Quantum Mechanics by Konstantin K. Likharev - HTML preview
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. (2.21)
k
x,0 e dx
exp
ikx dx,
where k
k k
1/4
1/2
2
0
2
2 (2 ) ( x
)
(2 x)
This Gaussian integral may be worked out by the following standard method, which will be used many times in this course. Let us complement the exponent to the full square of a linear combination of x and k, adding a compensating term independent of x:
2
x
~
1
~ 2
~
ikx
x 2 i ( x) k
k ( x) .
(2.22)
2
2
2
2 2
(2 x)
(2 x)
~
Since the integration in the right-hand side of Eq. (21) should be performed at constant k , in the infinite
~
limits of x, its result would not change if we replace dx with dx d[ x + 2 i( x)2 k ]. As a result, we get:6
1
1
x
a
k x
x
k
d
1/4
1/2
~
2
exp
2 2 exp
2 (2 ) ( x)
(2 x)2
(2.23)
1/ 2
~
1
1
2
k
exp
,
2
(2 )1/4 ( k)1/ 2
(2 k)2
so ak also has a Gaussian distribution, now along the k-axis, centered to the value k 0 (Fig. 1.6b), with the constant k defined as
k 1
2 x .
(2.24)
Thus we may represent the initial wave packet (16) as
5 For the notation brevity, from this point on the infinite limit signs will be dropped in all 1D integrals.
6 The fact that the argument’s shift is imaginary is not important. (Let me leave proof of this fact for the reader’s exercise.)
Chapter 2
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1 1/2
1
( k k 2
)
x 0
,
exp
.
(2.25)
1/4
1/ 2
0
eikx
2
dk
2
(2 ) ( k)
(2 k
)
From the comparison of this formula with Eq. (16), it is evident that the r.m.s. uncertainty of the wave number k in this packet is indeed equal to k defined by Eq. (24), thus justifying the notation. The comparison of the last relation with Eq. (1.35) shows that the Gaussian packet represents the ultimate case in which the product x p = x( k) has the lowest possible value (/2); for any other envelope’s shape, the uncertainty product may only be larger.
We could of course get the same result for k from Eq. (16) using the definitions (1.23), (1.33), and (1.34); the real advantage of Eq. (25) is that it can be readily generalized to t > 0. Indeed, we already know that the time evolution of the wavefunction is always given by Eq. (1.69), for our current case7
Gaussian
wave
1 1/ 2
1
( k k )2
2
0
packet:
ikx
k
x, t
.
(2.26)
1/4
1/ 2
arbitrary
exp
2
e
exp i
t dk
2
(2 ) ( k
)
(2 k
)
2 m
time
Fig. 1 shows several snapshots of the real part of the wavefunction (26), for a particular case k = 0.1 k 0.
1
v ph
x
)
t 0
t 2.2 v 0
Re
0
Fig. 2.1. Typical time evolution
of a 1D wave packet on (a)
v
1
gr
smaller and (b) larger time scales.
The dashed lines show the packet
5
0
5
10
15
20
x / x
envelopes, i.e. .
1
x
x
t 0
t 3
Re
t 20
v 0
v 0
0
1
10
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
x / x
The plots clearly show the following effects:
7 Note that Eq. (26) differs from Eq. (16) only by an exponent of a purely imaginary number, and hence this wavefunction is also properly normalized to 1 – see Eq. (3). Hence the wave packet introduction offers a natural solution to the problem of traveling de Broglie wave’s normalization, which was mentioned in Sec. 1.2.
Chapter 2
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(i) the wave packet as a whole (as characterized by its envelope) moves along the x-axis with a certain group velocity v gr,
(ii) the “carrier” quasi-sinusoidal wave inside the packet moves with a different, phase velocity v ph that may be defined as the velocity of the spatial points where the wave’s phase ( x, t) arg takes a certain fixed value (say, = /2 where Re vanishes), and
(iii) the wave packet’s spatial width gradually increases with time – the packet spreads.
All these effects are common for waves of any physical nature.8 Indeed, let us consider a 1D
wave packet of the type (26) but a more general one:
Arbitrary
x, t
a ei( kx t) dk
,
(2.27) 1D wave
k
packet
propagating in a medium with an arbitrary (but smooth!) dispersion relation ( k), and assume that the wave number distribution ak is narrow: k << k k 0 – see Fig. 1.6b. Then we may expand the function
( k) into the Taylor series near the central wave number k 0, and keep only three of its leading terms: d ~ 1 2
d ~2
~
( k)
k
k ,
where k k k , k
(2.28)
0
2
0
0
0 ,
dk
2 dk
where the derivatives have to be evaluated at the point k = k 0. In this approximation,9 the expression in the parentheses on the right-hand side of Eq. (27) may be rewritten as
kx k
~
d ~ 1 2
d
~ 2
t k x kx
k
k
t
0
0
2
dk
2
dk
(2.29)
d
d
k x t k x
t
k t
0
0
~
1 2 ~ 2 ,
dk 2
2
dk
and Eq. (27) becomes
i( k x t)
~
d 1 d 2 ~
0
0
2
~
x, t e
a exp i
k x
t
k t
k
d
.
(2.30)
k
dk 2
dk 2
First, let us neglect the last term in the square brackets (which is much smaller than the first term if the dispersion relation is smooth enough), and compare the result with the initial form of the wave packet (27):
~
~
x 0
, a eikxdk
0 , with
.
(2.31)
k
A x eik x
A x
a eikxdk
k
The comparison shows that in this approximation, Eq. (30) is reduced to
ik ( x v t
x, t
)
0
ph
(
A x v t) e
,
(2.32)
gr
where v gr and v ph are two constants with the dimension of velocity: d
v
,
v
.
(2.33a) Group
gr
ph
k k
k k
dk
and phase
0
k
0
velocities
8 See, e.g., brief discussions in CM Sec. 6.3 and EM Sec. 7.2.
9 By the way, in the particular case of de Broglie waves described by the dispersion relation (1.30), Eq. (28) is exact, because = E/ is a quadratic function of k = p/, and all higher derivatives of over k vanish for any k 0.
Chapter 2
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Clearly, Eq. (32) describes the effects (i) and (ii) listed above. For the particular case of the de Broglie waves, whose dispersion law is given by Eq. (1.30),
d
k
k
v
0
v
v ,
0
gr
v
.
(2.33b)
gr
k k
0
ph
0
k k 0
dk
m
k
2 m
2
We see that (very fortunately for the correspondence principle :-) the velocity of the wave packet’s envelope is equal to v 0 – the classical velocity of the same particle.
Next, the last term in the square brackets of Eq. (30) describes the effect (iii), the wave packet’s spread. It may be readily evaluated if the packet (27) is initially Gaussian, as in our example (25):
~2
k
a const exp
.
(2.34)
k
2 k
2
In this case, the integral (30) is Gaussian, and may be worked out exactly as the integral (21), i.e. by representing the merged exponents under the integral as a full square of a linear combination of x and k:
~2
k
~
2
i d ~
ik ( x v t)
2
k t
(2 k
)2
gr
2
2
dk
(2.35)
2
~
x v t
x v t
gr
gr 2
2
i d
( t)
2
k i
ik x
k t ,
2( t)
4( t)
0
2
2
0
dk
where I have introduced the following complex function of time:
1
i d 2
2
i d 2
t()
t ( x
)
t ,
(2.36)
(
4 k 2
)
2 dk 2
2 dk 2
~
and used Eq. (24). Now integrating over k , we get
( x v t 2
2
)
1
x, t
gr
d
2
exp
i k x
k t
.
