m
2
m
0
0
Both involved integrals are of the same well-known Gaussian type,89 giving
2
2
0
H
m
.
(2.273)
trial
2 m
8
As a function of , this expression has a single minimum at the value opt that may be found from the requirement (269), giving opt = m0/2. The resulting minimum of Htrial is exactly equal to ground-state energy following from Eq. (262),
Harmonic
oscillator:
0
E
.
(2.274)
0
ground state
2
energy
Such a coincidence of results of the WKB approximation and the variational method is rather unusual, and implies (though does not strictly prove) that Eq. (274) is exact. As a minimum, this coincidence gives a strong motivation to plug the trial wavefunction (270), with = opt, i.e.
Harmonic
1/ 4
2
oscillator:
m
m x
ground state
0
exp
0
,
(2.275)
0
wavefunction
2
and the energy (274), into the Schrödinger equation (261). Such substitution90 shows that the equation is indeed exactly satisfied.
According to Eq. (275), the characteristic scale of the wavefunction’s spatial spread91 is 89 See, e.g., MA Eqs. (6.9b) and (6.9c).
90 Actually, this is a twist of one of the tasks of Problem 1.12.
91 Quantitatively, as was already mentioned in Sec. 2.1, x 0 = 2 x = 2 x 21/2.
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1/ 2
Harmonic
x
.
(2.276)
oscillator:
0
m
spatial scale
0
Due to the importance of this scale, let us give its crude estimates for several representative systems:92
(i) For atom-bound electrons in solids and fluids, m ~ 10-30 kg, and 0 ~ 1015 s-1, giving x 0 ~ 0.3
nm, of the order of the typical inter-atomic distances in condensed matter. As a result, classical mechanics is not valid at all for the analysis of their motion.
(ii) For atoms in solids, m 10-24-10-26 kg, and 0 ~ 1013 s-1, giving x 0 ~ 0.01 – 0.1 nm, i.e.
somewhat smaller than inter-atomic distances. Because of that, the methods based on classical mechanics (e.g., molecular dynamics) are approximately valid for the analysis of atomic motion, though they may miss some fine effects exhibited by lighter atoms – e.g., the so-called quantum diffusion of hydrogen atoms, due to their tunneling through the energy barriers of the potential profile created by other atoms.
(iii) Recently, the progress of patterning technologies has enabled the fabrication of high-quality micromechanical oscillators consisting of zillions of atoms. For example, the oscillator used in one of the pioneering experiments in this field93 was a ~1-m thick membrane with a 60-m diameter, and had m ~ 210-14 kg and 0 ~ 31010 s-1, so that x 0 ~ 410-16 m. It is remarkable that despite such extreme smallness of x 0 (much smaller than not only any atom but even any atomic nucleus!), quantum states of such oscillators may be manipulated and measured, using their coupling to electromagnetic (in particular, optical) resonant cavities.94
Returning to the Schrödinger equation (261), in order to analyze its higher eigenstates, we will need more help from mathematics. Let us recast this equation into a dimensionless form by introducing the dimensionless variable x/ x 0. This gives
d
2
2
,
(2.277)
2
d
where 2 E/0 = E/ E 0. In this notation, the ground-state wavefunction (275) is proportional to exp{-
2/2}. Using this clue, let us look for solutions to Eq. (277) in the form
2
C exp
H ( ) ,
(2.278)
2
where H() is a new function, and C is the normalization constant. With this substitution, Eq. (277) yields
2
d H dH
2
( )
1 H 0 .
(2.279)
2
d
d
92 By order of magnitude, such estimates are valid even for the systems whose dynamics is substantially different from that of harmonic oscillators, if a typical frequency of quantum transitions is taken for 0.
93 A. O’Connell et al., Nature 464, 697 (2010).
94 See a review of such experiments by M. Aspelmeyer et al., Rev. Mod. Phys. 86, 1391 (2014), and also recent experiments with nanoparticles placed in much “softer” potential wells – e.g., by U. Delić et al., Science 367, 892
(2020).
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It is evident that H = const and = 1 is one of its solutions, describing the ground-state eigenfunction (275) and energy (274), but what are the other eigenstates and eigenvalues? Fortunately, the linear differential equation (274) was studied in detail in the mid-1800s by C. Hermite who has shown that all its eigenvalues are given by the set
1 2 ,
n with n = 0, 1, 2,…,
(2.280)
n
so that Eq. (262) is indeed exact for any n.95 The eigenfunction of Eq. (279), corresponding to the eigenvalue n, is a polynomial (called the Hermite polynomial) of degree n, which may be most conveniently calculated using the following explicit formula:
n
Hermite
d
polynomials
H
1 n
.
(2.281)
n
2
exp n
2
exp
d
It is easy to use this formula to spell out several lowest-degree polynomials – see Fig. 35a: H ,
1
H 2 , H 4 2
,
2
H 8 3
12, H 16
4
- 48 2
...
12,
(2.282)
0
1
2
3
4
10
n 2
(a)
n 1
H ( )
n
n 0
0
n 3
10 3
0
3
(b)
3
E 3
2
Fig. 2.35. (a) A few lowest Hermite
E 2
polynomials and (b) the corresponding
U ( x)
eigenenergies (horizontal dashed lines)
1
and eigenfunctions (solid lines) of the
E
harmonic oscillator. The black dashed
1
curve shows the potential profile U( x)
0
drawn on the same scale as the energies
En, so that its crossings with the energy
E 0
levels correspond to classical turning
points.
0
x / x 0
95 Perhaps the most important property of this energy spectrum is that it is equidistant: En+1 – En = 0 = const.
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The properties of these polynomials, which most important for applications, are as follows: (i) the function Hn() has exactly n zeros (its plot crosses the - axis exactly n times); as a result, the “parity” (symmetry-antisymmetry) of these functions alternates with n, and (ii) the polynomials are mutually orthonormal in the following sense:
H ( ) H ( )
d
n
(2.283)
n
exp
2
n
n'
1/ 2 2 !
.
n, n'
Using the last property, we may readily calculate, from Eq. (278), the normalized eigenfunctions n( x) of the harmonic oscillator – see Fig.35b:
Harmonic
1
2
x
x
( x)
.
(2.284) oscillator:
n
H n
eigen-
2 n n
exp
1/ 2
2
1/ 4 1/ 2
!
x
2 x
x
0
0
0
functions
At this point, it is instructive to compare these eigenfunctions with those of a 1D rectangular potential well, with its ultimately hard walls – see Fig. 1.8. Let us list their common features: (i) The wavefunctions oscillate in the classically allowed regions with En > U( x), while dropping exponentially beyond the boundaries of that region. (For the rectangular well with infinite walls, the latter regions are infinitesimally narrow.)
(ii) Each step up the energy level ladder increases the number of the oscillation half-waves (and hence the number of its zeros), by one.96
And here are the major features specific for soft (e.g., the quadratic-parabolic) confinement: (i) The spatial spread of the wavefunction grows with n, following the gradual widening of the classically allowed region.
(ii) Correspondingly, En exhibits a slower growth than the En n 2 law given by Eq.
