, i.e. the solutions proportional to exp{- i( n/) t}, where
n is a still unknown, q- dependent addition to the background energy En of the n th energy level. To satisfy the Bloch theorem (193) as well, such a solution should have the following form:
n
a ( t) a exp
.
(2.205)
j
iqx i
t const
j
Plugging this solution into Eq. (203) and canceling the common exponent, we get
Tight-
binding
limit:
E E E
iqa
iqa
2 cos
,
(2.206)
n
n
n
n e
e
E
qa
n
n
energy
bands
so that in this approximation, the energy band width En (see Fig. 26b) equals 4 n .
The relation (206), whose validity is restricted to n << En, describes the lowest energy bands plotted in Fig. 26b reasonably well. (For larger , the agreement would be even better.) So, this calculation explains what the energy bands really are: in the tight-binding limit they are best interpreted as isolated well’s energy levels En, broadened into bands by the interwell interaction. Also, this result gives clear proof that the energy band extremes correspond to qa = 2 l and qa = 2( l + ½), with integer l. Finally, the sign alteration of the coupling coefficient n (204) explains why the energy maxima of one band are aligned, on the qa axis, with energy minima of the adjacent bands – see Fig. 26.
(ii) Weak-potential limit. Amazingly, the energy-band structure is also compatible with a completely different physical picture that may be developed in the opposite limit. Let the particle’s energy E be so high that the periodic potential U( x) may be treated as a small perturbation. Naively, in 55 Below, we will see several other examples of this behavior. This alternation rule is also described by the Wilson-Sommerfeld quantization condition (110).
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this limit we could expect a slightly and smoothly deformed parabolic dispersion relation E = 2 k 2/2 m.
However, if we are plotting the stationary-state energy as a function of q rather than k, we need to add 2 l/ a, with an arbitrary integer l, to the argument. Let us show this by expanding all variables into the 1D-spatial Fourier series. For the potential energy U( x) that obeys Eq. (192), such an expansion is straightforward:56
2 x
U ( x) U exp i
l"
(2.207)
l
,
"
l"
a
where the summation is over all integers l” , from – to +. However, for the wavefunction we should show due respect to the Bloch theorem (193), which shows that strictly speaking, ( x) is not periodic.
To overcome this difficulty, let us define another function:
iqx
u( x) ( x) e
,
(2.208)
and study its periodicity:
u( x a) ( x a) iq( x a) e
( x) e iqx u( x) .
(2.209)
We see that the new function is a-periodic, and hence we can use Eqs. (208)-(209) to rewrite the Bloch theorem in a different form:
1D Bloch
theorem:
( x) u( x) iqx
e
, with u( x a) u( x) .
(2.210) alternative
form
Now it is safe to expand the periodic function u( x) exactly as U( x):
2 x
u( x) u exp i
l'
(2.211)
l
,
'
a
l '
so that, according to Eq. (210),
iqx
2 x
2
( x) e
u exp i
l'
u
i q
l' x
(2.212)
l'
exp
l'
.
l'
a
l '
a
The only nontrivial part of plugging Eqs. (207) and (212) into the stationary Schrödinger equation (53) is how to handle the product term,
2
U ( x) U u exp i q
l' l" x
(2.213)
l" l '
.
l ,' l"
a
At fixed l’, we may change the summation over l” to that over l l’ + l” (so that l” l – l’), and write:
2
U ( x) exp i q
l x
u U .
(2.214)
l'
l l'
l
a l'
Now plugging Eq. (212) (with the summation index l’ replaced with l) and Eq. (214) into the stationary Schrödinger equation (53), and requiring the coefficients of each spatial exponent to match, we get an infinite system of linear equations for ul:
56 The benefits of such an unusual notation of the summation index ( l” instead of, say, l) will be clear in a few lines.
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2
2
2
U u
.
(2.215)
'
'
E
q
l u
l l
l
l
l '
2 m
a
(Note that by this calculation we have essentially proved that the Bloch wavefunction (210) is indeed a solution of the Schrödinger equation, provided that the quasimomentum q is selected in a way to make the system of linear equation (215) compatible, i.e. is a solution of its characteristic equation.) So far, the system of equations (215) is an equivalent alternative to the initial Schrödinger equation, for any potential’s strength.57 In the weak-potential limit, i.e. if all Fourier coefficients Un are small,58 we can complete all the calculations analytically.59 Indeed, in the so-called 0th approximation we can ignore all Un, so that in order to have at least one ul different from 0, Eq. (215) requires that 2
2
2 l
E E
.
(2.216)
l
q
2 m
a
( ul itself should be obtained from the normalization condition). This result means that in this approximation, the dispersion relation E( q) has an infinite number of similar quadratic branches numbered by integer l – see Fig. 28.
(2)
2
E
l 0
l 1
l 2
l 1
l 0
Fig. 2.28. The energy band/gap
)
1
(
picture in the weak potential limit ( n
E
1
<< E( n)), with the shading showing the
1st Brillouin zone.
0
1
qa /
2
On every branch, such eigenfunction has just one Fourier coefficient, i.e. is a monochromatic traveling wave
ikx
2 l
u e
u exp
.
(2.217)
l
l
l
i q
x
a
Next, the above definition of El allows us to rewrite Eq. (215) in a more transparent form 57 By the way, the system is very efficient for fast numerical solution of the stationary Schrödinger equation for any periodic profile U( x), even though to describe potentials with large Un, this approach may require taking into account a correspondingly large number of Fourier amplitudes ul.
58 Besides, possibly, a constant potential U 0, which, as was discussed in Chapter 1, may be always taken for the energy reference. As a result, in the following calculations, I will take U 0 = 0 to simplify the formulas.
59 This method is so powerful that its multi-dimensional version is not much more complex than the 1D version described here – see, e.g., Sec. 3.2 in the classical textbook by J. Ziman, Principles of the Theory of Solids, 2nd ed., Cambridge U. Press, 1979.