(2.37)
0
2
0
4( t)
2 dk
The imaginary part of the ratio 1/( t) in this exponent gives just an additional contribution to the wave’s phase and does not affect the resulting probability distribution
*
( x v t)2
gr
1
w x, t exp
Re
.
(2.38)
2
( t)
This is again a Gaussian distribution over the x-axis, centered to point x = v gr t, with the variance 1
2
2
1
1 d
1
x'
2 Re
x
2
t
.
(2.39a)
2
2
( t)
2
dk
( x
)
In the particular case of de Broglie waves, d 2/ dk 2 = / m, so Wave
2
packet’s
t
1
spread
x'
2 x
2
.
(2.39b)
2
2 m ( x
)
Chapter 2
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The physics of the packet spreading is very simple: if d 2/ dk 2 0, the group velocity d/ dk of each small group dk of the monochromatic components of the wave is different, resulting in the gradual (eventually, linear) accumulation of the differences of the distances traveled by the groups. The most curious feature of Eq. (39) is that the packet width at t > 0 depends on its initial width x’(0) = x in a non-monotonic way, tending to infinity at both x→ 0 and x → ∞. Because of that, for a given time interval t, there is an optimal value of x that minimizes x’: 1/ 2
t
x'
2 x
.
(2.40)
min
opt m
This expression may be used to estimate the spreading effect’s magnitude. Due to the smallness of the Planck constant on the human scale of things, for macroscopic bodies the spreading is extremely small even for very long time intervals; however, for light particles, it may be very noticeable: for an electron ( m = m e 10-30 kg), and t = 1 s, Eq. (40) yields ( x’)min ~ 1 cm.
Note also that for any t 0, the wave packet retains its Gaussian envelope, but the ultimate relation (24) is not satisfied, x’ p > /2, due to a gradually accumulated phase shift between the component monochromatic waves.
The last remark on this topic: in quantum mechanics, the wave packet spreading is not a ubiquitous effect! For example, in Chapter 5 we will see that in a quantum oscillator, the spatial width of a Gaussian packet (for that system, called the Glauber state of the oscillator) does not grow monotonically but rather either stays constant or oscillates in time.
Now let us briefly discuss the case when the initial wave packet is not Gaussian but is described by an arbitrary initial wavefunction. To make the forthcoming result more aesthetically pleasing, it is beneficial to generalize our calculations to an arbitrary initial time t 0; it is evident that if U does not depend on time explicitly, it is sufficient to replace t with ( t – t 0) in the above formulas. With this replacement, Eq. (27) becomes
i kx ( t t )
x, t
0
a e
dk
,
(2.41)
k
and the reciprocal transform (21) reads
1
a
( , )
.
(2.42)
k
x t e ikxdx
2
0
If we want to express these two formulas with one relation, i.e. plug Eq. (42) into Eq. (41), we should give the integration variable x some other name, e.g., x 0. (Such notation is appropriate because this variable describes the coordinate argument in the initial wave packet.) The result is 1
i k x x t t
x, t
dk dx x , t e
.
(2.43)
0
0 0
0
0
2
Changing the order of integration, this expression may be represented in the following general form: 1D
x, t G x, t; x , t x , t dx
,
(2.44) propagator:
0
0
0 0 0
definition
Chapter 2
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where the function G, usually called kernel in mathematics, in quantum mechanics is called the propagator.10 Its physical sense may be understood by considering the following special initial condition:11
x , t x x' ,
(2.45)
0
0
(
)
0
where x’ is a certain point within the particle’s motion domain. In this particular case, Eq. (44) gives Ψ x, t G x, t; x' , t .
(2.46)
0
Hence, the propagator, considered as a function of its arguments x and t only, is just the wavefunction of the particle, at the -functional initial conditions (45). Thus, just as Eq. (41) may be understood as a mathematical expression of the linear superposition principle in the momentum (i.e., reciprocal) space domain, Eq. (44) is an expression of this principle in the direct space domain: the system’s “response”
( x, t) to an arbitrary initial condition ( x 0, t 0) is just a sum of its responses to elementary spatial
“slices” of this initial function, with the propagator G( x, t; x 0, t 0) representing the weight of each slice in the final sum.
According to Eqs. (43) and (44), in the case of a free particle, the propagator is equal to 1
G x, t; x , t
,
(2.47)
0
0
i k x x t t
e
0
0 dk
2
Calculating this integral, one should remember that here is not a constant but a function of k, given by the dispersion relation for the partial waves. In particular, for the de Broglie waves, with = 2 k 2/2 m, 1
k 2
G x, t; x , t
.
(2.48)
0
0
exp i k x x 0
t
( t )
0 dk
2
2 m
This is a Gaussian integral again, and it may be readily calculated just it was done (twice) above, by completing the exponent to the full square. The result is
Free
m
1/2
m( x x )2
particle’s
G x, t; x , t
.
(2.49)
0
0
exp
0
propagator
2 i
( t t )
i t t
0
2 ( )
0
Please note the following features of this complex function:
(i) It depends only on the differences ( x – x 0) and ( t – t 0). This is natural because the free-particle propagation problem is translation-invariant both in space and time.
(ii) The function’s shape (Fig. 2) does not depend on its arguments – they just rescale the same function: as a function of x, it just becomes broader and lower with time. It is curious that the spatial broadening scales as ( t – t 0)1/2 – just as at the classical diffusion, indicating a deep mathematical analogy between quantum mechanics and classical statistics – to be discussed further in Chapter 7.
10 Its standard notation by letter G stems from the fact that the propagator is essentially the spatial-temporal Green’s function of the Schrödinger equation (1), defined very similarly to Green’s functions of other ordinary and partial differential equations describing various physics systems – see, e.g., CM Sec. 5.1 and/or EM Sec. 2.7
and 7.3.
11 Note that this initial condition is mathematically not equivalent to a -functional initial probability density (3).
Chapter 2
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(iii) In accordance with the uncertainty relation, the ultimately compressed wave packet (45) has an infinite width of momentum distribution, and the quasi-sinusoidal tails of the free-particle’s propagator, clearly visible in Fig. 2, are the results of the free propagation of the fastest (highest-momentum) components of that distribution, in both directions from the packet’s center.
0.5
Re G( x, t; x , t )
0
0
Im
m / ( t t )
0 1/ 2
0
Fig. 2.2. The real (solid line)
and imaginary (dotted line)
parts of the 1D free
particle’s propagator (49).
0.5 10
0
10
( x x ) / ( t t ) / m
0
0
1/2
In the following sections, I will mostly focus on monochromatic wavefunctions (which, for unconfined motion, may be interpreted as wave packets of a very large spatial width x), and only rarely discuss wave packets. My best excuse is the linear superposition principle, i.e. our conceptual ability to restore the general solution from that of monochromatic waves of all possible energies. However, the reader should not forget that, as the above discussion has illustrated, mathematically such restoration is not always trivial.
2.3. Particle reflection and tunneling
Now, let us proceed to the cases when a 1D particle moves in various potential profiles U( x) that are constant in time. Conceptually, the simplest of such profiles is a potential step – see Fig. 3.
classically accessible classically forbidden
E
U ( x)
Fig. 2.3. Classical 1D motion in a potential
x c
profile U( x).
classical turning point
As I am sure the reader knows, in classical mechanics the particle’s kinetic energy p 2/2 m cannot be negative, so if the particle is incident on such a step (in Fig. 3, from the left), it can only move within the classically accessible region where its (conserved) full energy,
2
p
E
U ( x) ,
(2.50)
2 m
is larger than the local value U( x). Let, for example, the initial velocity v = p/ m be positive, i.e. directed toward the step. Before it has reached the classical turning point x c , defined by equality Chapter 2
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U ( x ) E ,
(2.51)
c
the particle’s kinetic energy p 2/2 m is positive, so it continues to move in the initial direction. On the other hand, a classical particle cannot penetrate that classically forbidden region x > x c, because there, its kinetic energy would be negative. Hence when the particle reaches the point x = x c, its velocity has to change its sign, i.e. the particle is reflected back from the classical turning point.