(1.85), because the gradual reduction of spatial confinement moderates the kinetic energy’s growth.
Unfortunately, the “brute-force” approach to the harmonic oscillator problem, discussed above, is not too appealing. First, the proof of Eq. (281) is rather longish – so I do not have time/space for it.
More importantly, it is hard to use Eq. (284) for the calculation of the expectation values of observables including the so-called matrix elements of the system – as we will see in Chapter 4, virtually the only numbers important for most applications. Finally, it is also almost evident that there has to be some straightforward math leading to any formula as simple as Eq. (262) for En. Indeed, there is a much more efficient, operator-based approach to this problem; it will be described in Sec. 5.4.
2.10. Exercise problems
2.1. The initial wave packet of a free 1D particle is described by Eq. (20):
x,0 a eikxdk .
k
96 In mathematics, a slightly more general statement, valid for a broader class of ordinary linear differential equations, is frequently called the Sturm oscillation theorem, and is a part of the Sturm-Liouville theory of such equations – see, e.g., Chapter 10 in the handbook by G. Arfken et al., cited in MA Sec. 16.
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(i) Obtain a compact expression for the expectation value p of the particle's momentum at an arbitrary moment t > 0.
(ii) Calculate p for the case when the function ak2 is symmetric with respect to some value k 0.
2.2. Calculate the function ak defined by Eq. (20), for the wave packet with a rectangular spatial envelope:
C
exp ik x
a
x
a
0
for
,
/ 2 / ,
2
( x )
0
,
otherwise.
,
0
Analyze the result in the limit k 0 a .
2.3. Prove Eq. (49) for the 1D propagator of a free quantum particle, starting from Eq. (48).
2.4. Express the 1D propagator defined by Eq. (44), via the eigenfunctions and eigenenergies of a particle moving in an arbitrary stationary potential U( x).
2.5. Calculate the change of the wavefunction of a 1D particle, resulting from a short pulse of an external classical force that may be approximated by the delta function:97
F t P t.
2.6. Calculate the transparency T of the rectangular potential barrier (68),
for
,
0
x d / ,
2
U ( x) U ,
for d / 2 x d / ,
2
0
,0 for
d / 2 x,
for a particle of energy E > U 0. Analyze and interpret the result, taking into account that U 0 may be either positive or negative. (In the latter case, we are speaking about the particle’s passage over a rectangular potential well of a finite depth U 0.)
2.7. Prove Eq. (117) for the case TWKB << 1, using the connection formulas (105).
2.8. Spell out the stationary wavefunctions of a harmonic oscillator in the WKB approximation, and use them to calculate the expectation values x 2 and x 4 for the eigenstate number n >> 1.
2.9. Use the WKB approximation to express the expectation value of the kinetic energy of a 1D
particle confined in a soft potential well, in its n th stationary state, via the derivative dEn/ dn, for n >> 1.
2.10. Use the WKB approximation to calculate the transparency T of the following triangular potential barrier:
,
0
for
x ,
0
U ( x)
U Fx, for x ,
0
0
97 The constant P is called the force’s impulse. (In higher dimensionalities, it is a vector – just as the force is.) Chapter 2
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with F, U 0 > 0, as a function of the incident particle’s energy E.
Hint: Be careful with the sharp potential step at x = 0.
2.11.* Prove that the symmetry of the 1D scattering matrix S describing an arbitrary time-independent scatterer, allows its representation in the form (127).
2.12. Prove the universal relations between elements of the 1D transfer matrix T of a stationary (but otherwise arbitrary) scatterer, mentioned in Sec. 5.
2.13. A 1D particle had been localized in a very narrow and deep potential well, with the “energy area” U( x) dx equal to – W, where W > 0. Then (say, at t = 0) the well’s bottom is suddenly lifted up, so that the particle becomes completely free. Calculate the probability density, w( k), to find the particle in a state with a certain wave number k at t > 0, and the total final energy of the system.
2.14. Calculate the lifetime of the metastable localized state of a 1D particle in the potential U x W x Fx,
with W 0 ,
using the WKB approximation. Formulate the condition of validity of the result.
U( x)
2.15. Calculate the energy levels and the corresponding eigenfunctions
of a 1D particle placed into a flat-bottom potential well of width 2 a, with infinitely-high hard walls, and a transparent, short potential barrier in the
W ( x)
middle – see the figure on the right. Discuss particle dynamics in the limit when W is very large but still finite.
a
0
a x
2.16.* Consider a symmetric system of two potential wells of
U x
the type shown in Fig. 21, but with U(0) = U() = 0 – see the figure on the right. What is the sign of the well interaction force due
to their sharing a quantum particle of mass m, for the cases when
0
x
the particle is in:
(i) a symmetric localized eigenstate: S(– x) = S( x)?
(ii) an antisymmetric localized eigenstate: A(– x) = –A( x)?
Use an alternative approach to verify your result for the particular case of delta-functional wells.
2.17. Derive and analyze the characteristic equation for localized
U x
eigenstates of a 1D particle in a rectangular potential well of a finite depth
(see the figure on the right):
a / 2
a / 2
0
x
U
for
,
x a/ 2 ,
U ( x)
0
U
otherwise.
,
0
0
In particular, calculate the number of localized states as a function of well’s width a, and explore the limit U 0 << 2/2 ma 2.
Chapter 2
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2.18. Calculate the energy of a 1D particle localized in a potential well of an arbitrary shape U( x), provided that its width a is finite, and the average depth is very small: 2
1
U
,
where U
U x dx
2
2 ma
a well
2.19. A particle of mass m is moving in a field with the following potential:
U x U
,
0 x W x
where U 0( x) is a smooth, symmetric function with U 0(0) = 0, growing monotonically at x .
(i) Use the WKB approximation to derive the characteristic equation for the particle’s energy spectrum, and
(ii) semi-quantitatively describe the spectrum’s evolution at the increase of W , for both signs of this parameter.
Make both results more specific for the quadratic-parabolic potential (111): U
2
0( x) = m0 x 2/2.
2.20. Prove Eq. (189), starting from Eq. (188).
2.21. For the problem discussed at the beginning of Sec. 7, i.e. the 1D particle’s motion in an infinite Dirac comb potential shown in Fig. 24,
U x W
x ja,
with W 0 ,
j
(where j takes integer values), write explicit expressions for the eigenfunctions at the very bottom and at the very top of the lowest energy band. Sketch both functions.
2.22. A 1D particle of mass m moves in an infinite periodic system of very narrow and deep potential wells that may be described by delta functions:
U x W
x ja,
with W 0 .
j
(i) Sketch the energy band structure of the system for very small and very large values of the potential well’s “weight” W, and
(ii) calculate explicitly the ground state energy of the system in these two limits.
2.23. For the system discussed in the previous problem, write explicit expressions for the eigenfunctions of the system, corresponding to:
(i) the bottom of the lowest energy band,
(ii) the top of that band, and
(iii) the bottom of each higher energy band.
Sketch these functions.
2.24.* The 1D “crystal” analyzed in the last two problems, now extends only to x > 0, with a sharp step to a flat potential plateau at x < 0:
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U x W
x ja, with W ,0
for x
,
0
j1
U
for
,
0
x .