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U u E E u ,
(2.218)
l l ' l'
l l
l ' l
which may be formally solved for ul:
1
u
U u .
(2.219)
l
l
E
l' l
E
'
l l ' l
This formula shows that if the Fourier coefficients Un are non-zero but small, the wavefunctions do acquire other Fourier components (besides the main one, with the index corresponding to the branch number), but these additions are all small, besides narrow regions near the points El = El’ where two branches (216) of the dispersion relation E( q), with some specific numbers l and l’, cross. According to Eq. (216), this happens when
2
2
q
l q
l' ,
(2.220)
a
a
i.e. at q qm m/ a (with the integer m l + l’) 60 corresponding to 2
2 2
Weak-
E E
( l l' ) 2
n
l
n E ,
(2.221) potential
l
l '
2
2
( )
2
2
2 ma
2 ma
limit:
energy gap
with integer n l – l’. (According to their definitions, the index n is just the number of the branch positions crossing on the energy scale, while the index m numbers the position of the crossing points on the q-axis
– see Fig. 28.) In such a region, E has to be close to both El and El’, so that the denominator in just one of the infinite number of terms in Eq. (219) is very small, making the term substantial despite the smallness of Un. Hence we can take into account only one term in each of the sums (written for l and l’): U u ( E E ) u ,
n l '
l
l
(2.222)
U u ( E E ) u .
n l
l '
l '
Taking into account that for any real function U( x), the Fourier coefficients in its Fourier expansion (207) have to be related as U
*
-n = Un , Eq. (222) yields the following simple characteristic equation
E E
U
l
n
0
*
,
(2.223)
U
E E
n
l'
with the following solution:
1/ 2
Weak-
2
E E
E E
potential
l
l '
*
l
l '
( n)
E E
.
(2.224) limit:
U U ,
with E
E
ave
n
n
ave
2
2
level
anticrossing
According to Eq. (216), close to the branch crossing point qm = ( l + l’)/ a, the fraction participating in this result may be approximated as61
2
( n
E E
dE
n 2 aE )
l
l '
~
l
q,
with
,
q~
and
q q
q q
,
(2.225)
m
m
2
dq
ma
n
60 Let me hope that the difference between this new integer and the particle’s mass, both called m, is absolutely clear from the context.
61 Physically, / ( n/ a)/ m = k( n)/ m is just the velocity of a free classical particle with energy E( n).
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while the parameters E
*
ave = E( n) and UnUn = Un2 do not depend on q
~ , i.e. on the distance from the
central point qm. This is why Eq. (224) may be plotted as the famous level anticrossing (also called
“avoided crossing”, or “intended crossing”, or “non-crossing”) diagram (Fig. 29), with the energy gap width n equal to 2 Un, i.e. just twice the magnitude of the n-th Fourier harmonic of the periodic potential U( x). Such anticrossings are also clearly visible in Fig. 28, which shows the result of the exact solution of Eq. (198) for the particular case = 0.5.62
( n)
E E
E
2 U n
0
E E
l
l ' q qm
2
E
Fig. 2.29. The level anticrossing diagram.
We will run into the anticrossing diagram again and again in the course, notably at the discussion of spin-½ and other two-level systems. It is also repeatedly met in classical mechanics, for example at the calculation of frequencies of coupled oscillators.63,64 In our current case of the weak potential limit of the band theory, the diagram describes the interaction of two traveling de Broglie waves (217), with oppositely directed wave vectors, l and – l’ , via the ( l – l’)th (i.e. the n th) Fourier harmonic of the potential profile U( x).65 This effect exists also in the classical wave theory and is known as the Bragg reflection, describing, for example, the 1D model of the X-wave reflection by a crystal lattice (see, e.g.
Fig. 1.5) in the limit of weak interaction between the incident wave and each atom.
The anticrossing diagram shows that rather counter-intuitively, even a weak periodic potential changes the topology of the initially parabolic dispersion relation radically, connecting its different branches, and thus creating the energy gaps. Let me hope that the reader has enjoyed the elegant description of this effect, discussed above, as well as one more illustration of the wonderful ability of physics to give completely different interpretations (and different approximate approaches) to the same effect in opposite limits.
So, we have explained analytically (though only in two limits) the particular band structure shown in Fig. 26. Now we may wonder how general this structure is, i.e. how much of it is independent of the Dirac comb model (Fig. 24). For that, let us represent the band pattern, such as that shown in Fig.
62 From that figure, it is also clear that in the weak potential limit, the width En of the n th energy band is just E( n)
– E( n – 1) – see Eq. (221). Note that this is exactly the distance between the adjacent energy levels of the simplest 1D potential well of infinite depth – cf. Eq. (1.85).
63 See, e.g., CM Sec. 6.1 and in particular Fig. 6.2.
64 Actually, we could readily obtain this diagram in the previous section, for the system of two weakly coupled potential wells (Fig. 21), if we assumed the wells to be slightly dissimilar.
65 In the language of the de Broglie wave scattering, to be discussed in Sec. 3.3, Eq. (220) may be interpreted as the condition that each of these waves, scattered on the n th Fourier harmonic of the potential profile, constructively interferes with its counterpart, leading to a strong enhancement of their interaction.
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26b (plotted for a particular value of the parameter , characterizing the potential barrier strength) in a more condensed form, which would allow us to place the results for a range of values on a single comprehensible plot. The way to do this should be clear from Fig. 26b: since the dependence of energy on the quasimomentum in each energy band is not too eventful, we may plot just the highest and the smallest values of the particle’s energy E = 2 k 2/2 m as functions of maW/2 – see Fig. 30, which may be obtained from Eq. (198) with qa = 0 and qa = .
100
80
band
60
E
gap
E )
0
40
band
Fig. 2.30. Characteristic curves of the
Schrödinger equation for the infinite
20
Dirac comb (Fig. 24).