In order to see what the wave mechanics says about this situation, let us start from the simplest, sharp potential step shown with the bold black line in Fig. 4:
at
,
0
x ,
0
U ( x) U ( x)
(2.52)
0
U
0
at
,
.
x
0
For this choice, and any energy within the interval 0 < E < U 0, the classical turning point is x c = 0.
U( x), E
A
x
C
U 0
B
Fig. 2.4. Reflection of a
monochromatic de Broglie wave
from a potential step U 0 > E.
(This particular wavefunction’s
E
shape is for U 0 = 5 E.) The
wavefunction is plotted with the
0
x
same schematic vertical offset by
E as those in Fig. 1.8.
Let us represent the incident particle with a wave packet so long that the spread k ~ 1/ x of its wave-number spectrum is sufficiently small to make the energy uncertainty E = = ( d/ dk) k negligible in comparison with its average value E < U 0, as well as with ( U 0 – E). In this case, E may be considered as a given constant, the time dependence of the wavefunction is given by Eq. (1.62), and we can calculate its spatial factor ( x) from the 1D version of the stationary Schrödinger equation (1.65):12
2
d
2
U ( x
)
E .
(2.53)
2 m dx 2
At x < 0, i.e. at U = 0, the equation is reduced to the Helmholtz equation (1.78), and may be satisfied with either of two traveling waves, proportional to, respectively, exp{+ ikx} and exp{- ikx}, with k satisfying the dispersion equation (1.30):
mE
2
2
k
.
(2.54)
2
Thus the general solution of Eq. (53) in this region may be represented as
Incident
and
reflected
x
Ae ikx
Be ikx
x
.
(2.55)
,
for 0
waves
12 Note that this is not an eigenproblem like the one we have solved in Sec. 1.4 for a potential well. Indeed, now the energy E is considered given – e.g., by the initial conditions that launch a long wave packet upon the potential step – in Fig. 4, from the left side.
Chapter 2
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The second term on the right-hand side of Eq. (55) evidently describes a (formally, infinitely long) wave packet traveling to the left, arising because of the particle’s reflection from the potential step. If B = – A, Eq. (55) is reduced to Eq. (1.84) for a potential well with infinitely high walls, but for our current case of a finite step height U 0, the relation between the coefficients B and A may be different.
To show this, let us solve Eq. (53) for x > 0, where U = U 0 > E. In this region, the equation may be rewritten as
d
2
2
,
(2.56)
2
dx
where is a real and positive constant defined by a formula similar in structure to Eq. (54): m U E
2
2 (
)
0
0.
(2.57)
2
Decay in
The general solution of Eq. (56) is the sum of exp{+ x} and exp{– x}, with arbitrary pre-exponential classically coefficients. However, in our particular case the wavefunction should be finite at x +, so only the forbidden region
latter exponent is acceptable:
x
Ce x
x
.
(2.58)
,
for 0
Such penetration of the wavefunction into the classically forbidden region, and hence a non-zero probability to find the particle there, is one of the most fascinating predictions of quantum mechanics, which has been repeatedly observed in experiment – for example, via tunneling experiments, to be discussed in the next section.13 From Eq. (58), it is evident that the constant , defined by Eqs. (57), may be interpreted as the reciprocal penetration depth. Even for the lightest particles, this depth is usually very small. Indeed, for any E << U 0 that relation yields 1
.
(2.59)
E0
2 mU 0 1/2
For example, let us consider a conduction electron in a typical metal, which runs, at the metal’s surface, into a sharp potential step whose height is equal to the metal’s workfunction U 0 5 eV – see the discussion of the photoelectric effect in Sec. 1.1. In this case, according to Eq. (59), is close to 0.1
nm, i.e. is close to a typical size of an atom. For heavier elementary particles (e.g., protons) the penetration depth is correspondingly lower, and for macroscopic bodies, it is hardly measurable.
Returning to Eqs. (55) and (58), we still should relate the coefficients B and C to the amplitude A of the incident wave, using the boundary conditions at x = 0. Since E is a finite constant, and U( x) is a finite function, Eq. (53) says that d 2/ dx 2 should be finite as well. This means that the first derivative should be continuous:
2
d
d
d
2
m
lim
x
x lim
dx
lim
U( x) E dx 0
0
0
. (2.60)
2
2
0
dx
dx
dx
Repeating such calculation for the wavefunction ( x) itself, we see that it also should be continuous at all points, including the border point x = 0, so the boundary conditions in our problem are 13 Note that this effect is also pertinent to classical waves of any type, including mechanical waves (see, e.g., CM
Secs. 6.4 and 7.7) and electromagnetic waves (see, e.g., EM Secs. 7.3-7.7), but not classical particles.
Chapter 2
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d
d
(0) (0),
(0)
(0) .
(2.61)
dx
dx
Plugging Eqs. (55) and (58) into Eqs. (61), we get a system of two linear equations A B C,
ikA ikB
C
,
(2.62)
whose (easy :-) solution allows us to express B and C via A : k i
2 k
B A
,
C A
.
(2.63)
k i
k i
We immediately see that the numerator and denominator in the first of these fractions have equal moduli, so B = A . This means that, as we could expect, a particle with energy E < U 0 is totally reflected from the step – just as in classical mechanics. As a result, our solution (55) for x < 0 may be represented as a standing wave:
1
k
iAei
2
sin( kx ),
with tan
.
(2.64)
Note that the shift x / k = (tan-1 k/)/ k of the standing wave to the right, due to the partial penetration of the wavefunction under the potential step, is commensurate with, but generally not equal to the penetration depth 1/. The red line in Fig. 4 shows the exact behavior of the wavefunction, for a particular case E = U 0/5, at which k/ [ E/( U 0– E)]1/2= 1/2.
According to Eq. (59), as the particle’s energy E is increased to approach U 0, the penetration depth 1/ diverges. This raises an important question: what happens at E > U 0, i.e. if there is no classically forbidden region in the problem? In classical mechanics, the incident particle would continue to move to the right, though with a reduced velocity corresponding to the new kinetic energy E – U 0, so there would be no reflection. In quantum mechanics, however, the situation is different. To analyze it, it is not necessary to the whole problem again; it is sufficient to note that all our calculations, and hence Eqs. (63) are still valid if we take14
m E U
2
2 (
)
ik'
,
with
0
k'
0 .
(2.65)
2
With this replacement, Eq. (63) becomes15
k k'
2 k
B A
, C A
.
(2.66)
k k'
k k'
The most important result of this change is that now the particle’s reflection is not total: B <
A . To evaluate this effect quantitatively, it is fairer to use not the B/ A or C/ A ratios, but rather that of the probability currents (5) carried by the de Broglie waves traveling to the right, with amplitudes C and A, in the corresponding regions (respectively, for x > 0 and x < 0): 14 Our earlier discarding of the particular solution exp{ x}, now becoming exp{ –ik’x}, is still valid, but now on different grounds: this term would describe a wave packet incident on the potential step from the right, and this is not the problem under our current consideration.