0
0
Prove that the system can have a set of the so-called Tamm states, localized near the “surface” x = 0, and calculate their energies in the limit when U 0 is very large but finite. (Quantify this condition.) 98
2.25. Calculate the whole transfer matrix of the rectangular potential barrier, specified by Eq.
(68), for particle energies both below and above U 0.
2.26. Use the results of the previous problem to calculate the transfer matrix of one period of the periodic Kronig-Penney potential shown in Fig. 31b.
2.27. Using the results of the previous problem, derive the characteristic equations for particle’s motion in the periodic Kronig-Penney potential, for both E < U 0 and E > U 0. Try to bring the equations to a form similar to that obtained in Sec. 7 for the delta-functional barriers – see Eq. (198). Use the equations to formulate the conditions of applicability of the tight-binding and weak-potential approximations, in terms of the system’s parameters, and the particle’s energy E.
2.28. For the Kronig-Penney potential, use the tight-binding approximation to calculate the widths of the allowed energy bands. Compare the results with those of the previous problem (in the corresponding limit).
2.29. For the same Kronig-Penney potential, use the weak-potential limit formulas to calculate the energy gap widths. Again, compare the results with those of Problem 27, in the corresponding limit.
2.30. 1D periodic chains of atoms may exhibit what is called the Peierls instability, leading to the Peierls transition to a phase in which atoms are slightly displaced, from the exact periodicity, by alternating displacements xj = (-1) j x, with x << a, where j is the atom’s number in the chain, and a is its initial period. These displacements lead to the alternation of the coupling amplitudes n (see Eq.
(204)) between close values +
-
n and n . Use the tight-binding approximation to calculate the resulting change of the n th energy band, and discuss the result.
2.31.* Use Eqs. (1.73)-(1.74) of the lecture notes to derive Eq. (252), and discuss the relation between these Bloch oscillations and the Josephson oscillations of frequency (1.75).
2.32. A 1D particle of mass m is placed to the following triangular potential well: U x , for x ,
0
with F 0 .
Fx,
for x ,
0
(i) Calculate its energy spectrum using the WKB approximation.
98 In applications to electrons in solid-state crystals, the delta-functional potential wells model the attractive potentials of atomic nuclei, while U 0 represents the workfunction, i.e. the energy necessary for the extraction of an electron from the crystal to the free space – see, e.g., Sec. 1.1(ii), and also EM Sec. 2.6 and SM Sec. 6.3.
Chapter 2
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Essential Graduate Physics
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(ii) Estimate the ground state energy using the variational method, with two different trial functions.
(iii) Calculate the three lowest energy levels, and also the 10th level, with an accuracy better than 0.1%, from the exact solution of the problem.
(iv) Compare and discuss the results.
Hint: The values of the first zeros of the Airy function, necessary for Task (iii), may be found in many math handbooks, for example, in Table 9.9.1 of the online version of the collection edited by Abramowitz and Stegun – see MA Sec. 16(i).
2.33. Use the variational method to estimate the ground state energy E g of a particle in the following potential well:
U x U
exp
2
x
U .
0
, with
,
0
and
0
0
Spell out the results in the limits of small and large U 0, and give their interpretation.
2.34. For a 1D particle of mass m, placed to a potential well with the following profile, U x
2
ax s , with a
and
,
0
s 0 ,
(i) calculate its energy spectrum using the WKB approximation, and
(ii) estimate the ground state energy using the variational method.
Compare the ground-state energy results for the parameter s equal to 1, 2, 3, and 100.
2.35. Use the variational method to estimate the 1st excited state of the 1D harmonic oscillator.
2.36. Assuming the quantum effects to be small, calculate the lower part of
the energy spectrum of the following system: a small bead of mass m, free to move without friction along a ring of radius R, which is rotated about its vertical diameter with a constant angular velocity – see the figure on the right. Formulate a quantitative condition of validity of your results.
R
Hint: This system was used as the analytical mechanics’ “testbed problem”
in the CM part of this series, and the reader is welcome to use any relations derived
mg
there.
2.37. A 1D harmonic oscillator, with mass m and frequency 0, had been in its ground state; then an additional force F was suddenly applied, and after that kept constant. Calculate the probability of the oscillator staying in its ground state.
2.38. A 1D particle of mass m has been placed to a quadratic potential well (111), 2
2
m x
U ( x)
0
,
2
and allowed to relax into the ground state. At t = 0, the well was fast accelerated to move with velocity v, without changing its profile, so that at t 0 the above formula for U is valid with the replacement x
x’ x – vt. Calculate the probability for the system to still be in the ground state at t > 0.
Chapter 2
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2.39. Initially, a 1D harmonic oscillator was in its ground state. At a certain moment of time, its spring constant is abruptly increased so that its frequency 0 = (/ m)1/2 is increased by a factor of , and then is kept constant at the new value. Calculate the probability that after the change, the oscillator is still in its ground state.
2.40. A 1D particle is placed into the following potential well:
,
for
x ,
0
U ( x)
2
2
m x / ,
2
for x 0.
0
(i) Find its eigenfunctions and eigenenergies.
(ii) This system had been let to relax into its ground state, and then the potential wall at x < 0
was rapidly removed so that the system was instantly turned into the usual harmonic oscillator (with the same m and 0). Find the probability for the oscillator to remain in its ground state.
2.41. Prove the following formula for the propagator of the 1D harmonic oscillator:
1/ 2
m
im
G( x, t; x , t )
0
exp
0
x x
t t
xx
0
0
2 20cos[ (
)] 2
0
0
0 .
2 i
sin[ ( t t )]
2sin[ ( t t )]
0
0
0
0
Discuss the relation between this formula and the propagator of a free 1D particle.
2.42. In the context of the Sturm oscillation theorem mentioned in Sec. 9, prove that the number of eigenfunction’s zeros of a particle confined in an arbitrary but finite potential well always increases with the corresponding eigenenergy.
Hint: You may like to use the suitably modified Eq. (186).
2.43.* Use the WKB approximation to calculate the lifetime of the metastable ground state of a
1D particle of mass m in the “pocket” of the potential profile
2
m
U ( x)
0
2
3
x x .
2
Contemplate the significance of this problem.
Chapter 2
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Chapter 3. Higher Dimensionality Effects
The description of the basic quantum-mechanical effects, given in the previous chapter, may be extended to higher dimensions in an obvious way. This is why this chapter is focused on the phenomena ( such as the AB effect and the Landau levels) that cannot take place in one dimension due to topological reasons, and also on a few key 3D problems ( such as the Born approximation in the scattering theory, and the axially- and spherically-symmetric systems) that are important for numerous applications.
