1
2
Chapter 1
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ˆ A
ˆ
ˆ
ˆ
ˆ
c c A c A c c
A
c
A
,
(1.54)
1
1
2
2
1 1 2 2 1 1 2 2
where n are arbitrary wavefunctions, while cn are arbitrary constants (in quantum mechanics, frequently called c-numbers, to distinguish them from operators and wavefunctions). The most important examples of linear operators are given by:
(i) the multiplication by a function, such as for the operator rˆ given by Eq. (26), and (ii) the spatial or temporal differentiation, such as in Eqs. (25)-(27).
Next, it is of key importance that the Schrödinger equation (25) is also linear. (This fact was already used in the discussion of wave packets in the last section.) This means that if each of several functions n are (particular) solutions of Eq. (25) with a certain Hamiltonian, then their arbitrary linear combination,
c ,
(1.55)
n
n
n
is also a solution of the same equation.41
Let us use the linearity to accomplish an apparently impossible feat: immediately find the general solution of the Schrödinger equation for the most important case when the system’s Hamiltonian does not depend on time explicitly – for example, like in Eq. (41) with time-independent potential energy U = U(r), when the Schrödinger equation has the form
Potential
2
i
field:
2 U (r) .
(1.56)
t
2 m
Schrödinger
equation
First of all, let us prove that the following product,
a ( t) (r) ,
(1.57) Variable
n
n
n
separation
qualifies as a (particular) solution of such an equation. Indeed, plugging Eq. (57) into Eq. (25) with any time-independent Hamiltonian, using the fact that in this case
ˆ
ˆ
Ha ( t) (r) a ( t) H (r) ,
(1.58)
n
n
n
n
and dividing both parts of the equation by an n, we get
i da
H ˆ
n
n
.
(1.59)
a dt
n
n
The left-hand side of this equation may depend only on time, while the right-hand side, only on coordinates. These facts may be only reconciled if we assume that each of these parts is equal to (the same) constant of the dimension of energy, which I will denote as En.42 As a result, we are getting two separate equations for the temporal and spatial parts of the wavefunction:
41 At the first glance, it may seem strange that the linear Schrödinger equation correctly describes quantum properties of systems whose classical dynamics is described by nonlinear equations of motion (e.g., an anharmonic oscillator – see, e.g., CM Sec. 5.2). Note, however, that statistical equations of classical dynamics (see, e.g., SM Chapters 5 and 6) also have this property, so it is not specific to quantum mechanics.
42 This argumentation, leading to variable separation, is very common in mathematical physics – see, e.g., its discussion in CM Sec. 6.5, and in EM Sec. 2.5 and beyond.
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Stationary
Schrödinger
H ˆ E ,
(1.60)
equation
n
n
n
da
n
i
E a .
(1.61a)
n n
dt
The latter of these equations, rewritten in the form
da
E
n i n
dt ,
(1.61b)
an
is readily integrable, giving
Stationary
state:
E
ln a i
t const,
a
that
so
const exp i t ,
ω
with
.
(1.62)
time
n
n
n
n
n
n
evolution
Now plugging Eqs. (57) and (62) into Eq. (22), we see that in the quantum state described by Eqs. (57)-
(62), the probability w of finding the particle at a certain location does not depend on time: w *
.
(1.63)
n r
n r
w r
With the same substitution, Eq. (23) shows that the expectation value of any operator that does not depend on time explicitly is also time-independent:
A
*
3
ˆ
r r
= const.
(1.64)
n A
n
d r
Due to this property, the states described by Eqs. (57)-(62) are called stationary; they are fully defined by the possible solutions of the stationary (or “time-independent”) Schrödinger equation (60).43
Note that for the time-independent Hamiltonian (41), the stationary Schrödinger equation (60), Potential field:
2
stationary
2
U (r
)
E ,
(1.65)
Schrödinger
n
n
n
n
2 m
equation
is a linear, homogeneous differential equation for the function n, with a priory unknown parameter En.