15 These formulas are completely similar to those describing the partial reflection of classical waves from a sharp interface between two uniform media, at normal incidence (see, e.g., CM Sec. 6.4 and EM Sec. 7.4), with the effective impedance Z of de Broglie waves being proportional to their wave number k.
Chapter 2
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2
I
k' C
Potential
C
4 kk'
4 E( E U )01/2
T
.
(2.67) step’s
2
2
I
A
k A
( k k' )
1/2
E
E U
transparency
0 1/ 2
2
(The parameter T so defined is called the transparency of the system, in our current case of the potential step of height U 0, at particle’s energy E.) The result given by Eq. (67) is plotted in Fig. 5a as a function of the U 0/ E ratio. Note its most important features:
(i) At U 0 = 0, the transparency is full, T = 1 – naturally, because there is no step at all.
(ii) At U 0 E, the transparency drops to zero, giving a proper connection to the case E < U 0.
(iii) Nothing in our solution’s procedure prevents us from using Eq. (67) even for U 0 < 0, i.e. for the step-down (or “cliff”) potential profile – see Fig. 5b. Very counter-intuitively, the particle is (partly) reflected even from such a cliff, and the transmission diminishes (though rather slowly) at U 0 –.
(a)
(b)
A
C
1
E 0
B
0.8
0
U 0
0.6
T
U
0.4
0
Fig. 2.5. (a) The transparency of a potential step with U
0.2
0
< E as a function of its height, according to Eq. (67), and
0
(b) the “cliff” potential profile, with U 0 < 0.
1
0
1
U / E
0
The most important conceptual conclusion of this analysis is that the quantum particle is partly reflected from a potential step with U 0 < E, in the sense that there is a non-zero probability T < 1 to find it passed over the step, while there is also some probability, (1 – T) > 0, to have it reflected.
The last property is exhibited, but for any relation between E and U 0, by another simple potential profile U( x), the famous potential (or “tunnel”) barrier. Fig. 6 shows its simple flat-top (“rectangular”) version:
,
0
for
x d /
,
2
U ( x) U
,
for d / 2 x d / ,
2
(2.68)
0
,0
for
d / 2 x
.
To analyze this problem, it is sufficient to look for the solution to the Schrödinger equation in the form (55) at x – d/2. At x > + d/2, i.e., behind the barrier, we may use the arguments presented above (no wave source on the right!) to keep just one traveling wave, now with the same wave number: ikx
( x) Fe .
(2.69)
However, under the barrier, i.e. at – d/2 x +d/2, we should generally keep both exponential terms,
x
x
( x) Ce
De ,
(2.70)
b
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because our previous argument used in the potential step problem’s solution is no longer valid. (Here k and are still defined, respectively, by Eqs. (54) and (57).) In order to express the coefficients B, C, D, and F via the amplitude A of the incident wave, we need to plug these solutions into the boundary conditions similar to Eqs. (61), but now at two boundary points, x = d/2.
U U 0
A
C
F
E
B
D
Fig. 2.6. A rectangular potential
U 0
barrier, and the de Broglie waves
taken into account in its analysis.
x
d / 2
d / 2
Solving the resulting system of 4 linear equations, we get 4 ratios B/ A, C/ A, etc.; in particular,
F
i
k
1 ikd
cosh d
sinh d
e
,
(2.71a)
A
2 k
and hence the barrier’s transparency
1
1
Rectangular
2
2
2
2
2
2
2
tunnel
F
k
k
barrier’s
2
2
2
T
cosh d
sinh
d
1
sinh
d
.
(2.71b)
transparency
A
2
k
2
k
So, quantum mechanics indeed allows particles with energies E < U 0 to pass “through” the potential barrier – see Fig. 6 again. This is the famous effect of quantum-mechanical tunneling. Fig. 7a shows the barrier transparency as a function of the particle energy E, for several characteristic values of its thickness d, or rather of the ratio d/, with defined by Eq. (59).16
(a)
(b)
0.01
d
d / 3.0
0.8
0 3
.
1 1
0 6
2
1 1
0 10
10
0.6
1 1
0 14
T
1.0
T
0.4
1 1
0 18
30
1 1
0 22
0.2
3.0
1 1
0 26
1 1
0 30
0
1
2
3
0
0.2
0.4
0.6
0.8
E / U
1 E / U 0 1/2
0
Fig. 2.7. The transparency of a rectangular potential barrier as a function of the particle’s energy E.
16 The branches with E > U 0 have been computed from Eq. (71b) by using the replacement (65).
Chapter 2
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The plots show that generally, the transparency grows with the particle’s energy. This growth is natural because the penetration constant decreases with the growth of E, i.e., the wavefunction penetrates more and more into the barrier, so more and more of it is “picked up” at the second interface ( x = +d/2) and transferred into the wave F exp{ ikx} propagating behind the barrier.
Now let us consider the important limit of a very thin and high rectangular barrier with d << , E
<< U 0, giving k << << 1/ d. In this limit, the second form of Eq. (71b) yields 2
1
k 2
2
d
m
T
,
where
d
U d ,
(2.72)
2
1
2 k
2 k
2
k 0
The last product, U 0 d, is just the weight (or the “energy area”) W U( x) dx
(2.73)
U ( x) E
of the barrier for our particular case (68). This fact implies that the very simple result (72) may be correct for a barrier of any shape, provided that it is sufficiently thin and high.
To examine this guess, let us consider the tunneling problem for a very thin barrier with d, kd
<< 1, by approximating it with the Dirac’s -function (Fig. 8):
U ( x) W ( x) ,
(2.74)
so that the parameter W satisfies Eq. (73).
U ( x) W ( x)
A
F
E
B
Fig. 2.8. A delta-functional potential
x
barrier.
0
The solutions of the tunneling problem at all points but x = 0 still may be taken in the form of Eqs. (55) and (69), so we only need to analyze the boundary conditions at that point. However, due to the special character of the -function, we should be careful here. Indeed, instead of Eq. (60) we now get
2
d
d
d
2
m
lim
lim
dx lim
x
x
U( x) E
0
0
dx
2
0 2
dx
dx
dx
(2.75)
2 m
W ( ).
0
2
According to this relation, at a finite W, the derivatives d/ dx are also finite, so the wavefunction itself is still continuous:
d
lim
x x lim
dx .
0
0
0
(2.76)
dx
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Using these two boundary conditions, we readily get the following system of two linear equations, 2 m
A B F,
ikF ikA
(
ikB)
W
F ,
(2.77)
2
whose solution yields
B
i
F
1
m
,
,
where
W
.
(2.78)
A 1 i
A
1
2
i
k
(Taking Eq. (73) into account, this definition of coincides with that in Eq. (72).) For the barrier transparency T F/ A2, this result again gives the first of Eqs. (72), which is therefore general for such thin barriers. That formula may be recast to give the following simple expression (valid only for E << U max):
2
1
E
Transparency:
T
,
where
W
m
E
,
(2.79)
2
0
2
thin barrier
1
E E
2
0
which shows that as energy becomes larger than the constant E 0, the transparency approaches 1 – as it eventually does for any tunnel barrier. Another general behavior of Eq. (79) is that at E 0 (i.e. k 0), T E. Indeed, at ka << 1, any barrier of a finite width a may be well approximated with Eq. (74).
As Eq. (71) shows, in the opposite limit of relatively thick barriers ( d >> ), the transparency is dominated by what is called the tunnel exponent:
Transparency:
4 k
2
2 d
thick barrier
T
e
(2.80)
k 2 2
– the behavior which may be clearly seen as the straight-line segments in semi-log plots (Fig. 7b) of T
as a function of the combination (1 – E/ U 0)1/2 which is proportional to – see Eq. (57).