3.1. Quantum interference and the AB effect
In the past two chapters, we have already discussed some effects of the de Broglie wave interference. For example, standing waves inside a potential well, or even on the top of a potential barrier, may be considered as a result of interference of incident and reflected waves. However, there are some remarkable new effects made possible by spatial separation of such waves, and such separation requires a higher (either 2D or 3D) dimensionality. A good example of wave separation is provided by the Young-type experiment (Fig. 1) in which particles, emitted by the same source, are passed through two narrow holes (or slits) is an otherwise opaque partition.
x
l '
l ''
1
1
1
z
W (
w r)
2
C
particle
l ''
l '
2
detector
2
particle
Fig. 3.1. The scheme of the “two-slit”
source
(Young-type) interference experiment.
partition
with 2 slits
According to Eq. (1.22), if particle interactions are negligible (which is always true if the emission rate is sufficiently low), the average rate of particle counting by the detector is proportional to the probability density w(r, t) = (r, t) *(r, t) to find a single particle at the detector’s location r, where (r, t) is the solution of the single-particle Schrödinger equation (1.25) for the system. Let us calculate the rate for the case when the incident particles may be represented by virtually-monochromatic waves of energy E (e.g., very long wave packets), so that their wavefunction may be taken in the form given by Eqs. (1.57) and (1.62): (r, t) = (r) exp{- iEt/}. In this case, in the free-space parts of the system, where U(r) = 0, (r) satisfies the stationary Schrödinger equation (1.78a): 2
2
E .
(3.1a)
2 m
With the standard definition k (2 mE)1/2/, it may be rewritten as the 3D Helmholtz equation: 3D
Helmholtz
2
2
k 0 .
(3.1b)
equation
© K. Likharev
QM: Quantum Mechanics
The opaque parts of the partition may be well described as classically forbidden regions, so if their size scale a is much larger than the wavefunction penetration depth described by Eq. (2.59), we may use on their surface S the same boundary conditions as for the well’s walls of infinite height:
0 .
(3.2)
S
Eqs. (1) and (2) describe the standard boundary problem of the theory of propagation of scalar waves of any nature. For an arbitrary geometry, this problem does not have a simple analytical solution. However, for a conceptual discussion of wave interference, we may use certain natural assumptions that will allow us to find its particular, approximate solution.
First, let us discuss the wave emission, into free space, by a small-size, isotropic source located at the origin of our reference frame. Naturally, the emitted wave should be spherically symmetric: (r)
= ( r). Using the well-known expression for the Laplace operator in spherical coordinates,1 we may reduce Eq. (1) to the following ordinary differential equation:
1 d
d
2
2
r
k 0 .
(3.3)
2
r dr
dr
Let us introduce a new function, f( r) r( r). Plugging the reciprocal relation = f/ r into Eq. (3), we see that it is reduced to the 1D wave equation,
2
d f
2
k f 0 .
(3.4)
2
dr
As was discussed in Sec. 2.2, for a fixed k, the general solution of Eq. (4) may be represented in the form of two traveling waves:
ikr
ikr
f f e
f e
(3.5)
so that the full wavefunction is
f
2
ikr
f
ikr
f i( kr t)
f
i
( kr t)
E
k
(r)
e
e
,
i.e. r, t
e
e
,
with
. (3.6)
r
r
r
r
2 m
If the source is located at point r ’ 0, the obvious generalization of Eq. (6) is f
i kR
t
f
(r, t)
(
)
i( kR t
)
e
e
,
with R R , R r r' .
(3.7)
R
R
The first term of this solution describes a spherically-symmetric wave propagating from the source outward, while the second one, a wave converging onto the source point r ’ from large distances.
Though the latter solution is possible at some very special circumstances (say, when the outgoing wave is reflected back from a spherical shell), for our current problem, only the outgoing waves are relevant, so that we may keep only the first term (proportional to f+) in Eq. (7). Note that the factor R is the denominator (that was absent in the 1D geometry) has a simple physical sense: it provides the independence of the full probability current I = 4 R 2 j( R), with j( R) k* 1/ R 2, of the distance R
between the observation point and the source.
1 See, e.g., MA Eq. (10.9) with / = / = 0.
Chapter 3
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Now let us assume that the partition’s geometry is not too complicated – for example, it is either planar as shown in Fig. 1, or nearly-planar, and consider the region of the particle detector location far behind the partition (at z >> 1/ k), and at a relatively small angle to it: x << z. Then it should be physically clear that the spherical waves (7) emitted by each point inside the slit cannot be perturbed too much by the opaque parts of the partition, and their only role is the restriction of the set of such emitting points to the area of the slits. Hence, an approximate solution of the boundary problem is given by the following Huygens principle: the wave behind the partition looks as if it was the sum of contributions (7) of point sources located in the slits, with each source’s strength f+ proportional to the amplitude of the wave arriving at this pseudo-source from the real source – see Fig. 1. This principle finds its confirmation in the strict wave theory, which shows that with our assumptions, the solution of the boundary problem (1)-(2) may be represented as the following Kirchhoff integral:2
Kirchhoff
( ' ) ikR 2
k
( ) c
e
d r' ,
c
with
integral
r
r
.
(3.8)
R
2 i
slits
If the source is also far from the partition, its wave’s front is almost parallel to the slit plane, and if the slits are not too broad, we can take (r ’) constant (1,2) at each slit, so that Eq. (8) is reduced to cA
(r) a" exp ikl" a" exp ikl" , with a"
,
(3.9)
1
1
2
2
,
1 2
,
1 2
,
1 2
l" ,12
where A 1,2 are the slit areas, and l” 1,2 are the distances from the slits to the detector. The wavefunctions on the slits may be calculated approximately3 by applying the same Eq. (7) to the region before the slits:
1,2 ( f+/ l’ 1,2)exp{ ikl’ 1,2}, where l’ 1,2 are the distances from the source to the slits – see Fig. 1. As a result, Eq. (9) may be rewritten as
Wave-
c f A
function
(r) a exp ikl a exp ikl ,
with l
l' l'' ; a
.
(3.10)
1
1 2 2
,
1 2
,
1 2
,
1 2
,
1 2
,
1 2
superposition
l' l"
,
1 2
,
1 2
(As Fig. 1 shows, each of l 1,2 is the full length of the classical path of the particle from the source, through the corresponding slit, and further to the observation point r.)
According to Eq. (10), the resulting rate of particle counting at point r is proportional to Quantum
interference
2
2
(
w r) (r) *
(r) a a
2 a a cos ,
(3.11)
1
2
1 2
12
where
k( l l )
(3.12)
12
2
1
is the difference of the total wave phase accumulations along each of two alternative paths. The last expression may be evidently generalized as
2 For the proof and a detailed discussion of Eq. (8), see, e.g., EM Sec. 8.5.
3 A possible (and reasonable) concern about the application of Eq. (7) to the field in the slits is that it ignores the effect of opaque parts of the partition. However, as we know from Chapter 2, the main role of the classically forbidden region is reflecting the incident wave toward its source (i.e. to the left in Fig. 1). As a result, the contribution of this reflection to the field inside the slits is insignificant if A 1,2 >> 2, and even in the opposite case provides just some rescaling of the amplitudes a 1,2, which is not important for our conceptual discussion.
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Quantum
,
(3.13) interference:
12
k dr
phase
C
difference
with integration along the virtually closed contour C (see the dashed line in Fig. 1), i.e. from point 1, in the positive (i.e. counterclockwise) direction all the way to point 2. (From our discussion of the 1D
WKB approximation in Sec. 2.4, we may expect such generalization to be valid even if k changes, sufficiently slowly, along the paths.)