Such equations fall into the mathematical category of eigenproblems,44 whose eigenfunctions n and eigenvalues En should be found simultaneously, i.e. self-consistently.45
Mathematics46 tells us that for such equations with space-confined eigenfunctions n, tending to zero at r , the spectrum of eigenvalues is discrete. It also proves that the eigenfunctions corresponding to different eigenvalues are orthogonal, i.e. that space integrals of the products n n’*
vanish for all pairs with n n’. Due to the Schrödinger equation’s linearity, each of these functions may be multiplied by a proper constant coefficient to make their set orthonormal:
*
n n'
3
for
,
1
,
d r
(1.66)
n
n'
n, n'
for
,
0
n n' .
43 In contrast, the full Schrödinger equation (25) is frequently called time-dependent or non-stationary.
44 From the German root eigen, meaning “particular” or “characteristic”.
45 Eigenvalues of energy are frequently called eigenenergies, and it is often said that the eigenfunction n and the corresponding eigenenergy En together determine the n th stationary eigenstate of the system.
46 See, e.g., Sec. 9.3 of the wonderful handbook by G. Korn and T. Korn, listed in MA Sec. 16(ii).
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Moreover, the eigenfunctions n(r) form a full set, meaning that an arbitrary function (r), in particular the actual wavefunction of the system in the initial moment of its evolution (which I will always, with a few clearly marked exceptions, take for t = 0) may be represented as a unique expansion over the eigenfunction set:
(r )
0
, c (r) .
(1.67)
n
n
n
The expansion coefficients cn may be readily found by multiplying both sides of Eq. (67) by *n’, integrating the results over the space, and using Eq. (66). The result is
c
*
3
(r) (r,0)
.
(1.68)
n
d r
n
Now let us consider the following wavefunction
En
(r, t) c a t() (r) c exp i
t (r) .
(1.69) General
n k
k
n
n
solution
n
n
for U = U(r)
Since each term of the sum has the form (57) and satisfies the Schrödinger equation, so does the sum as the whole. Moreover, if the coefficients cn are derived in accordance with Eq. (68), then the solution (69) satisfies the initial conditions as well. At this moment we can use one more bit of help from mathematicians, who tell us that the linear, partial differential equation of type (65), with fixed initial conditions, may have only one ( unique) solution. This means that in our case of time-independent potential Hamiltonian, Eq. (69) gives the general solution of the Schrödinger equation (25).
So, we have succeeded in our apparently over-ambitious goal. Now let us pause this mad mathematical dash for a minute, and discuss this key result.
1.6. Time evolution
For the time-dependent factor, an( t), of each component (57) of the general solution (69), our procedure gave a very simple and universal result (62), describing a linear change of the phase n
arg( an) of this complex function in time, with the constant rate
d
E
n
n
,
(1.70)
n
dt
so that the real and imaginary parts of an oscillate sinusoidally with this frequency. The relation (70) coincides with Einstein’s conjecture (5), but could these oscillations of the wavefunctions represent a physical reality? Indeed, for photons, described by Eq. (5), E may be (and as we will see in Chapter 9, is) the actual, well-defined energy of one photon, and is the frequency of the radiation so quantized.
However, for non-relativistic particles, described by wave mechanics, the potential energy U, and hence the full energy E, are defined to an arbitrary constant, because we may measure them from an arbitrary reference level. How can such a change of the energy reference level (which may be made just in our mind) alter the frequency of oscillations of a variable?
According to Eqs. (22)-(23), this time evolution of a wavefunction does not affect the particle’s probability distribution, or even any observable (including the energy E, provided that it is always referred to the same origin as U), in any stationary state. However, let us combine Eq. (5) with Bohr’s assumption (7):
Chapter 1
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Essential Graduate Physics
QM: Quantum Mechanics
E E .
(1.71)
nn'
n'
n
The difference nn’ of the eigenfrequencies n and n’, participating in this formula, is evidently independent of the energy reference, and as will be proved later in the course, determines the measurable frequency of the electromagnetic radiation (or possibly of a wave of a different physical nature) emitted or absorbed at the quantum transition between the states.