00
2
4
6
8
10
These plots (in mathematics, commonly called characteristic curves, while in applied physics, band-edge diagrams) show, first of all, that at small , all energy gap widths are equal and proportional to this parameter, and hence to W. This feature is in a full agreement with the main conclusion (224) of our general analysis of the weak-potential limit, because for the Dirac comb potential (Fig. 24), U x W
δ x ja const,
(2.226)
j
all Fourier harmonic amplitudes, defined by Eq. (207), are equal by magnitude: Ul = W/ a. As is further increased, the gaps grow and the allowed energy bands shrink, but rather slowly. This is also natural, because, as Eq. (79) shows, transparency T of the delta-functional barriers separating the quasi-localized states (and hence the coupling parameters n T1/2 participating in the general tight-binding limit’s theory) decrease with W very gradually.
These features may be compared with those for more realistic and relatively simple periodic functions U( x), for example the sinusoidal potential U( x) = A cos(2 x/ a) – see Fig. 31a. For this potential, the stationary Schrödinger equation (53) takes the following form:
2
d
2
2 x
A cos
E .
(2.227)
2 m dx 2
a
By introduction of dimensionless variables66
x
E
A
,
, 2
,
(2.228)
)
1
(
)
1
(
a
E
E
66 Note that this definition of is quantitatively different from that for the Dirac comb (226), but in both cases, this parameter is proportional to the amplitude of the potential modulation.
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where E(1) is defined by Eq. (221) with n = 1, Eq. (227) is reduced to the canonical form of the well-studied Mathieu equation 67
2
d
Mathieu
( 2 cos 2 ) .
0
(2.229)
2
equation
d
(a)
(b)
a
U( x)
d
U( x)
U 0
A
0
a
x
A
0
x
Fig. 2.31. Two other simple periodic potential profiles: (a) the sinusoidal (“Mathieu”) potential and (b) the Kronig-Penney potential.
Figure 32 shows the characteristic curves of this equation. We see that now at small the first energy gap grows much faster than the higher ones: n n. This feature is in accord with the weak-coupling result 1 = 2 U 1, which is valid only in the linear approximation in Un, because for the Mathieu potential, Ul = A( l,+1 + l,-1)/2. Another clearly visible feature is the exponentially fast shrinkage of the allowed energy bands at 2 > (in Fig. 32, on the right from the dashed line), i.e. at E < A. It may be readily explained by our tight-binding approximation result (206): as soon as the eigenenergy drops significantly below the potential maximum U max = A (see Fig. 31a), the quantum states in the adjacent potential wells are connected only by tunneling through relatively high potential barriers separating these wells, so that the coupling amplitudes n become exponentially small – see, e.g., Eq. (189).
Fig. 2.32. Characteristic curves of the
Mathieu equation. The dashed line
corresponds to the equality = 2, i.e. E
band
= A U max, separating the regions of
gap
under-barrier tunneling and over-barrier
motion. Adapted from Fig. 28.2.1 at
http://dlmf.nist.gov. (Contribution by US
Government, not subject to copyright).
Another simple periodic profile is the Kronig-Penney potential, shown in Fig. 31b, which gives relatively simple analytical expressions for the characteristic curves. Its advantage is a more realistic law of the decrease of the Fourier harmonics Ul at l >> 1, and hence of the energy gaps in the weak-potential limit:
67 This equation, first studied in the 1860s by É. Mathieu in the context of a rather practical problem of vibrating elliptical drumheads (!), has many other important applications in physics and engineering, notably including the parametric excitation of oscillations – see, e.g., CM Sec. 5.5.
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U
0
( n)
2 U
, at E ~ E
U .
(2.230)
n
n
0
n
Leaving a detailed analysis of the Kronig-Penney potential for the reader’s exercise, let me conclude this section by addressing the effect of potential modulation on the number of eigenstates in 1D systems of a large but finite length l >> a, k-1. Surprisingly, the Bloch theorem makes the analysis of this problem elementary, for arbitrary U( x). Indeed, let us assume that l is comprised of an integer number of periods a, and its ends are described by similar boundary conditions – both assumptions evidently inconsequential for l >> a. Then, according to Eq. (210), the boundary conditions impose, on the quasimomentum q, exactly the same quantization condition as we had for k for a free 1D motion.
Hence, instead of Eq. (1.100), we can write
l
dN
dq ,
(2.231) 1D number
2
of states
with the corresponding change of the summation rule:
l
f ( q)
f ( q) dk .
(2.232)
q
2
As a result, the density of states in the 1D q-space, dN/ dq = l/2, does not depend on the potential profile at all! Note, however, that the profile does affect the density of states on the energy scale, dN/ dE. As an extreme example, on the bottom and at the top of each energy band we have dE/ dq
0, and hence
dN dN dE l dE .
(2.233)
dE
dq
dq
2
dq
This effect of state concentration at the band/gap edges (which survives in higher spatial dimensionalities as well) has important implications for the operation of several important electronic and optical devices, in particular semiconductor lasers and light-emitting diodes.
2.8. Periodic systems: Particle dynamics
The band structure of the energy spectrum of a particle moving in a periodic potential has profound implications not only for its density of states but also for its dynamics. Indeed, let us consider the simplest case of a wave packet composed of the Bloch functions (210), all belonging to the same (say, n th) energy band. Similarly to Eq. (27) for a free particle, we can describe such a packet as i qx q t
( x, t)
a u ( x) e
dq ,
(2.234)
q q
where the a-periodic functions u( x), defined by Eq. (208), are now indexed to emphasize their dependence on the quasimomentum, and ( q) En( q)/ is the function of q describing the shape of the corresponding energy band – see, e.g., Fig. 26b or Fig. 28. If the packet is narrow in the q-space, i.e. if the width q of the distribution aq is much smaller than all the characteristic q-scales of the dispersion relation ( q), in particular of / a, we may simplify Eq. (234) exactly as it was done in Sec. 2 for a free particle, despite the presence of the periodic factors uq( x) under the integral. In the linear approximation of the Taylor expansion, we get a full analog of Eq. (32), but now with q rather than k, and Chapter 2
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d
v
,
and v
,
(2.235)
gr
q q
ph
q q
0
0
dq
q
where q 0 is the central point of the quasimomentum distribution. Despite the formal similarity with Eqs.