The exponential dependence of the barrier’s transparency on its thickness is the most important factor for various applications of quantum-mechanical tunneling – from the field emission of electrons to vacuum17 to the scanning tunneling microscopy.18 Note also substantial negative implications of the effect on the electronic technology progress. Most importantly, it imposes limits on the so-called Dennard scaling of field-effect transistors in semiconductor integrated circuits (which was the technological basis of the well-known Moore’s law), due to the increase of tunneling both through the gate oxide and along the channel of the transistors, from source to drain.19
Finally, one more feature visible in Fig. 7a (for case d = 3) are the oscillations of the transparency as a function of energy, at E > U 0, with T = 1, i.e. the reflection completely vanishing, at some points.20 This is our first glimpse at one more interesting quantum effect: resonant tunneling. This 17 See, e.g., G. Fursey , Field Emission in Vacuum Microelectronics, Kluwer, New York, 2005.
18 See, e.g., G. Binning and H. Rohrer, Helv. Phys. Acta 55, 726 (1982).
19 See, e.g., V. Sverdlov et al., IEEE Trans. on Electron Devices 50, 1926 (2003), and references therein. (A brief discussion of the field-effect transistors, and literature for further reading, may be found in SM Sec. 6.4.) 20 Let me mention in passing the curious case of the potential well U( x) = –(2/2 m)( + 1)/cosh2( x/ a), with any positive integer and any real a, which is reflection-free (T = 1) for the incident de Broglie wave of any energy E, and hence for any incident wave packet. (The well is called the Pöschl-Teller potential, though it was first described in a 1930 paper by P. Epstein, before the 1933 publication by G. Pöschl and E. Teller.) Chapter 2
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effect will be discussed in more detail in Sec. 5 below, by using another potential profile where it is more clearly pronounced.
2.4. Motion in soft potentials21
Before moving on to explore other quantum-mechanical effects, let us examine how the results discussed in the previous section are modified in the opposite limit of the so-called soft (or “smooth”) potential profiles, like the one sketched in Fig. 3.22 The most efficient analytical tool to study this limit is the so-called WKB (or “JWKB”, or “quasiclassical”) approximation developed by H. Jeffrey, G.
Wentzel, A. Kramers, and L. Brillouin in 1925-27. In order to derive its 1D version, let us rewrite the Schrödinger equation (53) in a simpler form
2
d
2
k ( x) 0 ,
(2.81)
2
dx
where the local wave number k( x) is defined similarly to Eq. (65),
Local
2
2
m E U ( x)
k ( x)
,
(2.82) wave
2
number
besides that now it may be a function of x. We already know that for k( x) = const, the fundamental solutions of this equation are A exp{+ ikx} and B exp{– ikx}, which may be represented in a single form i( x)
( x) e
,
(2.83)
where ( x) is a complex function, in these two simplest cases being equal, respectively, to ( kx – i ln A) and (– kx – i ln B) . This is why we may try to use Eq. (83) to look for a solution of Eq. (81) even in the general case when k( x) const. Differentiating Eq. (83) twice, we get 2
2
2
d
d
d
i
d
d
i
i
e ,
i
e .
(2.84)
2
2
dx
dx
dx
dx
dx
Plugging the last expression into Eq. (81) and canceling the common factor exp{ i( x)} 0, we get 2
2
d
d
2
i
k ( x) 0 .
(2.85)
2
dx
dx
This is still an exact, general equation. Superficially, it looks even harder to solve than the initial equation (81) because Eq. (85) is nonlinear. However, it is ready for simplification in the limit when the potential profile is soft, dU/ dx 0. Indeed, for a uniform potential, d 2/ dx 2 = 0. Hence, in the so-called 0th approximation, ( x) 0( x), we may try to keep that equality, so Eq. (85) is reduced to
d
2
0
2
x
d
k ( x),
i.e.
0 k( x),
,
(2.86)
0 x i k( x' ) dx'
dx
dx
so its general solution is a linear superposition of the two functions (83), with replaced with 0: 21 Following tradition, I will frequently use this shorthand for “potential energy”, returning to the full term in cases where there is any chance of confusion between this notion and another (say, electrostatic) potential.
22 Quantitative conditions of the “softness” will be formulated later in this section.
Chapter 2
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x
x
( x) A exp
,
(2.87)
0
i k( x' ) dx' B exp i k( x' ) dx'
where the choice of the lower limits of integration affects only the constants A and B. The physical sense of this result is simple: it is a sum of the forward- and back-propagating de Broglie waves, with the coordinate-dependent local wave number k( x) adjusted to the potential profile.
Let me emphasize the non-trivial nature of this approximation.23 First, any attempt to address the problem with the standard perturbation approach (say, = 0 + 1 +…, with n proportional to the n th power of some small parameter) would fail for most potentials, because as Eq. (86) shows, even a slight but persisting deviation of U( x) from a constant leads to a gradual accumulation of the phase 0, impossible to describe by any small perturbation of . Second, the dropping of the term d 2/ dx 2 in Eq.
(85) is not too easy to justify. Indeed, since we are committed to the “soft potential limit” dU/ dx 0, we should be ready to assume the characteristic length a of the spatial variation of to be large, and neglect the terms that are the smallest ones in the limit a . However, both first terms in Eq. (85) are apparently of the same order in a, namely O( a-2); why have we neglected just one of them?
The price we have paid for such a “sloppy” treatment is substantial: Eq. (87) does not satisfy the fundamental property of solutions of the stationary Schrödinger equation, namely the probability current’s conservation: I( x) = const. However, this is not true for any component of Eq. (87); for example for the first, forward-propagating component on its right-hand side, Eq. (5) yields I ( x)
2
A k( x) ,
(2.88)
0
m
evidently not a constant if k( x) const. The brilliance of the WKB theory is that the problem may be fixed without a full revision of the 0th approximation, just by amending it. Indeed, let us explore the next, 1st approximation:
x
( x) ( x) ( x) ,
(2.89)
WKB
0
1
where 0 still obeys Eq. (86), while 1 describes a 0th approximation’s correction that is small in the following sense:24
d
d
1
0
k( x) .
(2.90)
dx
dx
Plugging Eq. (89) into Eq. (85), with the account of the definition (86), we get 2
2
d
d d d
d
0
1
1
i
2
0
1
0
.
(2.91)
2
2
dx
dx
dx dx
dx
Using the condition (90), we may neglect d 21/ dx 2 in comparison with d 20/ dx 2 inside the first parentheses, and d1/ dx in comparison with 2 d0/ dx inside the second parentheses. As a result, we get the following (still approximate!) result:
23 Philosophically, this space-domain method is very close to the time-domain van der Pol method in classical mechanics, and the very similar rotating wave approximation (RWA) in quantum mechanics – see, e.g., CM Secs.
5.2-5.5, and also Secs. 6.5, 7.6, 9.2, and 9.4 of this course.
24 For certainty, I will use the discretion given by Eq. (82) to define k( x) as the positive root of its right-hand side.