As another but related example, consider two similar particles 1 and 2, each in the same (say, the lowest-energy) eigenstate, but with their potential energies (and hence the ground state energies E 1,2) different by a constant U U 1 – U 2. Then, according to Eq. (70), the difference 1 – 2 of their wavefunction phases evolves in time with the reference-independent rate
d
U
.
(1.72)
dt
Certain measurement instruments, weakly coupled to the particles, may allow observation of this evolution, while keeping the particle’s quantum dynamics virtually unperturbed, i.e. Eq. (70) intact.
Perhaps the most dramatic measurement of this type is possible using the Josephson effect in weak links between two superconductors – see Fig. 7.47
I sin( )
1
2
exp i1
exp i2
V
Fig. 1.7. The Josephson effect in a weak link
between two bulk superconductor electrodes.
As a brief reminder,48 superconductivity may be explained by a specific coupling between conduction electrons in solids, that leads, at low temperatures, to the formation of the so-called Cooper pairs. Such pairs, each consisting of two electrons with opposite spins and momenta, behave as Bose particles and form a coherent Bose-Einstein condensate.49 Most properties of such a condensate may be described by a single, common wavefunction , evolving in time just as that of a free particle, with the effective potential energy U = q = –2 e, where is the electrochemical potential,50 and q = –2 e is the electric charge of a Cooper pair. As a result, for the system shown in Fig. 7, in which externally applied voltage V fixes the difference 1 – 2 between the electrochemical potentials of two superconductors, Eq. (72) takes the form
Eq. (72) for
d
e
2
Josephson
V .
(1.73)
effect
dt
If the link between the superconductors is weak enough, the electric current I of the Cooper pairs (called the supercurrent) through the link may be approximately described by the following simple relation, 47 The effect was predicted in 1962 by Brian Josephson (then a graduate student!) and observed soon after that.
48 For a more detailed discussion, including the derivation of Eq. (75), see e.g. EM Chapter 6.
49 A detailed discussion of the Bose-Einstein condensation may be found, e.g., in SM Sec. 3.4.
50 For more on this notion see, e.g. SM Sec. 6.3.
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Josephson
I I sin,
(1.74)
c
supercurrent
where I c is some constant, dependent on the weak link’s strength.51 Now combining Eqs. (73) and (74), we see that if the applied voltage V is constant in time, the current oscillates sinusoidally, with the so-called Josephson frequency
2 e
V ,
(1.75)
J
as high as ~484 MHz per microvolt of applied dc voltage. This effect may be readily observed experimentally: though its direct detection is a bit tricky, it is easy to observe the phase locking (synchronization)52 of the Josephson oscillations by an external microwave signal of frequency . Such phase locking results in the relation J = n fulfilled within certain dc current intervals, and hence in the formation, on the weak link’s dc I-V curve, of virtually vertical current steps at dc voltages
V
n
,
(1.76)
n
e
2
where n is an integer.53 Since frequencies may be stabilized and measured with very high precision, this effect is being used in highly accurate standards of dc voltage.
1.7. Spatial dependence
In contrast to the simple and universal time dependence (62) of the stationary states, the spatial distributions of their wavefunction n(r) need to be calculated from the problem-specific stationary Schrödinger equation (65). The solution of this equation for various particular cases is a major focus of the next two chapters. For now, let us consider just the simplest example, which nevertheless will be the basis for our discussion of more complex problems: let a particle be confined inside a rectangular hard-wall box. Such confinement may be described by the following potential energy profile:54
,
0
0
for x a , 0 y a ,
and 0 z a ,
Hard-wall box:
U (r)
x
y
z
(1.77) potential
.
otherwise
,
The only way to keep the product U(r) n in Eq. (65) finite outside the box, is to have = 0 in these regions. Also, the function has to be continuous everywhere, to avoid the divergence of the 51 In some cases, the function I() may somewhat deviate from Eq. (74), but these deviations do not affect its fundamental 2-periodicity, and hence the fundamental relations (75)-(76). (No corrections to them have been found yet.)