(33) for the free particle, this result is much more eventful. For example, as evident from the dispersion relation’s topology (see Figs. 26b, 28), the group velocity vanishes not only at q = 0, but at all values of q that are multiples of (/ a), at the bottom and on the top of each energy band. Even more intriguing is that the group velocity’s sign changes periodically with q.
This group velocity alternation leads to fascinating, counter-intuitive phenomena if a particle placed in a periodic potential is the subject of an additional external force F( t). (For electrons in a crystal, this may be, for example, the force of the applied electric field.) Let the force be relatively weak, so that the product Fa (i.e. the scale of the energy increment from the additional force per one lattice period) is much smaller than both relevant energy scales of the dispersion relation E( q) – see Fig. 26b: Fa E
, .
(2.236)
n
n
This strong relationship allows one to neglect the force-induced interband transitions, so that the wave packet (234) includes the Bloch eigenfunctions belonging to only one (initial) energy band at all times.
For the time evolution of its center q 0, theory yields68 an extremely simple equation of motion Time
evolution
1
of quasi-
q F( t) .
(2.237)
0
momentum
This equation is physically very transparent: it is essentially the 2nd Newton law for the time evolution of the quasimomentum q under the effect of the additional force F( t) only, excluding the periodic force
– U( x)/ x of the background potential U( x). This is very natural, because as Eq. (210) implies, q is essentially the particle’s momentum averaged over the potential’s period, and the periodic force effect drops out at such an averaging.
Despite the simplicity of Eq. (237), the results of its solution may be highly nontrivial. First, let us use Eqs. (235) and (237) to find the instant group acceleration of the particle (i.e. the acceleration of its wave packet’s envelope):
dv
gr
d d q
d d q dq
d
q dq
d
0
0
2
( )
1 2
0
0
0
a
F( t) .
(2.238)
gr
2
2
q q 0
dt
dt dq
dq
dq
dt
dq
dt
dq
0
0
0
0
This means that the second derivative of the dispersion ( q) relation (specific for each energy band) plays the role of the effective reciprocal mass of the particle at this particular value of q 0: 2
Effective
mass
m
.
(2.239)
ef
2
2
2
2
d / dq
d E / dq
n
For the particular case of a free particle, for which Eq. (216) is exact, this expression is reduced to the original (and constant) mass m, but generally, the effective mass depends on the wave packet’s 68 The proof of Eq. (237) is not difficult, but becomes more compact in the bra-ket formalism, to be discussed in Chapter 4. This is why I recommend to the reader its proof as an exercise after reading that chapter. For a generalization of this theory to the case of essential interband transitions see, e.g., Sec. 55 in E. Lifshitz and L.
Pitaevskii, Statistical Physics, Part 2, Pergamon,1980.
Chapter 2
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momentum. According to Eq. (239), at the bottom of any energy band, m ef is always positive but depends on the strength of the particle’s interaction with the periodic potential. In particular, according to Eq. (206), in the tight-binding limit, the effective mass is very large:
2
E )1
(
m
m
m .
(2.240)
ef q
( / a) n
2 a 2
2
n
n
On the contrary, in the weak-potential limit, the effective mass is close to m at most points of each energy band, but at the edges of the (narrow) bandgaps, it is much smaller. Indeed, expanding Eq. (224) in the Taylor series near point q = qm, we get
2
2
1 dE
l
~2
~ 2
E
E
U
q U
q ,
(2.241)
( n)
ave
n
E E
2 U dq
n
2 U
n
q q
n
m
where and q~ are defined by Eq. (225), so that
2
U
m
U m
n
m .
(2.242)
ef
n
q qm
2
2 E ( n)
The effective mass effects in real atomic crystals may be very significant. For example, the charge carriers in silicon have m ef 0.19 m e in the lowest, normally-empty energy band (traditionally called the conduction band), and m ef 0.98 m e in the adjacent lower, normally-filled valence band. In some semiconducting compounds, the conduction-band mass may be even smaller – down to 0.0145 m e in InSb!
However, the effective mass’ magnitude is not the most surprising effect. A more fascinating corollary of Eq. (239) is that on the top of each energy band the effective mass is negative – please revisit Figs. 26b, 28, and 29 again. This means that the particle (or more strictly, its wave packet’s envelope) is accelerated in the direction opposite to the applied force. This is exactly what electronic engineers, working with electrons in semiconductors, call holes, characterizing them by a positive mass
m ef, but compensating this sign change by taking their charge e positive. If the particle stays in close vicinity of the energy band’s top (say, due to frequent scattering effects, typical for the semiconductors used in engineering practice), such double sign flip does not lead to an error in calculations of hole’s dynamics, because the electric field’s force is proportional to the particle’s charge, so that the particle’s acceleration a gr is proportional to the charge-to-mass ratio.69
However, in some phenomena such simple representation is unacceptable.70 For example, let us form a narrow wave packet at the bottom of the lowest energy band,71 and then exert on it a constant force F > 0 – say, due to a constant external electric field directed along the x-axis. According to Eq.
(237), this force would lead to linear growth of q 0 in time, so that in the quasimomentum space, the 69 More discussion of this issue may be found in SM Sec. 6.4.
70 The balance of this section describes effects that are not discussed in most quantum mechanics textbooks.
Though, in my opinion, every educated physicist should be aware of them, some readers may skip them at the first reading, jumping directly to the next Sec. 9.
71 Physical intuition tells us (and the theory of open systems, to be discussed in Chapter 7, confirms) that this may be readily done, for example, by weakly coupling the system to a relatively low-temperature environment, and letting it relax to the lowest possible energy.