Chapter 2
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Φ
2
d
i d
d
i d d
i d
d
1
0
0
ln
0
ln k( x) i
k
x ,
(2.92)
2
ln 1/2( )
dx
2 dx
dx
2 dx
dx 2 dx
dx
x
1
i Φ
i Φ i Φ i k( x' ) dx' ln
,
WKB
0
1
(2.93)
1/ 2
k
( x)
a
x
b
x
WKB
( x)
exp i k( x' )
dx'
exp i k( x' ) dx' .
for 2
k x (2.94)
WKB
.0
wave-
1/ 2
k
( x)
1/ 2
k ( x)
function
(Again, the lower integration limit is arbitrary, because its choice may be incorporated into the complex constants a and b.) This modified approximation overcomes the problem of current continuity; for example, for the forward-propagating wave, Eq. (5) gives
WKB
I
( x)
2
a const .
(2.95) probability
WKB
m
current
Physically, the factor k 1/2 in the denominator of the WKB wavefunction’s pre-exponent may be easily understood. The smaller the local group velocity (32) of the wave packet, v gr( x) = k( x)/ m, the
“easier” (more probable) it should be to find the particle within a certain interval dx. This is exactly the result that the WKB approximation gives: w( x) = * 1/ k( x) 1/ v gr. Another value of the 1st approximation is a clarification of the WKB theory’s validity condition: it is given by Eq. (90). Plugging into this relation the first form of Eq. (92), and estimating d 20/ dx 2 as d0/ dx/ a, where a is the spatial scale of a substantial change of d0/ dx = k( x), we may write the condition as WKB:
first
ka 1.
(2.96) condition
of validity
In plain English, this means that the region where U( x), and hence k( x), change substantially should contain many de Broglie wavelengths = 2/ k.
So far I have implied that k 2( x) E – U( x) is positive, i.e. particle moves in the classically accessible region. Now let us extend the WKB approximation to situations where the difference E –
U( x) may change sign, for example to the reflection problem sketched in Fig. 3. Just as we did for the sharp potential step, we first need to find the appropriate solution in the classically forbidden region, in this case for x > x c. For that, there is again no need to redo our calculations, because they are still valid if we, just as in the sharp-step problem, take k( x) = i( x), where 2 m U ( x) E
2
x
,
0
for x x ,
(2.97)
2
c
and keep just one of two possible solutions (with > 0), in analogy with Eq. (58). The result is c
x
( x)
exp ( x' ) dx' ,
for 2
k 0 ,
i.e.
2
κ ,
0
(2.98)
WKB
1/ 2
( x)
with the lower limit at some point with 2 > 0 as well. This is a really wonderful formula! It describes the quantum-mechanical penetration of the particle into the classically forbidden region and provides a natural generalization of Eq. (58) – leaving intact our estimates of the depth ~ 1/ of such penetration.
Chapter 2
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Now we have to do what had been done for the sharp-step problem in Sec. 2: use the boundary conditions at the classical turning point x = x c to relate the constants a, b, and c. However, now this operation is a tad more complex, because both WKB functions (94) and (98) diverge, albeit weakly, at the point, because here both k( x) and ( x) tend to zero. This connection problem may be solved in the following way. 25
Let us use our commitment to the potential’s “softness”, assuming that it allows us to keep just two leading terms in the Taylor expansion of the function U( x) at the point x c: dU
dU
U ( x) U ( x )
( x x ) E
( x x ) .
(2.99)
c
x x
c
x x
c
c
c
dx
dx
Using this truncated expansion, and introducing the following dimensionless variable for the coordinate’s deviation from the classical turning point,
1/ 3
2
x x
c
,
with x
(2.100)
0
x
2 m dU dx
0
,
/
x x
c
we reduce the Schrödinger equation (81) to the so-called Airy equation
2
Airy
d
equation
0 .
(2.101)
2
d
This simple linear, ordinary, homogenous differential equation of the second order has been very well studied. Its general solution may be represented as a linear combination of two fundamental solutions, the Airy functions, Ai( ) and Bi( ), shown in Fig. 9a.26
(a)
(b)
1
1
Bi( )
Ai
( )
WKB
Ai( )
Ai( )
0
0
)
1
1
10
0
10
3
0
3
Fig. 2.9. (a) The Airy functions Ai and Bi, and (b) the WKB approximation for the function Ai().
25 An alternative way to solve the connection problem, without involving the Airy functions but using an analytical extension of WKB formulas to the complex-argument plane, may be found, e.g., in Sec. 47 of the textbook by L. Landau and E. Lifshitz, Quantum Mechanics, Non-Relativistic Theory, 3rd ed. Pergamon, 1977.
26 Note the following (exact) integral formulas,
3
1
3
3
Ai( 1
)
cos d , Bi( )
exp
sin
d ,
3
3
3
0
0
frequently more convenient for practical calculations of the Airy functions than the differential equation (101).
Chapter 2
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The latter function diverges at +, and thus is not suitable for our current problem (Fig. 3), while the former function has the following asymptotic behaviors at >> 1:
1
2 3/2
exp
,
for
,
1
Ai( )
2
3
(2.102)
1/ 4
1/ 2
2
sin 3/2
, for .
3
4
Now let us apply the WKB approximation to the Airy equation (101). Taking the classical turning point ( = 0) for the lower limit, for > 0 we get
2
1/ 2
2
( ) ,
( ) ,
( ' )
3 / 2
d '
,
(2.103)
3
0
i.e. exactly the exponent in the top line of Eq. (102). Making a similar calculation for < 0, with the natural assumption b = a (full reflection from the potential step), we arrive at the following result:
2 3/2
c' exp
, for ,
0
1
Ai
(2.104)
WKB
3
1/ 4
2
a' sin 3/2
, for 0
.
3
This approximation differs from the exact solution at small values of , i.e. close to the classical turning point – see Fig. 9b. However, at >> 1, Eqs. (104) describe the Airy function exactly, provided that
a'
WKB:
,
c'
.
(2.105) connection
4
2
formulas
These connection formulas may be used to rewrite Eq. (104) as
2 3/2
exp
,
for ,
0
a'
3
Ai
(2.106)
WKB
1/ 4
2
1
2
3 / 2
2 3/2
exp i
i - exp i
i , for
,
0
i
3
4
3
4
and hence may be described by the following two simple mnemonic rules:
(i) If the classical turning point is taken for the lower limit in the WKB integrals in the classically allowed and the classically forbidden regions, then the moduli of the quasi-amplitudes of the exponents are equal.
(ii) Reflecting from a “soft” potential step, the wavefunction acquires an additional phase shift
= /2, if compared with its reflection from a “hard”, infinitely high potential wall located at point x c (for which, according to Eq. (63) with = , we have B = – A).
In order for the connection formulas (105)-(106) to be valid, deviations from the linear approximation (99) of the potential profile should be relatively small within the region where the WKB
approximation differs from the exact Airy function: ~ 1, i.e. x – x c ~ x 0. These deviations may be estimated using the next term of the Taylor expansion, dropped in Eq. (99): ( d 2 U/ d 2 x)( x – x c)2/2. As a result, the condition of validity of the connection formulas (i.e. of the “softness” of the reflecting Chapter 2
Page 22 of 76
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potential profile) may be expressed as d 2 U/ d 2 x x 0 << dU/ dx at x x c – meaning the ~ x 0–wide vicinity of the point x c). With the account of Eq. (100) for x 0, this condition becomes 3
4
2
Connection
d U
2 m dU
formulas’
.
(2.107)
2
2
validity
dx x x
dx x x
c
c
As an example of a very useful application of the WKB approximation, let us use the connection formulas to calculate the energy spectrum of a 1D particle in a soft 1D potential well (Fig. 10).