52 For the discussion of this very general effect, see, e.g., CM Sec. 5.4.
53 The size of these dc current steps may be readily calculated from Eqs. (73) and (74). Let me leave this task for the reader’s exercise.
54 Another common name for such potentials, especially of lower dimensionality, is the potential well, in our current case “rectangular” one: with a flat “bottom” and vertical, infinitely high “walls”. Note that sometimes, very unfortunately, such potential profiles are called “quantum wells”. (This term seems to imply that the particle’s confinement in such a well is a phenomenon specific for quantum mechanics. However, as we will repeatedly see in this course, the opposite is true: quantum effects do as much as they only can to overcome the particle’s confinement in a potential well, letting it partly penetrate in the “classically forbidden” regions beyond the well’s walls.)
Chapter 1
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kinetic-energy term (-2/2 m)2 n. Hence, in this case we may solve the stationary Schrödinger equation (60) just inside the box, i.e. with U = 0, so that it takes a simple form
2
2
E ,
(1.78a)
2
n
n
n
m
with zero boundary conditions on all the walls.55 For our particular geometry, it is natural to express the Laplace operator in the Cartesian coordinates { x, y, z} aligned with the box sides, with the origin at one of the corners of its rectangular ax ay az volume, so that our boundary problem becomes: 2
2
2
2
E ,
0
for x a , 0 y a ,
0
and z a ,
2
2
2
2
n
n
n
x
y
z
m x
y
z
(1.78b)
with
for
0
: x
and
0
a ; y
and
0
a ; z
and
0
a .
n
x
y
z
This problem may be readily solved using the same variable separation method as was used in Sec. 5 – now to separate the Cartesian spatial variables from each other, by looking for a partial solution of Eq. (78) in the form
(r) X ( x) Y ( y) Z ( z) .
(1.79)
(Let us postpone assigning function indices for a minute.) Plugging this expression into Eq. (78b) and dividing all terms by the product XYZ, we get
2
1 d 2 X
2
1 d 2 Y
2
1 d 2 Z
E .
(1.80)
2 m X dx 2
2 m Y dy 2
2 m Z dz 2
Now let us repeat the standard argumentation of the variable separation method: since each term on the left-hand side of this equation may be only a function of the corresponding argument, the equality is possible only if each of them is a constant – in our case, with the dimensionality of energy. Calling these constants Ex, etc., we get three similar 1D equations
2
1 2
2
d X
1 2
2
d Y
1 2
d Z
E ,
E ,
E ,
(1.81)
2
2
x
m X dx
2
2
y
m Y dy
2
2
z
m Z dx
with Eq. (80) turning into the following energy-matching condition:
E E E E .
(1.82)
x
y
z
All three ordinary differential equations (81), and their solutions, are similar. For example, for X( x), we have the following 1D Helmholtz equation
2
d X
2 mE
2
2
k X ,
0
with
x
k
,
(1.83)
2
x
x
2
dx
and simple boundary conditions: X(0) = X( ax) = 0. Let me hope that the reader knows how to solve this well-known 1D boundary problem – describing, for example, the usual mechanical waves on a guitar string. The problem allows an infinite number of sinusoidal standing-wave eigenfunctions,56
55 Rewritten as 2 f + k 2 f = 0, Eq. (78a) is just the Helmholtz equation, which describes waves of any nature (with the wave vector k) in a uniform, isotropic, linear medium – see, e.g., EM Secs. 7.5-7.9 and 8.5.
56 The front coefficient in the last expression for X ensures the (ortho)normality condition (66).
Chapter 1
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1/ 2
n
n
x
Rectangular
x
2
X sin k x,
with k
,
that
so
X
sin
x
, with n ,
1 ,...
2
, (1.84) potential well:
x
x
a
a
a
1D eigen-
x
x
x
x
functions
corresponding to the following eigenenergies:
2
2
2
2
2
2
E
k
n
E n .