Chapter 2
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packet’s center would slide, with a constant speed, along the q axis – see Fig. 33a. Close to the energy band’s bottom, this motion would correspond to a positive effective mass (possibly, somewhat different than the genuine particle’s mass m), and hence be close to the free particle’s acceleration. However, as soon as q 0 has reached the inflection point where d 2 E 1/ dq 2 = 0, the effective mass, and hence its acceleration (238) change signs to negative, i.e. the packet starts to slow down (in the direct space), while still moving ahead with the same velocity in the quasimomentum space. Finally, at the energy band’s top, the particle stops at a certain x max, while continuing to move forward in the q-space.
(a)
E
(b)
E( q)
E ( q)
2
1
x / F
0
1
1
E 1
E
E ( q)
1
1
x 0
0
qa
a
0
x
E / F
max
1
Fig. 2.33. The Bloch oscillations (red lines) and the Landau-Zener tunneling (blue arrows) represented in: (a) the reciprocal space of q, and (b) the direct space. On panel (b), the tilted gray strips show the allowed energy bands, while the bold red lines, the Wannier-Stark ladder’s steps.
Now we have two alternative ways to look at the further time evolution of the wave packet along the quasimomentum’s axis. From the extended zone picture (which is the simplest for this analysis, see Fig. 33a),72 we may say that the particle crosses the 1st Brillouin zone boundary and continues to go forward in q-space, i.e. down the lowest energy band. According to Eq. (235), this region (up to the next energy minimum at qa = 2) corresponds to a negative group velocity. After q 0 has reached that minimum, the whole process repeats again – and again, and again.
These are the famous Bloch oscillations – the effect which had been predicted, by the same F.
Bloch, as early as 1929 but evaded experimental observation until the 1980s (see below) due to the strong scattering effects in real solid-state crystals. The time period of the oscillations may be readily found from Eq. (237):
q
2 / a
2
t
,
(2.243)
B
dq / dt
F /
Fa
so that their frequency may be expressed by a very simple formula
72 This phenomenon may be also discussed from the point of view of the reduced zone picture, but then it requires the introduction of instant jumps between the Brillouin zone boundary points (see the dashed red line in Fig. 33) that correspond to physically equivalent states of the particle. Evidently, for the description of this particular phenomenon, this language is more artificial.
Chapter 2
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2
Fa
Bloch
,
(2.244)
B
oscillations:
t
B
frequency
and hence is independent of any peculiarities of the energy band/gap structure.
The direct-space motion of the wave packet’s center x 0( t) during the Bloch oscillation process may be analyzed by integrating the first of Eqs. (235) over some time interval t, and using Eq. (237): t
t
d( q )
t
d( q )
t t
0
0
x ( t) v dt
dt
d
q .
(2.245)
0
gr
0
dq
dq / dt
F
F
0
0
0
0
0
t0
If the interval t is equal to the Bloch oscillation period t B (243), the initial and final values of E( q 0) =
( q 0) are equal, giving x 0 = 0: in the end of the period, the wave packet returns to its initial position in space. However, if we carry out this integration only from the smallest to the largest values of ( q 0), i.e.
the adjacent points where the group velocity vanishes, we get the following Bloch oscillation swing: Bloch
E
x
.
(2.246) oscillations:
max
max
min
1
F
F
spatial
swing
This simple result may be interpreted using an alternative energy diagram (Fig. 33b), which results from the following arguments. The additional force F may be described not only via the 2nd Newton law’s version (237), but, alternatively, by its contribution – Fx to the Gibbs potential energy73
U ( x) U ( x) Fx
(2.247)
The exact solution of the Schrödinger equation (61) with such a potential may be hard to find directly, but if the force F is sufficiently weak, as we are assuming throughout this discussion, the second term in Eq. (247) may be considered as a constant on the scale of a << x max. In this case, our quantum-mechanical treatment of the periodic potential U( x) is still virtually correct, but with an energy shift depending on the “global” position x 0 of the packet’s center. In this approximation, the total energy of the wave packet is
E E( q ) Fx .
(2.248)
0
0
In a plot of such energy as a function of x 0 (Fig. 33b), the energy dependence on q 0 is hidden, but as was discussed above, it is rather uneventful and may be well characterized by the position of bandgap edges on the energy axis.74 In this representation, the Bloch oscillations keep the full energy E of the particle constant, i.e. follow a horizontal line in Fig. 33b, limited by the classical turning points corresponding to the bottom and the top of the allowed energy band. The distance x max between these points is evidently given by Eq. (246).
Besides this alternative look at the Bloch oscillation swing, the total energy diagram shown in Fig. 33b enables one more remarkable result. Let a wave packet be so narrow in the momentum space 73 Physically, this is just the relevant part of the potential energy of the total system comprised of our particle (in the periodic potential) and the source of the force F – see, e.g., CM Sec. 1.4.
74 In semiconductor physics and engineering, such spatial band-edge diagrams are virtually unavoidable components of almost every discussion/publication. In this series, a few more examples of such diagrams may be found in SM Sec. 6.4.
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that x ~ 1/ q >> x max; then it may be well represented by definite energy, i.e. by a horizontal line in Fig. 33b. But Eq. (247) is exactly invariant with respect to the following simultaneous translation of the coordinate and the energy:
x x a, E E Fa .
(2.249)
This means that it is satisfied by an infinite set of similar solutions, each corresponding to one of the horizontal red lines shown in Fig. 33b. This is the famous Wannier-Stark ladder,75 with the step height Wannier-Stark
E
Fa .