U( x)
En
Fig. 2.10. The WKB treatment of an eigenstate
of a particle in a soft 1D potential well.
x
0
x
x
L
R
As was discussed in Sec. 1.7, we may consider the standing wave describing an eigenfunction n (corresponding to an eigenenergy En) as a sum of two traveling de Broglie waves going back and forth between the walls, being sequentially reflected from each of them. Let us apply the WKB approximation to such traveling waves. First, according to Eq. (94), propagating from the left classical turning point x L
to the right such point x R, it acquires the phase change
x R
k( x) dx
.
(2.108)
x L
At the reflection from the soft wall at x R, according to the mnemonic rule (ii), the wave acquires an additional shift /2. Now, traveling back from x R to x L, the wave gets a shift similar to one given by Eq.
(108): = . Finally, at the reflection from x L, it gets one more /2-shift. Summing up all these contributions at the wave’s roundtrip, we may write the self-consistency condition (that the wavefunction “catches its own tail with its teeth”) in the form
x R
2 k( x) dx 2 n
, with n ,
1 ,...
2
(2.109)
total
2
2
x L
Rewriting this result in terms of the particle’s momentum p( x) = k( x), we arrive at the so-called Wilson-Sommerfeld (or, less fairly, “Bohr-Sommerfeld”) quantization rule
Wilson-
1
Sommerfeld
p( x) dx 2
n
quantization
,
(2.110)
C
2
rule
where the closed path C means the full period of classical motion.27
27 Note that at the motion in more than one dimension, a closed classical trajectory may have no classical turning points. In this case, the constant ½, arising from the turns, should be dropped from Eqs. (110) written for the scalar product p(r) dr, giving the so-called Bohr quantization rule. It was suggested by N. Bohr as early as 1913
as an interpretation of Eq. (1.8) for the circular motion of the electron around the proton, while its 1D
modification (110) is due to W. Wilson (1915) and A. Sommerfeld (1916).
Chapter 2
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Let us see what this quantization rule gives for the very important particular case of a quadratic potential profile of a harmonic oscillator of frequency 0. In this case,
m
2
2
U ( x)
x ,
(2.111)
0
2
and the classical turning points (where U( x) = E) are the roots of a simple equation m
En
2
2
1 2
1/ 2
x E ,
that
so
x
.
(2.112)
n
,
0
x x 0
2 0 c
R
L
R
m
0
Due to the potential’s symmetry, the integration required by Eq. (110) is also simple: x
x
x
1/ 2
R
R
R
2
x
p( x) dx 2
m E U ( x)
dx mE
dx
n
1/2
2 n 1/22 1
2
x
x
x
0
R
L
L
(2.113)
1
E
2 mE
x
d mE
x
n 1/ 2 2
1 2
n
R
1/2
2 n 1/22
,
R 4
0
0
so Eq. (110) yields
1
E
(2.114)
n
n' ,
with n' n 1
.
...
2,
1,
,
0
0
2
To estimate the validity of this result, we have to check the condition (96) at all points of the classically allowed region, and Eq. (107) at the turning points. The checkup shows that both conditions are valid only for n >> 1. However, we will see in Sec. 9 below that Eq. (114) is actually exactly correct for all energy levels – thanks to the special properties of the potential profile (111).
Now let us use the mnemonic rule (i) to examine the particle’s penetration into the classically forbidden region of an abrupt potential step of a height U 0 > E. For this case, the rule, i.e. the second of Eqs. (105), yields the following relation of the quasi-amplitudes in Eqs. (94) and (98): c = a /2. If we now naively applied this relation to the sharp step sketched in Fig. 4, forgetting that it does not satisfy Eq. (107), we would get the following relation of the full amplitudes defined by Eqs. (55) and (58): C
1 A
.
(WRONG!)
(2.115)
2 k
This result differs from the correct Eq. (63), and hence we may expect that the WKB approximation’s prediction for more complex potentials, most importantly for tunneling through a soft potential barrier (Fig. 11) should be also different from the exact result (71) for the rectangular barrier shown in Fig. 6.
U ( x)
d
x ' x
WKB
c
c
U max
a
c
f
E
b
d
Fig. 2.11. Tunneling through
a soft 1D potential barrier.
0
x
x
x '
x
c
m
c
Chapter 2
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To analyze tunneling through such a soft barrier, we need (just as in the case of a rectangular barrier) to take into consideration five partial waves, but now they should be taken in the WKB form:
x
x
a
b
exp i k x' dx'
i k x' dx'
x
x
1/ 2
( )
exp
1/ 2
( ) , for
,
k ( x)
k ( x)
c
c
x
d
x
x' dx'
x' dx'
x
x
x '
(2.116)
WKB
exp
1/ 2
( )
exp
1/ 2
( ) , for ,
c
c
( x)
( x)
x
f
exp i k x' dx'
x '
x
1/ 2
( ) ,
for
,
c
k ( x)
where the lower limits of integrals are arbitrary (each within the corresponding range of x). Since on the right of the left classical point, we have two exponents rather than one, and on the right of the second point, one traveling wave rather than two, the connection formulas (105) have to be generalized by using asymptotic formulas not only for Ai( ), but also for the second Airy function, Bi( ). The analysis, similar to that carried out above (though naturally a bit bulkier),28 gives a remarkably simple result: Soft
2
x '
x '
potential
f
c
c
2
1/ 2
barrier:
T
exp
2 ( )
exp
2
( )
,
(2.117)
WKB
x dx
m U x E
dx
transparency
a
x
x
c
c
with the pre-exponential factor equal to 1 – the fact that might be readily expected from the mnemonic rule (i) of the connection formulas.
This formula is broadly used in applied quantum mechanics, despite the approximate character of its pre-exponential coefficient for insufficiently soft barriers that do not satisfy Eq. (107). For example, Eq. (80) shows that for a rectangular barrier with thickness d >> , the WKB approximation (117) with d WKB = d underestimates T by a factor of [4 k/( k 2 + 2)]2 – equal, for example, 4, if k = , i.e.
if U 0 = 2 E. However, on the appropriate logarithmic scale (see Fig. 7b), such a factor, smaller than an order of magnitude, is just a small correction.
Note also that when E approaches the barrier’s top U max (Fig. 11), the points x c and x c ’ merge, so according to Eq. (117), TWKB 1, i.e. the particle reflection vanishes at E = U max. So, the WKB
approximation does not describe the effect of the over-barrier reflection at E > U max. (This fact could be noticed already from Eq. (95): in the absence of the classical turning points, the WKB probability current is constant for any barrier profile.) This conclusion is incorrect even for apparently smooth barriers where one could naively expect the WKB approximation to work perfectly. Indeed, near the point x = x m where the potential reaches maximum (i.e. U( x m) = U max), we may approximate almost any smooth function U( x) with the quadratic term of the Taylor expansion, i.e. with an inverted parabola: 2
m x x
0
m 2
U ( x) U
.
(2.118)
max
2
Calculating the derivatives dU/dx and d 2 U/dx 2 for this function and plugging the results into the condition (107), we may see that the WKB approximation is only valid if U max – E >> 0. Just for the 28 For the most important case TWKB << 1, Eq. (117) may be simply derived from Eqs. (105)-(106) – the exercise left for the reader.
Chapter 2
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reader’s reference, an exact analysis of tunneling through the barrier (118) gives the following Kemble formula:29
1
Kemble
T
,
(2.119)
1
exp 2 ( E U
) /
formula
max
0
valid for any sign of the difference ( E – U max). This formula describes a gradual approach of T to 1, i.e.
a gradual reduction of reflection, at the particle energy’s increase, with T = ½ at E = U max.
To conclude this section, one last remark: the WKB approximation opens a straight way toward an alternative formulation of quantum mechanics, based on the Feynman path integral. However, I will postpone its discussion until a more compact notation has been introduced in Chapter 4.
