(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
)
gr
1 d 2
2
( x, t) 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 axis x, centered to point x = v gr t, with the r.m.s. width 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 that Chapter 2
Page 7 of 76
QM: Quantum Mechanics
2
t
1
Wave
x'
2 x
2
.
(2.39b) packet’s
2
2 m ( x
)
spread
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 for estimates of the spreading effect. Due to the smallness of the Planck constant on the human scale of things, for macroscopic bodies this effect 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 all above formulas. With this replacement, Eq. (27) becomes
i kx ( t t )
0
( x, t) 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
0
0
( x, t)
dk dx ( x , t ) e
.
(2.43)
0
0
0
2
Changing the order of integration, this expression may be rewritten 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
Page 8 of 76
QM: Quantum Mechanics
where the function G, usually called kernel in mathematics, in quantum mechanics is called the propagator.9 Its physical sense may be understood by considering the following special initial condition:10
( x , t ) ( x x' ) ,
(2.45)
0
0
0
where x’ is a certain point within the domain of particle’s motion. 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 its 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 particular 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 )
exp
.
(2.48)
0
0
i k x x 0
t
( t )
0 dk
2
2 m
This is a Gaussian integral again, and 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 )
0
.
(2.49)
0
0
exp
propagator
2 i
( t t )
i t t
0
2 ( )
0
Please note the following features of this complex function (plotted in Fig. 2):
(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 does not depend on its arguments – they just rescale the same function: its snapshot (Fig. 2), if plotted as a function of un-normalized x, 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, as a result of a deep mathematical analogy between quantum mechanics and classical statistics – to be discussed further in Chapter 7.
9 Its standard notation by letter G stems from the fact that the propagator is essentially the spatial-temporal Green’s function, 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.
10 Note that such initial condition is mathematically not equivalent to a -functional initial probability density (3).
Chapter 2
Page 9 of 76
QM: Quantum Mechanics
(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 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 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 (that, 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 travel through 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 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
Page 10 of 76
QM: Quantum Mechanics
U ( x ) E ,
(2.51)
c
the particle’s kinetic energy p 2/2 m is positive, so that 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 does the wave mechanics say 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
C
( x)
U
0
B
Fig. 2.4. The reflection of a
monochromatic wave from a potential
step U
0
> E. (This particular
E
wavefunction’s shape is for U 0 = 5 E.)
The wavefunction is plotted with the
same schematic vertical offset by E as
0
x
those in Fig. 1.8.
Let us represent an 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):11
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 exp{+ ikx} and exp{- ikx} correspondingly, 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
.
(2.55)
x
ikx
ikx
Ae
Be
waves
11 Note that this is not the 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.
Chapter 2
Page 11 of 76
QM: Quantum Mechanics
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, this solution is reduced to Eq. (1.84) for the 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
The general solution of Eq. (56) is the sum of exp{+ x} and exp{– x}, with arbitrary coefficients. Decay in classically
However, in our particular case the wavefunction should be finite at x +, so only the latter exponent forbidden region
is acceptable:
x
x
Ce
.
(2.58)
Such penetration of the wavefunction to 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, and has been repeatedly observed in experiment – e.g., via tunneling experiments – see the next section.12 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 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 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 that the boundary conditions in our problem are 12 Note that this effect is pertinent to 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).
Chapter 2
Page 12 of 76
QM: Quantum Mechanics
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 that 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, at x < 0 our solution (55) 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 issue: 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 re-solve the whole problem; it is sufficient to note that all our calculations, and hence Eqs.
(63) are still valid if we take13
m E U
2
2 (
)
ik'
,
with
0
k'
0 .
(2.65)
2
With this replacement, Eq. (63) becomes14
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): 13 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.
14 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
Page 13 of 76
QM: Quantum Mechanics
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. (75), 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, “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
Chapter 2
Page 14 of 76
QM: Quantum Mechanics
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 four 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
Rectangular
2
2
2
2
tunnel
F
k
barrier’s
2
2
T
cosh d
sinh
d
.
(2.71b)
transparency
A
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).
(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 10 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
1 /
E / U
1 E / U 0 2
0
Fig. 2.7. The transparency of a rectangular potential barrier as a function of the particle’s energy E.
Chapter 2
Page 15 of 76
QM: Quantum Mechanics
The plots show that generally, the transparency grows gradually 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 that 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, d << , E << U 0, giving k << << 1/ d. In this limit, Eq. (71) yields F 2
2
1
1
1 k 2
2
1 d
m
T
,
where
d
U d , (2.72)
A
2
2
2
0
1 i
1
2
k
2 k
k
The last product, U 0 d, is just the “energy area” (or the “weight”)
W U( x) dx
(2.73)
U ( x) E
of the barrier. 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 confirm this guess, let us consider the tunneling problem for a very thin barrier with d, kd
<< 1, 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 that the wavefunction itself is still continuous:
d
lim
x x lim
dx .