(2.250)
WS
ladder
The importance of this alternative representation of the Bloch oscillations is due to the following fact. In most experimental realizations, the power of the electromagnetic radiation with frequency (244), that may be extracted from the oscillations of a charged particle, is very low, so that their direct detection represents a hard problem.76 However, let us apply to a Bloch oscillator an additional ac field at frequency B. As these frequencies are brought close together, the external signal should synchronize (“phase-lock”) the Bloch oscillations,77 resulting in certain changes of time-independent observables – for example, a resonant change of absorption of the external radiation. Now let us notice that the combination of Eqs. (244) and (250) yield the following simple relation:
E
.
(2.251)
WS
B
This means that the phase-locking at B allows for an alternative (but equivalent) interpretation – as the result of ac-field-induced quantum transitions78 between the steps of the Wannier-Stark ladder.
(Again, such occasions when two very different languages may be used for alternative interpretations of the same effect is one of the most beautiful features of physics.)
This phase-locking effect has been used for the first experimental confirmations of the Bloch oscillation theory.79 For this purpose, the natural periodic structures, solid-state crystals, are inconvenient due to their very small period a ~ 10-10 m. Indeed, according to Eq. (244), such structures require very high forces F (and hence very high electric fields E = F/ e) to bring B to an experimentally convenient range. This problem has been overcome using artificial periodic structures ( superlattices) of certain semiconductor compounds, such as Ga1- x Al x As with various degrees x of the gallium-to-aluminum atom replacement, whose layers may be grown over each other epitaxially, i.e., with very few crystal structure violations. Such superlattices, with periods a ~ 10 nm, have enabled a clear observation of the resonance at B, and hence a measurement of the Bloch oscillation frequency, in particular its proportionality to the applied dc electric field, predicted by Eq. (244).
75 This effect was first discussed in detail by Gregory Hugh Wannier in his 1959 monograph on solid-state physics, while the name of Johannes Stark is traditionally associated with virtually any electric field effect on atomic systems, after he had discovered the first of such effects in 1913.
76 In systems with many independent particles (such as electrons in semiconductors), the detection problem is exacerbated by the phase incoherence of the Bloch oscillations performed by each particle. This drawback is absent in atomic Bose-Einstein condensates whose Bloch oscillations (in a periodic potential created by standing optical waves) were eventually observed by M. Ben Dahan et al., Phys. Rev. Lett. 76, 4508 (1996).
77 A simple analysis of phase locking of a classical oscillator may be found, e.g., in CM Sec. 5.4. (See also the brief discussion of the phase locking of the Josephson oscillations at the end of Sec. 1.6 of this course.) 78 A quantitative theory of such transitions will be discussed in Sec. 6.6 and then in Chapter 7.
79 E. Mendez et al., Phys. Lev. Lett. 60, 2426 (1988).
Chapter 2
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Very soon after this discovery, the Bloch oscillations were observed80 in small Josephson junctions, where they result from the quantum dynamics of the Josephson phase difference in a 2-
periodic potential profile, created by the junction. A straightforward translation of Eq. (244) to this case (left for the reader’s exercise) shows that the frequency of such Bloch oscillations is
I
I
,
i.e. f
B
,
(2.252)
B
2 e
B
2
2 e
where I is the dc current passed through the junction – the effect not to be confused with the “classical”
Josephson oscillations with frequency (1.75). It is curious that Eq. (252) may be legitimately interpreted as a result of a periodic transfer, through the Josephson junction, of discrete Cooper pairs (of charge –
2 e), between two coherent Bose-Einstein condensates in the superconducting electrodes of the junction.81
Next, our discussion of the Bloch oscillations was based on the premise that the wave packet of the particle stays within one (say, the lowest) energy band. However, just one look at Fig. 28 shows that this assumption becomes unrealistic if the energy gap separating this band from the next one becomes very small, 1 0. Indeed, in the weak-potential approximation, which is adequate in this limit, U 1
0, the two dispersion curve branches (216) cross without any interaction, so that if our particle (meaning its the wave packet) is driven to approach that point, it should continue to move up in energy –
see the dashed blue arrow in Fig. 33a. Similarly, in the real-space representation shown in Fig. 33b, it is intuitively clear that at 1 0, the particle residing at one of the steps of the Wannier-Stark ladder should be able to somehow overcome the vanishing spatial gap x 0 = 1/ F and to “leak” into the next band – see the horizontal dashed blue arrow on that panel.
This process, called the Landau-Zener (or “interband”, or “band-to-band”) tunneling,82 is indeed possible. To analyze it, let us first take F = 0, and consider what happens if a quantum particle, described by an x-long (i.e. E-narrow) wave packet, is incident from free space upon a periodic structure of a large but finite length l = Na >> a – see, e.g., Fig. 22. If the packet’s energy E is within one of the energy bands, it may evidently propagate through the structure (though may be partly reflected from its ends). The corresponding quasimomentum may be found by solving the dispersion relation for q; for example, in the weak-potential limit, Eq. (224) (which is valid near the gap) yields
1
~
~
~
1/ 2
2
2
2
~
q q q, with q
(2.253)
m
E Un , for
2
U
E ,
n
~
where
( n)
E E E and = 2 aE( n)/ n – see the second of Eqs. (225).
Now, if the energy E is inside one of the energy gaps n, the wave packet’s propagation in an infinite periodic lattice is impossible, so that it is completely reflected from it. However, our analysis of the potential step problem in Sec. 3 implies that the packet’s wavefunction should still have an exponential tail protruding into the structure and decaying on some length – see Eq. (58) and Fig. 2.4.
80 D. Haviland et al. , Z. Phys. B 85, 339 (1991).
81 See, e.g., D. Averin et al., Sov. Phys. – JETP 61, 407 (1985). This effect is qualitatively similar to the transfer of single electrons, with a similar frequency f = I/ e, in tunnel junctions between normal (non-superconducting) metals – see, e.g., EM Sec. 2.9 and references therein.
82 It was predicted independently by L. Landau, C. Zener, E. Stueckelberg, and E. Majorana in 1932.
Chapter 2
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Indeed, a straightforward review of the calculations leading to Eq. (253) shows that it remains valid for energies within the gap as well, if the quasimomentum is understood as a purely imaginary number: 1/ 2
1
~
~
~
q i,
where
2
2
U
E
E U
.