Our result (11)-(12) shows that the particle counting rate oscillates as a function of the difference ( l 2 – l 1), which in turn changes with the detector’s position, giving the famous interference pattern, with the amplitude proportional to the product a 1 a 2, and hence vanishing if any of the slits is closed. For the wave theory, this is a well-known result,4 but for particle physics, it was (and still is :-) rather shocking. Indeed, our analysis is valid for a very low particle emission rate, so that there is no other way to interpret the pattern other than resulting from a particle’s interference with itself – or rather the interference of its de Broglie waves passing through each of two slits.5 Nowadays, such interference is reliably observed not only for electrons but also for much heavier particles: atoms and molecules, including very complex organic ones;6 moreover, atomic interferometers are used as ultra-sensitive instruments for measurements of gravity, rotation, and tilt.7
Let us now discuss a very interesting effect of magnetic field on quantum interference. To simplify our discussion, let us consider a slightly different version of the two-slit experiment, in which each of the two alternative paths is constricted to a narrow channel using partial confinement – see Fig.
2. (In this arrangement, moving the particle detector without changing channels’ geometry, and hence local values of k may be more problematic experimentally, so let us think about its position r as fixed.) In this case, because of the effect of the walls providing the path confinement, we cannot use Eqs. (10) for the amplitudes a 1,2. However, from the discussions in Sec. 1.6 and Sec. 2.2, it should be clear that the first of the expressions (10) remains valid, though maybe with a value of k specific for each channel.
region with B 0
channel 1
1
w (
w B)
C
2
channel 2
Fig. 3.2. The AB effect.
In this geometry, we can apply some local magnetic field B, say normal to the plane of particle motion, whose lines would pierce, but not touch the contour C drawn along the particle propagation 4 See, e.g., a detailed discussion in EM Sec. 8.4.
5 Here I have to mention the fascinating experiments (first performed in 1987 by C. Hong et al. with photons, and recently, in 2015, by R. Lopes et al., with non-relativistic particles – helium atoms) on the interference of de Broglie waves of independent but identical particles, in the same internal quantum state and virtually the same values of E and k. These experiments raise the important issue of particle indistinguishability, which will be discussed in Sec. 8.1.
6 See, e.g., the recent demonstration of the quantum interference of oligo-porphyrin molecules, consisting of
~2,000 atoms, with a total mass above 25,000 m p – Y. Fein et al., Nature Physics 15, 1242 (2019).
7 See, e.g., the review paper by A. Cronin, J. Schmiedmayer, and D. Pritchard, Rev. Mod. Phys. 81, 1051 (2009).
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channels – see the dashed line in Fig. 2. In classical electrodynamics,8 the external magnetic field’s effect on a particle with electric charge q is described by the Lorentz force F qv
,
(3.14)
B
B
where B is the field value at the point of its particle’s location, so that for the experiment shown in Fig.
2, F = 0, and the field would not affect the particle motion at all. In quantum mechanics, this is not so, B
and the field does affect the probability density w, even if B = 0 at all points where the wavefunction
(r) is not equal to zero.
In order to describe this surprising effect, let us first develop a general framework for an account of electromagnetic field effects on a quantum particle, which will also give us some by-product results important for forthcoming discussions. To do that, we need to calculate the Hamiltonian of a charged particle in electric and magnetic fields. For an electrostatic field, this is easy. Indeed, from classical electrodynamics we know that such field may be represented as a gradient of its electrostatic potential , E r,
(3.15)
so that the force exerted by the field on a particle with electric charge q,
F
,
(3.16)
E
E
q
may be described by adding the field-induced potential energy,
U r qr,
(3.17)
to other (possible) components of the full potential energy of the particle. As was already discussed in Sec. 1.4, such potential energy may be included in the particle’s Hamiltonian operator just by adding it to the kinetic energy operator – see Eq. (1.41).
However, the magnetic field’s effect is peculiar: since its Lorentz force (14) cannot do any work on a classical particle:
d W F dr F v dt q(v
(3.18)
B
B
B
B ) v dt ,
0
the field cannot be represented by any potential energy, so it may not be immediately clear how to account for it in the Hamiltonian. The crucial help comes from the analytical-mechanics approach to classical electrodynamics:9 in the non-relativistic limit, the Hamiltonian function of a particle in an electromagnetic field looks like that in the electric field only:
mv 2
p 2
H
U
q ;
(3.19)
2
2 m
however, the momentum p mv that participates in this expression is now the difference p P A
q .
(3.20)
Here A is the vector potential, defined by the well-known relations for the electric and magnetic fields:10
8 See, e.g., EM Sec. 5.1. Note that Eq. (14), as well as all other formulas of this course, are in the SI units.
9 See, e.g., EM Sec. 9.7, in particular Eq. (9.196).
10 See, e.g., EM Sec. 6.1, in particular Eqs. (6.7).
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A
E
,
B A ,
(3.21)
t
while P is the canonical momentum, whose Cartesian components may be calculated (in classics) from the Lagrangian function L using the standard formula of analytical mechanics,
L
P
.
(3.22)
j
v
j
To emphasize the difference between the two momenta, p = mv is frequently called the kinematic momentum (or “mv-momentum”). The distinction between p and P = p + qA becomes more clear if we notice that the vector potential is not gauge-invariant: according to the second of Eqs. (21), at the so-called gauge transformation
A A ,
(3.23)
with an arbitrary single-valued scalar gauge function = (r, t), the magnetic field does not change.
Moreover, according to the first of Eqs. (21), if we make the simultaneous replacement
,
(3.24)
t
the gauge transformation does not affect the electric field either. With that, the gauge function’s choice does not affect the classical particle’s equation of motion, and hence the velocity v and momentum p.
Hence, the kinematic momentum is gauge-invariant, while P is not, because according to Eqs. (20) and (23), the introduction of changes it by q.
Now the standard way of transfer to quantum mechanics is to treat the canonical rather than kinematic momentum as prescribed by the correspondence postulate discussed in Sec. 1.2. This means that in the wave mechanics, the operator of this variable is still given by Eq. (1.26):11
Canonical
Pˆ i
.
(3.25) momentum
operator
Hence the Hamiltonian operator corresponding to the classical function (19) is
1
2
iq 2
2
H ˆ
i qA
q
A
q ,
(3.26)
2 m
2 m
so that the stationary Schrödinger equation (1.60) of a particle moving in an electromagnetic field (but Charged particle
otherwise free) is
in EM field
2
2
iq
A
q
E ,
(3.27)
2 m
We may now repeat all the calculations of Sec. 1.4 for the case A 0, and get the following generalized expression for the probability current density:
11 The validity of this choice is clear from the fact that if the kinetic momentum was described by this differential operator, the Hamiltonian operator corresponding to the classical Hamiltonian function (19), and the corresponding Schrödinger equation would not describe the magnetic field effects at all.
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iq
1
q
j
A c.c
ˆp c.c
2
A .
(3.28)
2 im
2 m
m
We see that the current density is gauge-invariant (as required for any observable) only if the wavefunction’s phase changes as
q
.