0
0
0
(2.76)
dx
Chapter 2
Page 16 of 76
QM: Quantum Mechanics
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
Thin
1
E
barrier:
T
,
where
W
m
E
,
(2.79)
2
0
2
transparency
1
E E
2
0
which shows that as energy becomes larger than the constant E 0, the transparency approaches 1.
Now proceeding to another important limit of thick barriers ( d >> ), Eq. (71) shows that in this case, the transparency is dominated by what is called the tunnel exponent:
Thick
4 k
2
barrier:
2 d
T
e
(2.80)
transparency
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). This exponential dependence on the barrier thickness is the most important factor for various applications of quantum-mechanical tunneling – from the field emission of electrons to vacuum15 to the scanning tunneling microscopy.16 Note also very substantial negative implications of the effect for the electronic technology progress, most importantly imposing limits on the so-called Dennard scaling of field-effect transistors in semiconductor integrated circuits (which is 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.17
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.18 This is our first glimpse at one more interesting quantum effect: resonant tunneling. This effect will be discussed in more detail in Sec. 5 below, using another potential profile where it is more clearly pronounced.
15 See, e.g., G. Fursey , Field Emission in Vacuum Microelectronics, Kluwer, New York, 2005.
16 See, e.g., G. Binning and H. Rohrer, Helv. Phys. Acta 55, 726 (1982).
17 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.) 18 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 Broigle wave of any energy E, and hence for any incident wave packet. Unfortunately, a proof of this fact would require more time/space than I can afford. (Note that it was first described in a 1930 paper by Paul Sophus Epstein, before the 1933 publication by G. Pöschl and E. Teller, which is responsible for the common name of this Pöschl-Teller potential.) Chapter 2
Page 17 of 76
QM: Quantum Mechanics
2.4. Motion in soft potentials
Before moving on to exploring other quantum-mechanical effects, let us see how the results discussed in the previous section are modified in the opposite limit of the so-called soft (also called
“smooth”) potential profiles, like the one sketched in Fig. 3.19 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 use Eq. (83) to look for solution of Eq. (81) even in the general case, 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 requiring the factor before exp{ i( x)} to vanish, we get 2
2
d
d
2
i
k ( x) 0 .
(2.85)
2
dx
dx
This is still an exact, general equation. At the first sight, it looks 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 very 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 result, so that 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 that its general solution is a linear superposition of two functions (83), with replaced with 0:
x
x
( x) A exp
,
(2.87)
0
i k( x' ) dx' B exp i k( x' ) dx'
19 Quantitative conditions of the “softness” will be formulated later in this section.
Chapter 2
Page 18 of 76
QM: Quantum Mechanics
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) that self-adjusts to the potential profile.
Let me emphasize the non-trivial nature of this approximation.20 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 the Schrödinger equation solutions, the probability current’s conservation.
Indeed, since Eq. (81) describes a fixed-energy (stationary) spatial part of the general Schrödinger equation, its probability density w = * =*, and should not depend on time. Hence, according to Eq. (6), we should have 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:21
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:
20 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.
21 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
Page 19 of 76
QM: Quantum Mechanics
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 is easy to understand. 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
Page 20 of 76
QM: Quantum Mechanics
Now we have to do what we have done for the sharp-step problem in Sec. 2: use the boundary conditions at 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. 22
Let us use our commitment of 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.23
(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().
22 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.
23 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
Page 21 of 76
QM: Quantum Mechanics
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 = 0, 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
QM: Quantum Mechanics
potential profile) may be expressed as d 2 U/ d 2 x<< dU/ dxat 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 “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.24
24 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 – 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
Page 23 of 76
QM: Quantum Mechanics
Let us see what does this quantization rule give 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 that 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 special properties of the potential profile (111).
Now let us use the mnemonic rule (i) to examine 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.
d
x ' x
U ( 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
Page 24 of 76
QM: Quantum Mechanics
In order to analyze tunneling through such a soft barrier, we need (just as in the case of a rectangular barrier) to take unto 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 waves rather than two, the connection formulas (105) have to be generalized, using asymptotic formulas not only for Ai( ), but also for the second Airy function, Bi( ). The analysis, absolutely similar to that carried out above (though naturally a bit bulkier),25 gives a remarkably simple result:
2
x '
x '
Soft
f
c
c
2
1/ 2
potential
T
exp
2 ( )
exp
2
( )
,
(2.117)
WKB
barrier:
x dx