(2.254)
n
2
2
,
for
n
With this replacement, the Bloch solution (193b) indeed describes an exponential decay of the wavefunction at length ~ 1/.
Returning to the effects of weak force F, in the real-space approach described by Eq. (248) and illustrated in Fig. 33b, we may recast Eq. (254) as
1/
1
( x)
2
~ 2
U
( x
F )
,
(2.255)
n
2
where x~ is the particle’s (i.e. its wave packet center’s) deviation from the mid-gap point. Thus the gap creates a potential barrier of a finite width x 0 = 2 Un/ F, through which the wave packet may tunnel with a non-zero probability. As we already know, in the WKB approximation (in our case requiring
x 0 >> 1) this probability is just the potential barrier’s transparency T, which may be calculated from Eq. (117):
xc
1
1/ 2
2
2 U
lnT 2 ( x) dx
2
U
~ 2
( x
F
.
(2.256)
n
~
)
x
d
n 2 x 1 2
d
c
1/2
2
( x) 0
x
0
c
where x c x 0/2 = Un / F are the classical turning points. Working out this simple integral (or just noticing that it is a quarter of the unit circle’s area, and hence is equal to /4), we get Landau-Zener
U 2
n
tunneling
T exp
.
(2.257)
probability
F
This famous result may be also obtained in a more complex way, whose advantage is a constructive proof that Eq. (257) is valid for an arbitrary relation between F and Un 2, i.e. arbitrary T, while our simple derivation was limited to the WKB approximation, valid only at T << 1.83 Using Eq.
(225), we may rewrite the product F participating in Eq. (257), as
1 d E E
dq
d E E
l
l
l
l
u
'
0
'
F
,
(2.258)
2
dq
( n)
dt
2
dt
( n)
2
0
E E E
E E E
l
l'
l
l'
where u has the meaning of the “speed” of the energy level crossing in the absence of the gap. Hence, Eq. (257) may be rewritten in the form
U 2
2
n
T exp
,
(2.259)
u
which is more transparent physically. Indeed, the fraction 2 Un / u = nu gives the time scale t of the energy’s crossing the gap region, and according to the Fourier transform, its reciprocal, max ~ 1/ t 83 In Chapter 6 below, Eq. (257) will be derived using a different method, based on the so-called Golden Rule of quantum mechanics, but also in the weak-potential limit, i.e. for hyperbolic dispersion law (253).
Chapter 2
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gives the upper cutoff of the frequencies essentially involved in the Bloch oscillation process. Hence Eq.
(259) means that
ln
n
T
.
(2.260)
max
This formula allows us to interpret the Landau-Zener tunneling as the system’s excitation across the energy gap n by the highest-energy quantum max available from the Bloch oscillation process. This interpretation remains valid even in the opposite, tight-binding limit, in which, according to Eqs. (206) and (237), the Bloch oscillations are purely sinusoidal, so that the Landau-Zener tunneling is completely suppressed at B < 1.
Interband tunneling is an important ingredient of several physical phenomena and even some practical electron devices, for example, the tunneling (or “Esaki”) diodes. This simple device is just a junction of two semiconductor electrodes, one of them so strongly n-doped by electron donors that some electrons form a degenerate Fermi gas at the bottom of the conduction band. 84 Similarly, the counterpart semiconductor electrode is p-doped so strongly that the Fermi level in the valence band is shifted below the band edge – see Fig. 34.
(a)
(b)
(c)
I
n-doped
eV
p-doped
eV
0
/ e
V
Fig. 2.34. The tunneling (“Esaki”) diode: (a) the band-edge diagram of the device at zero bias; (b) the same diagram at a modest positive bias eV ~ /2, and (c) the I-V curve of the device (schematically). Dashed lines on panels (a) and (b) show the Fermi level positions.
In thermal equilibrium, and in the absence of external voltage bias, the Fermi levels of the two electrodes self-align, leading to the build-up of the contact potential difference / e, with a bit larger than the energy bandgap – see Fig. 34a. This potential difference creates an internal electric field that tilts the energy bands (just as the external field did in Fig. 33b), and leads to the formation of the so-called depletion layer, in which the Fermi level is located within the energy gap and hence there are no charge carriers ready to move. In the usual p-n junctions, this layer is broad and prevents any current at applied voltages V lower than ~/ e. In contrast, in a tunneling diode the depletion layer is so thin (below
~10 nm) that the interband tunneling is possible and provides a substantial Ohmic current at small applied voltages – see Fig. 34c. However, at larger positive biases, with eV ~ /2, the conduction band is aligned with the middle of the energy gap in the p- doped electrode, and electrons cannot tunnel there.
Similarly, there are no electrons in the n-doped semiconductor to tunnel into the available states just above the Fermi level in the p-doped electrode – see Fig. 34b. As a result, at such voltages the current 84 Here I have to rely on the reader’s background knowledge of basic semiconductor physics; it will be discussed in more detail in SM Sec. 6.4.
Chapter 2
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drops significantly, to grow again only when eV exceeds ~, enabling electron motion within each energy band. Thus the tunnel junction’s I-V curve has a part with a negative differential resistance ( dV/ dI < 0) – see Fig. 34c. This phenomenon, equivalent in its effect to negative kinematic friction in mechanics, may be used for amplification of weak analog signals, for self-excitation of electronic oscillators85 (i.e. an ac signal generation), and for signal swing restoration in digital electronics.
2.9. Harmonic oscillator: Brute force approach
To complete our review of the basic 1D wave mechanics, we have to consider the famous harmonic oscillator, i.e. a 1D particle moving in the quadratic-parabolic potential (111), so that the stationary Schrödinger equation (53) is
2
d
2
m 2 x 2
0
E .