(3.29)
This may be a point of conceptual concern: since quantum interference is described by the spatial dependence of the phase , can the observed interference pattern depend on the gauge function’s choice? (That would not make any sense, because we may change the gauge in our mind.) Fortunately, this is not true, because the spatial phase difference between two interfering paths, participating in Eq.
(12), is gauge-transformed as
q
(3.30)
12
12
2
.
1
But has to be a single-valued function of coordinates, hence in the limit when the points 1 and 2
coincide, 1 = 2, so that gauge-invariant, and so is the interference pattern.
However, the difference may be affected by the magnetic field, even if it is localized outside the channels in which the particle propagates. Indeed, in this case, the field cannot affect the particle’s velocity v and the probability current density j:
(
j r)
j r
,
(3.31)
B
0
( ) B0
so that the last form of Eq. (28) yields
q
r
( )
r
(
A .
(3.32)
B 0
) B0
Integrating this equation along the contour C (Fig. 2), for the phase difference between points 1 and 2
we get
q
,
(3.33)
12 B0
12 B0
A dr
C
where the integral should be taken along the same contour C as before (in Fig. 2, from point 1, counterclockwise along the dashed line to point 2). But from classical electrodynamics we know12 that as points 1 and 2 tend to each other, i.e. the contour C becomes closed, the last integral is just the magnetic flux B nd 2 r through any smooth surface limited by the contour, so that Eq. (33) may be rewritten as
AB
q
effect
Φ .
(3.34a)
12 B 0
12 B 0
In terms of the interference pattern, this means a shift of interference fringes, proportional to the magnetic flux (Fig. 3).
12 See, e.g., EM Sec. 5.3.
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(a)
(b)
Fig. 3.3. Typical results of a two-paths interference experiment by A. Tonomura et al., Phys. Rev.
Lett. 56, 792 (1986), showing the AB effect for electrons well shielded from the applied magnetic field. In this particular experimental geometry, the AB effect produces a relative shift of the interference patterns inside and outside the dark ring. (a) = 0 ’/2, (b) = 0 ’. © 1986 APS.
This phenomenon is usually called the “Aharonov-Bohm” (or just the AB) effect.13 For particles with a single elementary charge, q = e, this result is frequently represented as
2
,
(3.34b)
B
0
12
12
B 0
' 0
where the fundamental constant 0 ’ 2/ e 4.1410-15 Wb has the meaning of the magnetic flux necessary to change 12 by 2, i.e. to shift the interference pattern (11) by one period, and is called the normal magnetic flux quantum – “normal” because of the reasons we will soon discuss.
The AB effect may be “almost explained” classically, in terms of Faraday’s electromagnetic induction. Indeed, a change of magnetic flux in time induces a vortex-like electric field E around it. That field is not restricted to the magnetic field’s location, i.e. may reach the particle’s trajectories.
The field’s magnitude (or rather of its integral along the contour C) may be readily calculated by integration of the first of Eqs. (21):
d Φ
V
Δ ΔE d
r
.
(3.35)
dt
C
I hope that in this expression the reader readily recognizes the integral (“undergraduate”) form of Faraday’s induction law.14 To calculate the effect of this electric field of the particles, let us assume that the variable separation described by Eq. (1.57) may be applied to the end points 1 and 2 of particle’s alternative trajectories as two independent systems,15 and that the magnetic flux’ change by a certain amount does not change the spatial factors 1,2, with the phases 1,2 included into the time-dependent factors a 1,2. Then we may repeat the arguments that were used in Sec. 1.6 at the discussion of 13 I prefer the latter, less personable name, because the effect had been actually predicted by Werner Ehrenberg and Raymond Siday in 1949, before it was rediscovered (also theoretically) by Y Aharonov and D. Bohm in 1959. To be fair to Aharonov and Bohm, it was their work that triggered a wave of interest in the phenomenon, resulting in its first experimental observation by Robert G. Chambers in 1960 and several other groups soon after that. Later, the experiments were improved using ferromagnetic cores and/or superconducting shielding to provide a better separation between the electrons and the applied field – as in the work whose result is shown in Fig. 3.
14 See, e.g., EM Sec. 6.1.
15 This assumption may seem a little bit of a stretch, but the resulting relation (37) may be indeed proven for a rather realistic model, though that would take more time/space than I can afford.
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the Josephson effect, and since the change (35) leads to the change of the potential energy difference U
= q V between the two points, we may rewrite Eq. (1.72) as
d
U
q
q d
12
V
.
(3.36)
dt
dt
Integrating this relation over the time of the magnetic field’s change, we get
q
,
(3.37)
12
- superficially, the same result as given by Eq. (34).
However, this interpretation of the AB effect is limited. Indeed, it requires the particle to be in the system (on the way from the source to the detector) during the flux change, i.e. when the induced electric field E may affect its dynamics. On the contrary, Eq. (34) predicts that the interference pattern would shift even if the field change has been made when there was no particle in the system, and hence the field E could not be felt by it. Experiment confirms the latter conclusion. Hence, there is something in the space where a particle propagates (i.e., outside of the magnetic field region), that transfers the information about even the static magnetic field to the particle. The standard interpretation of this surprising fact is as follows: the vector potential A is not just a convenient mathematical tool, but a physical reality (just as its scalar counterpart ), despite the large freedom of choice we have in prescribing specific spatial and temporal dependences of these potentials without affecting any observable – see Eqs. (23)-(24).
To conclude this section, let me briefly discuss the very interesting form taken by the AB effect in superconductivity. To be applied to this case, our results require two changes. The first one is simple: since superconductivity may be interpreted as the Bose-Einstein condensate of Cooper pairs with electric charge q = –2 e, 0 ’ has to be replaced by the so-called superconducting flux quantum 16
Super-
conducting
Φ
07
.
2
10
15 Wb 07
.
2
10 7 Gs cm2.
(3.38)
flux
0
e
quantum
Second, since the pairs are Bose particles and are all condensed in the same (ground) quantum state, described by the same wavefunction, the total electric current density, proportional to the probability current density j, may be extremely large – in practical superconducting materials, up to
~1012 A/m2. In these conditions, one cannot neglect the contribution of that current into the magnetic field and hence into its flux , which (according to the Lenz rule of the Faraday induction law) tries to compensate for changes in external flux. To see possible results of this contribution, let us consider a closed superconducting loop (Fig. 4). Due to the Meissner effect (which is just another version of the flux self-compensation), the current and magnetic field penetrate into a superconductor by only a small distance (called the London penetration depth) L ~ 10-7 m.17 If the loop is made of a superconducting
“wire” that is considerably thicker than L, we may draw a contour deep inside the wire, at that the current density is negligible. According to the last form of Eq. (28), everywhere at the contour, q
A 0 .
(3.39)
16 One more bad, though common term: a metallic wire may (super)conduct, but a quantum hardly can!
17 For more detail, see EM Sec. 6.4.
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Integrating this equation along the contour as before (in Fig. 4, from some point 1, all the way around the ring to the virtually coinciding point 2), we need to have the phase difference 12 equal to 2 n, because the wavefunction exp{ i} in the initial and final points 1 and 2 should be “essentially” the same, i.e. produce the same observables. As a result, we get
Φ A d
r
2 n
.
n
(3.40) Flux
0
q
quantization
C
This is the famous flux quantization effect,18 which justifies the term “magnetic flux quantum” for the constant 0 given by Eq. (38).