(1.85)
x
x
2
x
1
2 m
2
x
x
max
Figure 8 shows these simple results, using a somewhat odd but very graphic and common representation, in that the eigenenergy values (frequently called the energy levels) are used as horizontal axes for plotting the eigenfunctions – despite their completely different dimensionality.
E / E
x
1
x
X ( x)
n 3
x
9
n 2
x
5
Fig. 1.8. The lowest eigenfunctions (solid lines) and
4
eigenvalues (dashed lines) of Eq. (83) for a potential well
n 1
x
of length ax. Solid black lines show the effective potential
10
energy profile for this 1D eigenproblem.
0
a
x
x
Due to the similarity of all Eqs. (81), Y( y) and Z( z) are absolutely similar functions of their arguments, and may also be numbered by integers (say, ny and nz) independent of nx, so that the spectrum of values of the total energy (82) is
2
2
2
2
2
n
n
n
Rectangular
E
x y z
n , n , n
.
(1.86) potential well:
x
y
z
2
2
2
2 m a
a
a
energy levels
x
y
z
Thus, in this 3D problem, the role of the index n in the general Eq. (69) is played by a set of three independent integers { nx, ny, nz}. In quantum mechanics, such integers play a key role and thus have a special name, the quantum numbers. Using them, that general solution, for our current simple problem may be represented as the sum
E
Rectangular
n
x
n
y
n
z
n n n
potential
x
y
z
, ,
x
y
z
Ψ(r, t)
cn , n , sin
sin
sin
exp
n
i
t ,
(1.87) well:
x
y
z
a
a
a
general
n , n , n
x
y
z
1
x
y
z
solution
with the front coefficients that may be readily calculated from the initial wavefunction (r, 0), using Eq. (68) – again with the replacement n { nx, ny, nz}.
This simplest problem is a good illustration of typical results the wave mechanics gives for spatially-confined motion, including the discrete energy spectrum, and (in this case, evidently) orthogonal eigenfunctions. Perhaps most importantly, its solution shows that the lowest value of the particle’s kinetic energy (86), reached in the so-called ground state (in our case, the state with nx = ny =
nz = 1) is above zero for any finite size of the confining box.
Chapter 1
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An example of the opposite case of a continuous spectrum for the unconfined motion of a free particle is given by the plane waves (29). With the account of relations E = and p = k, such wavefunction may be viewed as the product of the time-dependent factor (62) by the eigenfunction, Free
particle:
eigen-
a exp i ,
(1.88)
k
k
k r
functions
which is the solution of the stationary Schrödinger equation (78a) if it is valid in the whole space.57 The reader should not be worried too much by the fact that the fundamental solution (86) in free space is a traveling wave (having, in particular, a non-zero value of the probability current j), while those inside a quantum box are standing waves, with j = 0, even though the free space may be legitimately considered as the ultimate limit of a quantum box with volume V = ax ay az . Indeed, due to the linearity of wave mechanics, two traveling-wave solutions (88) with equal and opposite values of the momentum (and hence with the same energy) may be readily combined to give a standing-wave solution,58 for example, exp{ ikr} + exp{- ikr} = 2cos(kr), with the net current j = 0. Thus, depending on the convenience for a particular problem, we may represent its general solution as a sum of either traveling-wave or standing-wave eigenfunctions. Since in the unlimited free space, there are no boundary conditions to satisfy, the Cartesian components of the wave vector k in Eq. (88) can take any real values. (This is why it is more convenient to label these wavefunctions, and the corresponding eigenenergies,
Free
2
2
particle:
k
E
0 ,
(1.89)
eigen-
k
2 m
energies
with their wave vector k rather than an integer index.)