(2.261)
2 m dx 2
2
Conceptually, on the background of the fascinating quantum effects discussed in the previous sections, this is not a very interesting system: Eq. (261) is a standard 1D eigenproblem, resulting in a discrete energy spectrum En, with smooth eigenfunctions n( x) vanishing at x (because the potential energy tends to infinity there).86 However, as we will repeatedly see later in the course, this problem’s solutions have an enormous range of applications, so we have to know their basic properties.
The direct analytical solution of the problem is not very simple (see below), so let us start by trying some indirect approaches to it. First, as was discussed in Sec. 4, the WKB-approximation-based Wilson-Sommerfeld quantization rule (110), applied to this potential, yields the eigenenergy spectrum (114). With the common quantum number convention, this result is
Harmonic
oscillator:
1
energy
E
,
(2.262)
n
n
,
with n
...
2,
1,
0,
0
levels
2
so that (in contrast to the 1D rectangular potential well) the ground-state energy corresponds to n = 0.
However, as was discussed in the end of Sec. 4, for the quadratic potential (111) the WKB
approximation’s conditions are strictly satisfied only at En >> 0, so that so far we can only trust Eq.
(262) for high levels, with n >> 1, rather than for the (most important) ground state.
This is why let me use Eq. (261) to demonstrate another approximate approach, called the variational method, whose simplest form is aimed at finding ground states. The method is based on the following observation. (Here I am presenting its 1D wave mechanics form, though the method is much more general.) Let n be the exact, full, and orthonormal set of stationary wavefunctions of the system under study, and En the set of the corresponding energy levels, satisfying Eq. (1.60): H ˆ E .
(2.263)
n
n
n
Then we may use this set for the unique expansion of an arbitrary trial wavefunction trial : 85 See, e.g., CM Sec. 5.4.
86 The stationary state of the harmonic oscillator (which, as will be discussed in Secs. 5.4 and 7.1, may be considered as the state with a definite number of identical bosonic excitations) are sometimes called Fock states –
after Vladimir Aleksandrovich Fock. (This term is also used in a more general sense, for definite-particle-number states of systems with indistinguishable bosons of any kind – see Sec. 8.3.)
Chapter 2
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,
that
so
*
* *
(2.264)
n
n
,
trial
trial
n
n
n
n
where n are some (generally, complex) coefficients. Let us require the trial function to be normalized, using the condition (1.66) of orthonormality of the eigenfunctions n:
*
3
* *
3
*
*
3
*
d x
d x
d x
W
, (2.265)
n
n
n'
n'
n'
n n n'
n'
n
n,n'
1
trial
trial
n
n, n'
n, n'
n, n'
n
where each of the coefficients Wn, defined as
2
*
W
,
0
(2.266)
n
n
n
n
may be interpreted as the probability for the particle, in the n th trial state, to be found in the n th genuine stationary state. Now let us use Eq. (1.23) for a similar calculation of the expectation value of the system’s Hamiltonian in the trial state:87
*
ˆ
3
* * ˆ
3
*
*
3
H
H d x H d x E d x n
n
n'
n'
trial
trial
n'
n
n'
trial
n
n'
n, n'
n, n'
(2.267)
*
E W E
n'
n
n'
n,n'
.
n
n
n, n'
n
Since the exact ground state energy E g is, by definition, the lowest one of the set En, i.e. En E g, Eqs.
(265) and (267) yield the following inequality:
Variational
H
W E E W E .
(2.268)
n
g
g
n
g
method’s
trial
n
n
justification
Thus, the genuine ground state energy of the system is always lower than (or equal to) its energy in any trial state. Hence, if we make several attempts with reasonably selected trial wavefunctions, we may expect the lowest of the results to approximate the genuine ground state energy reasonably well.
Even more conveniently, if we select some reasonable class of trial wavefunctions dependent on a free parameter , then we may use the necessary condition of the minimum of Htrial,
H
trial 0 ,
(2.269)
to find the closest of them to the genuine ground state. Even better results may be obtained using trial wavefunctions dependent on several parameters. Note, however, that the variational method does not tell us how exactly the trial function should be selected, or how close its final result is to the genuine ground-state function. In this sense, this method has “uncontrollable accuracy”, and differs from both the WKB approximation and the perturbation methods (to be discussed in Chapter 6), for which we have certain accuracy criteria. Because of this drawback, the variational method is typically used as the last resort – though sometimes (as in the example that follows) it works remarkably well.88
87 It is easy (and hence left for the reader) to show that the uncertainty H in any state of a Hamiltonian system, including the trial state (264), vanishes, so that the Htrial may be interpreted as the definite energy of the state.
For our current goals, however, this fact is not important.
88 The variational method may be used also to estimate the first excited state (or even a few lowest excited states) of the system, by requiring the new trial function to be orthogonal to the previously calculated eigenfunctions of the lower-energy states. However, the method’s error typically grows with the state number.
Chapter 2
Page 66 of 76
QM: Quantum Mechanics
Let us apply this method to the harmonic oscillator. Since the potential (111) is symmetric with respect to point x = 0, and continuous at all points (so that, according to Eq. (261), d 2/ dx 2 has to be continuous as well), the most natural selection of the ground-state trial function is the Gaussian function
x C 2
exp x ,
(2.270)
trial
with some real > 0. The normalization coefficient C may be immediately found either from the standard Gaussian integration of trial2, or just from the comparison of this expression with Eq. (16), in which = 1/(2 x)2, i.e. x = 1/21/2, giving C 2 = (2/)1/2. Now the expectation value of the particle’s Hamiltonian,
ˆ 2
2
2
2
2
p
d
m x
ˆ
H
U x
0
,
(2.271)
2 m
2
2
m dx
2
in the trial state, may be calculated as
2
2
2
2
*
d
m x
0
H
dx
trial
trial
2
2
m dx
2
trial
(2.272)
2 1/2
2
m
2
2 2
exp 2 2
x
0
2
2
dx
x
exp 2 2
x dx.