B
C
1
2
Fig. 3.4. The magnetic flux quantization in a
superconducting loop (schematically).
Unfortunately, in this course I have no space/time to discuss the very interesting effects of
“partial flux quantization” that arise when a superconductor loop is closed with a Josephson junction, forming the so-called Superconductor QUantum Interference Device – “SQUID”. Such devices are used, in particular, for supersensitive magnetometry and ultrafast, low-power computing.19
3.2. Landau levels and quantum Hall effect
In the last section, we have used the Schrödinger equation (27) for an analysis of static magnetic field effects in “almost-1D”, circular geometries shown in Figs. 1, 2, and 4. However, this equation describes very interesting effects in fully-higher-dimensions as well, especially in the 2D case. Let us consider a quantum particle free to move in the [ x, y] plane only (say, due to its strong confinement in the perpendicular direction z – see the discussion in Sec. 1.8). Taking the confinement energy for the reference, we may reduce Eq. (27) to a similar equation, but with the Laplace operator acting only in the directions x and y:
2
2
q
n
n
i A
E .
(3.41)
2 m x
x
y
y
Let us find its solutions for the simplest case when the applied static magnetic field is uniform and perpendicular to the motion plane:
B B n .
(3.42)
z
18 It was predicted in 1949 by Fritz London and experimentally discovered (independently and virtually simultaneously) in 1961 by two experimental groups: B. Deaver and W. Fairbank, and R. Doll and M. Näbauer.
19 A brief review of these effects, and recommendations for further reading may be found in EM Sec. 6.5.
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According to the second of Eqs. (21), this relation imposes the following restriction on the choice of the vector potential:
Ay A
x B ,
(3.43)
x
y
but the gauge transformations still give us a lot of freedom in its choice. The “natural” axially-symmetric form, A = nB/2, where = ( x 2 + y 2)1/2 is the distance from some z-axis, leads to cumbersome math. In 1930, L. Landau realized that the energy spectrum of Eq. (41) may be obtained by making a much simpler, though very counter-intuitive choice:
A ,
0
A B
(3.44)
x
y
x x 0 ,
(with arbitrary x 0), which evidently satisfies Eq. (43), though ignores the physical symmetry of the x and y directions for the field (42).
Now, expanding the eigenfunction into the Fourier integral in the y-direction:
( x, y) X ( x
) exp
,
(3.45)
k
ik y y 0 dk
we see that for each component of this integral, Eq. (41) yields a specific equation
2
d
q
2
n
in k B x x
.
(3.46)
0 X EX
x
y
k
k
2 m dx
Since the two vectors inside the curly brackets are mutually perpendicular, its square has no cross-terms, so that Eq. (46) reduces to
2 d 2
q 2
2
2
k
X
,
where
.
(3.47)
2
k
B x x ' 0 X
EX
x '
x
2 m dx
2 m
k
k
0
0
q B
But this 1D Schrödinger equation is identical to Eq. (2.261) for a 1D harmonic oscillator,20 with the center at point x 0 ’, and frequency 0 equal to
q B
.
(3.48)
c
m
In the last expression, it is easy to recognize the cyclotron frequency of the classical particle’s rotation in the magnetic field. (It may be readily obtained using the 2nd Newton law for a circular orbit of radius r, v 2
m
F qv ,
(3.49)
B
B
r
and noting that the resulting ratio v/ r = q B / m is just the radius-independent angular velocity c of the particle’s rotation.) Hence, the energy spectrum for each Fourier component of the expansion (45) is the same:
20 This result may become a bit less puzzling if we recall that at the classical circular cyclotron motion of a particle, each of its Cartesian coordinates, including x, performs sinusoidal oscillations with frequency (48), just as a 1D harmonic oscillator with this frequency.
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1
E
n
,
(3.50) Landau
n
c
2
levels
independent of either x 0, or y 0, or k.
This is a good example of a highly degenerate system: for each eigenvalue En, there are many similar eigenfunctions that differ only by the positions { x 0, y 0} of their centers, and the rate k of their phase change along the y-axis. They may be used to assemble a large variety of linear combinations, including 2D wave packets whose centers move along classical circular orbits. Note, however, that the radius of such rotation cannot be smaller than the so-called Landau radius,
1/ 2
1/ 2
r
,
(3.51) Landau
L
radius
mc
B
q
which characterizes the minimum size of the wave packet, and follows from Eq. (2.276) after the replacement 0 c. This radius is remarkably independent of the particle mass, and may be interpreted in the following way: the scale B A min of the applied magnetic field’s flux through the effective area A
2
min = 2 r L of the smallest wave packet is just one normal flux quantum 0 ’ 2/ q .
A detailed analysis of such wave packets (for which we would not have time in this course), in particular proves the virtually evident fact: the applied magnetic field does not change the average density dN 2/ dE of different 2D states on the energy scale, following from Eq. (1.99), but just
“assembles” them on the Landau levels (see Fig. 5a), so that the number of different orbital states on each Landau level (per unit area) is
N
1 dN
1
2
dN
d k
1
1
A
1
B
q
2
2
n
Δ
2
E
Δ E
k
(3.52)
L
B 0
2
B 0
A
A dE
A d k
dk dE / dk
A 2 2
.
2
2
k /
c
m
2
This expression may again be interpreted in terms of magnetic flux quanta: n L0 ’ = B, i.e. there is one particular state on each Landau level per each normal flux quantum.
En
B = 0
(a)
(b)
B 0
electrodes
E
c
F
Fig. 3.5. (a) The “assembly” of
2D states on Landau levels, and
(b) filling the levels with
c
electrons at the quantum Hall
effect.
0
The most famous application of the Landau levels picture is the explanation of the quantum Hall effect 21. It is usually observed in the “Hall bar” geometry sketched in Fig. 6, where electric current I is passed through a rectangular conducting sample placed into magnetic field B perpendicular to the 21 It was first observed in 1980 by a group led by Klaus von Klitzing, while the classical version (54) of the effect was first observed by Edwin Hall a century earlier – in 1879.
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sample’s plane. The classical analysis of the effect is based on the notion of the Lorentz force (14). As the magnetic field is turned on, this force starts to deviate the effective charge carriers (electrons or holes) from their straight motion between the electrodes, bending them toward the insulated sides of the bar (in Fig. 6, parallel to the x-axis). Here the carriers accumulate, generating a gradually increasing electric field E until its force (16) exactly balances the Lorentz force (14):
E
q
qv
,
(3.53)
y
x B
where vx is the drift velocity of the carriers along the bar (Fig. 6), providing the sustained balance condition E y/ vx = B at each point of the sample.
y
w
E
I
v, j
B
Fig. 3.6. The Hall bar geometry. Darker
rectangles show external (3D) electrodes.
0
x
l
With n 2 carriers per unit area, in a sample of width w, this condition yields the following classical expression for the so-called Hall resistance R H, remarkably independent of w and l: Classical
V
E w
Hall
R
y
y
B
.