2.5. Resonant tunneling, and metastable states
Now let us move to other, conceptually different quantum effects taking place in more elaborate potential profiles. Neither piecewise-constant nor smooth-potential models of U( x) are convenient for their quantitative description because they both require “stitching” partial de Broglie waves at each classical turning point, which may lead to cumbersome calculations. However, we may get a very good insight into the physics of quantum effects that may take place in such profiles, while avoiding overcomplicated math, by approximating them with sets of Dirac’s delta functions.
Big help in this way is provided by the notions of scattering and transfer matrices, very useful for other purposes as well. Let us consider an arbitrary but finite-length potential “bump” (formally called a scatterer), localized somewhere between points x 1 and x 2, on a flat potential background, say U
= 0 (Fig. 12).
U ( x)
A
A
1
2
E
B
B
1
2
Fig. 2.12. De Broglie wave amplitudes
near a single 1D scatterer.
x
x
x
1
0
2
From Sec. 2, we know that the general solutions of the stationary Schrödinger equation, with a certain energy E, in the regions outside the interval [ x 1, x 2], are sets of two sinusoidal waves, traveling in opposite directions. Let us represent them in the form
ik( x x )
ik( x x )
j
j
A e
B e
,
(2.120)
j
j
j
where the index j equals either 1 or 2 (for now), and ( k)2/2 m = E. Note that each of the two wave pairs (129) has, in this notation, its own reference point xj, because this will be very convenient for what 29 This formula was derived (in a more general form, valid for an arbitrary soft potential barrier) by E. Kemble in 1935. In some communities, it is known as the “Hill-Wheeler formula”, after D. Hill and J. Wheeler’s 1953 paper in that the Kemble formula was spelled out for the quadratic profile (118). Note that mathematically Eq. (119) is similar to the Fermi distribution in statistical physics, with an effective temperature T ef = 0/2 k B. This coincidence has some curious implications for the Fermi particle tunneling statistics.
Chapter 2
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follows. As we have already discussed, if the de Broglie wave/particle is incident from the left (i.e. B 2 =
0), the solution of the linear Schrödinger equation within the scatterer range ( x 1 < x < x 2) can provide only linear expressions of the transmitted ( A 2) and reflected ( B 1) wave amplitudes via the incident wave amplitude A 1:
A S A , B S A ,
(2.121)
2
21 1
1
11 1
where S 11 and S 21 are certain (generally, complex) coefficients. Alternatively, if a wave with amplitude B 2 is incident on the scatterer from the right (i.e. if A 1 = 0), it can induce a transmitted wave ( B 1) and a reflected wave ( A 2), with amplitudes
B S B , A S B ,
(2.122)
1
12
2
2
22
2
where the coefficients S 22 and S 12 are generally different from S 11 and S 21. Now we can use the linear superposition principle to argue that if the waves with amplitudes A 1 and B 2 are simultaneously incident on the scatterer (say, because the wave B 2 has been partly reflected back by some other scatterer located at x > x 2), the resulting scattered wave amplitudes A 2 and B 1 are just the sums of their values for separate incident waves:
B S A S B ,
1
11 1
12
2
(2.123)
A S A S B .
2
21 1
22
2
These two relations may be conveniently represented using the so-called scattering matrix S: Scattering
B
A
S
S
1
1
11
12
matrix:
S
,
S
with
definition
.
(2.124)
A
B
S
S
2
2
21
22
Scattering matrices, duly generalized, are an important tool for the analysis of wave scattering in higher dimensions than one; for 1D problems, however, another matrix is often more convenient to represent the same linear relations (123). Indeed, let us solve this system for A 2 and B 2. The result is Transfer
A T A T B ,
A
A
matrix:
2
11 1
12
1 i.e.
2
T 1 ,
(2.125)
definition
B T A T B ,
B
B
2
21 1
22
1
2
1
where T is the transfer matrix, with the following elements:
S S
S
S
1
11 22
T S
,
22
T
,
11
T
, T
.
(2.126)
11
21
12
21
22
S
S
S
S
12
12
21
12
The matrices S and T have some universal properties, valid for an arbitrary (but time-independent) scatterer; they may be readily found from the probability current conservation and the time-reversal symmetry of the Schrödinger equation. Let me leave finding these relations for the reader’s exercise. The results show, in particular, that the scattering matrix may be rewritten in the following form:
i
re
t
S i
e
,
(2.127a)
i
t
re
where the four real parameters r, t, , and satisfy the following universal relation: 2
2
r t 1,
(2.127b)
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so only three of these parameters are independent. As a result of this symmetry, T 11 may be also represented in a simpler form similar to T
*
*
22: T 11 = exp{ i}/ t = 1/ S 12 = 1/ S 21 . The last form allows a ready expression of the scatterer’s transparency via just one coefficient of the transfer matrix: 2
A
2
2
2
T
S
T
.
(2.128)
21
11
A 1 B 0
2
In our current context, the most important property of the 1D transfer matrices is that to find the total transfer matrix T of a system consisting of several (say, N) sequential arbitrary scatterers (Fig. 13), it is sufficient to multiply their matrices.
A
A
A
A
1
2
3
N 1
B
B
B
B
1
2
3
N 1
Fig. 2.13. A sequence of several 1D
scatterers.
x
x
x
x
x
1
2
3
N 1
Indeed, extending the definition (125) to other points xj ( j = 1, 2, …, N + 1), we can write
A
A
A
A
A
2
T
1 ,
3
T
2
T T
1 , etc.
(2.129)
1
2
2 1
B
B
B
B
B
2
1
3
2
1
(where the matrix indices correspond to the scatterers’ order on the x-axis), so
A
A
N
1
T T
T
...
1 .
(2.130)
N
N 1
1
B
B
N 1
1
But we can also define a total transfer matrix similarly to Eq. (125), i.e. as
AN
A
1
T 1 ,
(2.131)
B
B
N 1
1
so comparing Eqs. (130) and (131) we get
Transfer
T T T
T
... .
(2.132) matrix:
N
N 1
1
composite
scatterer
This formula is valid even if the flat-potential gaps between component scatterers are shrunk to zero, so it may be applied to a scatterer with an arbitrary profile U( x), by fragmenting its length into many small segments x = xj+1 – xj, and treating each fragment as a rectangular barrier of the average height ( Uj)ef = [ U( xj+1) – U( xj)]/2 – see Fig. 14. Since very efficient numerical algorithms are readily available for fast multiplication of matrices (especially as small as 2×2 in our case), this approach is broadly used in practice for the computation of transparency of 1D potential barriers with complicated profiles U( x). (This procedure is much more efficient than the direct numerical solution of the stationary Schrödinger equation.)
Chapter 2
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U ( x)
A
( U )
j ef
A
1
N 1
B
B
1
N 1
Fig. 2.14. The transfer matrix approach
to a potential barrier with an arbitrary
profile.
x
x
x x
2
x
x
1
j
j 1
N 1
In order to apply this approach to several particular, conceptually important systems, let us calculate the transfer matrices for a few elementary scatterers, starting from the delta-functional barrier located at x = 0 – see Fig. 8. Taking x 1, x 2 0, we can merely change the notation of the wave amplitudes in Eq. (78) to get
i
1
S
,
S
.
(2.133)
11
1
21
i
1 i
An absolutely similar analysis of the wave incidence from the left yields
i
1
S
,
S
,
(2.134)
22
1
12
i
1 i
and using Eqs. (126), we get
Transfer
1 i
i
matrix:
T
short
.