(3.54)
H
effect
I
qn v w
qn
x
2 x
2
This formula is broadly used in practice for the measurement of the 2D density n 2 of the charge carriers, and of the carrier type – electrons with q = – e < 0, or holes with the effective charge q = + e > 0.
However, in experiments with high-quality (low-defect) 2D well structures, at sub-kelvin temperatures22 and high magnetic fields, the linear growth of R H with B, described by Eq. (54), is interrupted by virtually horizontal plateaus (Fig. 7). Most remarkably, the experimental values of R H on these plateaus are reproduced with extremely high accuracy (up to ~10-9) from experiment to experiment and, even more remarkably, from sample to sample.23 They are described by the following formula: Quantum
1
2
Hall
R R ,
where R
,
(3.55)
H
K
K
2
effect
i
e
so that
R 25.812 807 459
k
304... ,
(3.56)
K
and i is (only until the end of this section, following tradition!) the plateau number, i.e. a real integer.
22 In some systems, such as the graphene (virtually perfect 2D sheets of carbon atoms – see Sec. 4 below), the effect may be more stable to thermal fluctuations, due to their topological properties, so that it may be observed even at room temperature – see, e.g., K. Novoselov et al., Science 315, 1379 (2007). Also note that in some thin ferromagnetic layers, the quantum Hall effects may be observed in the absence of an external magnetic field – see, e.g., M. Götz et al., Appl. Phys. Lett. 112, 072102 (2018) and references therein.
23Due to this high accuracy (which is a rare exception in solid-state physics!), since 2018 the von Klitzing constant R K is used in metrology for the “legal” ohm’s definition, with its value (56) considered fixed – see Appendix CA: Selected Physical Constants.
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R H
Fig. 3.7. A typical record of the integer
quantum Hall effect. The lower trace (with
sharp peaks) shows the diagonal element,
Vx/ Ix, of the resistance tensor. (Adapted from
https://www.nobelprize.org/nobel_prizes/phy
sics/laureates/1998/press.html ).
B
tesla
This effect may be explained using the Landau level picture. The 2D sample is typically in a weak contact with 3D electrodes whose conductivity electrons, at low temperatures, fill all states with energies below a certain Fermi energy E F – see Fig. 5b. According to Eqs. (48) and (50), as B is increased, the spacing c between the Landau levels increases proportionately, so that fewer and fewer of these levels are below E F (and hence all their states are filled in equilibrium), and within certain ranges of field variations, the number i of the filled levels is constant. (In Fig. 5b, i = 2.) So, plugging n 2
= in L and q = – e into Eq. (54), and using Eq. (52) for n L, we get 1
1 2
R
B
,
(3.57)
H
2
i qn
i e
L
i.e. exactly the experimental result (55).
This admittedly oversimplified explanation of the quantum Hall effect does not take into account at least two important factors:
(i) the nonuniformity of the background potential U( x, y) in realistic Hall bar samples, and the role of the quasi-1D edge channels this nonuniformity produces;24 and
(ii) the Coulomb interaction of the electrons, in high-quality samples leading to the formation of R H plateaus with not only integer but also fractional values of i (1/3, 2/5, 3/7, etc.).25
Unfortunately, a thorough discussion of these very interesting features is well beyond the framework of this course.26,27
24 Such quasi-1D regions, with the width of the order of r L, form along the lines where the Landau levels cross the Fermi surface, and are actually responsible for all the electron transfer at the quantum Hall effect (giving the pioneering example of what is nowadays called the topological insulators). The particle motion along these channels is effectively one-dimensional; because of this, it cannot be affected by modest unintentional nonuniformities of the potential U( x, y). This fact is responsible for the extraordinary accuracy of Eq. (55).
25 This fractional quantum Hall effect was discovered in 1982 by D. Tsui, H. Stormer, and A. Gossard. In contrast, the effect described by Eq. (55) with an integer i (Fig. 7) is now called the integer quantum Hall effect.
26 For a comprehensive discussion of these effects, I can recommend, e.g., either the monograph by D. Yoshioka, The Quantum Hall Effect, Springer, 1998, or the review by D. Yennie, Rev. Mod. Phys. 59, 781 (1987). (See also the later publications cited above.)
27 Note also that the quantum Hall effect is sometimes discussed in terms of the so-called Berry phase, one of the geometric phases – the notion apparently pioneered by S. Pancharatnam in 1956. However, in the “usual”
Chapter 3
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QM: Quantum Mechanics
3.3. Scattering and diffraction
The second class of quantum effects, which becomes richer in multi-dimensional spaces, is typically referred to as either diffraction or scattering – depending on the context. In classical physics, these two terms are used to describe very different effects. The term “diffraction” is used for the interference of the waves re-emitted by elementary components of extended objects, under the effect of a single incident wave.28 On the other hand, the term “scattering” is used in classical mechanics to describe the result of the interaction of a beam of incident particles 29 with such an extended object, called the scatterer – see Fig. 8.
1
r
a,
k
detector
k
a
k i
scattered particles
incident
scatterer
Fig. 3.8. Scattering (schematically).
particles
Most commonly, the detector of the scattered particles is located at a large distance r >> a from the scatterer. In this case, the main observable independent of r is the flux (the number per unit time) of particles scattered in a certain direction, i.e. their flux per unit solid angle . Since it is proportional to the incident flux of particles per unit area, the efficiency of scattering in a particular direction may be characterized by the ratio of these two fluxes. This ratio has is called the differential cross-section of the scatterer:
Differential
d
of
flux
angle
solid
unit
per
particles
scatterd
cross-
.
(3.58)
section
d
of
flux
area
unit
per
particles
incident
Such terminology and notation stem from the fact that the integral of d/ d over all scattering angles, Total
d
of
flux
total
particles
scattered
cross-
Ω
section
d
,
(3.59)
Ω
d
area
unit
per
per
flux
incident
evidently having the dimensionality of area, has a simple interpretation as the total cross-section of scattering. For the simplest case when a solid object scatters all classical particles hitting its surface, but does not affect the particles flying by it, is just the geometric area of the scatterer, as observed from the direction of the incident particles. In classical mechanics, we first calculate the particle’s scattering quantum Hall effect the Berry phase equals zero, and I believe that this concept should be saved for the discussion of more topologically involved systems. Unfortunately, I will have no time/space for a discussion of such systems in this course, and have to refer the interested reader to special literature – see, e.g., either the key papers collected by A. Shapere and F. Wilczek, Geometric Phases in Physics, World Scientific, 1992, or the monograph by A.
Bohm et al., The Geometric Phase in Quantum Systems, Springer, 2003.
28 The notion of interference is very close to diffraction, but the former term is typically reserved for the wave re-emission by just a few components, such as two slits in the Young experiment – see Figs. 1 and 2. A detailed discussion of diffraction and interference of electromagnetic waves may be found in EM Secs. 8.3-8.8.
29 In the context of classical waves, the term “scattering” is typically reserved for wave interaction with disordered sets of small objects – see, e.g., EM Sec. 8.3.
Chapter 3
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QM: Quantum Mechanics