However, one aspect of continuous-spectrum systems requires a bit more caution with mathematics: the summation (69) should be replaced by the integration over a continuous index or indices – in our current case, the three Cartesian components of the vector k. The main rule of such replacement may be readily extracted from Eq. (84): according to this relation, for standing-wave solutions, the eigenvalues of kx are equidistant, i.e. separated by equal intervals kx = / ax, with similar relations for other two Cartesian components of vector k. Hence the number of different eigenvalues of the standing-wave vector k (with kx, ky, kz 0), within a volume d 3 k >> 1/ V of the k space is dN =
d 3 k/( kx kx kx) = ( V/3) d 3 k. Frequently, it is more convenient to work with traveling waves (88); in this case we should take into account that, as was just discussed, there are two different traveling wave numbers (say, + kx and – kx) corresponding to each standing wave vector’s kx > 0. Hence the same number of physically different states corresponds to a 23 = 8-fold larger k space or, equivalently, to an 8-fold smaller number of states per unit volume d 3 k:
Number
V
of 3D states
dN
3
d k.
(1.90)
3
2
57 In some systems (e.g., a particle interacting with a potential well of a finite depth), a discrete energy spectrum within a certain energy interval may coexist with a continuous spectrum in a complementary interval. However, the conceptual philosophy of eigenfunctions and eigenvalues remains the same even in this case.
58 This is, of course, the general property of waves of any physical nature, propagating in a linear medium – see, e.g., CM Sec. 6.5 and/or EM Sec. 7.3.
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For dN >> 1, this expression is independent of the boundary conditions, and is frequently represented as the following summation rule
V
Summation
3
lim
(k)
(k)
(k)
,
(1.91)
over
k V
3
f f dN
f
d k
3D states
k
2 3
where f(k) is an arbitrary function of k. Note that if the same wave vector k corresponds to several internal quantum states (such as spin – see Chapter 4), the right-hand side of Eq. (91) requires its multiplication by the corresponding degeneracy factor of orbital states.59
1.8. Dimensionality reduction
To conclude this introductory chapter, let me discuss the conditions when the spatial dimensionality of a wave-mechanical problem may be reduced.60 Naively, one may think that if the particle’s potential energy depends on just one spatial coordinate, say U = U( x, t), then its wavefunction has to be one-dimensional as well: = ( x, t). Our discussion of the particular case U = const in the previous section shows that this assumption is wrong. Indeed, though this potential61 is just a special case of the potential U( x, t), most of its eigenfunctions, given by Eqs. (87) or (88), do depend on the other two coordinates. This is why the solutions ( x, t) of the 1D Schrödinger equation
2
1D time-
2
i
U ( x, t) ,
(1.92) dependent
t
2 m 2
x
Schrödinger
equation
which follows from Eq. (65) by assuming / y = / z = 0, are insufficient to form the general solution of Eq. (65) for this case.
This fact is easy to understand physically for the simplest case of a stationary 1D potential: U =
U( x). The absence of the y- and z-dependence of the potential energy U may be interpreted as a potential well that is flat in two directions, y and z. Repeating the arguments of the previous section for this case, we see that the eigenfunctions of a particle in such a well have the form
r X ( x)
exp i k y k z ,
(1.93)
y
z
where X( x) is an eigenfunction of the following stationary 1D Schrödinger equation: 2
d 2 X
U ( x) X EX ,
(1.94)
2 m dx 2
ef
where U ef( x) is not the full potential energy of the particle, as it would follow from Eq. (92), but rather its effective value including the kinetic energy of the lateral motion:
2
U U E E U
k k .
(1.95)
ef
y z
2 2
y
2
z
m
59 Such factor is similar to the front factor 2 in Eq. (1) for the number of electromagnetic wave modes, in that case describing two different polarizations of the waves with the same wave vector.
60 Most textbooks on quantum mechanics jump to the formal solution of 1D problems without such discussion, ignoring the fact that such dimensionality restriction is adequate only under very specific conditions.
61 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 of this notion with another (say, electrostatic) potential.
Chapter 1
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In plain English, the particle’s partial wavefunction X( x) and its full energy, depend on its transverse momenta, which have continuous spectrum – see the discussion of Eq. (89). This means that Eq. (92) is adequate only if the condition ky = kz = 0 is somehow enforced, and in most physical problems, it is not.
For example, if a de Broglie (or any other) plane wave ( x, t) is incident on a potential step, it would be reflected exactly back, i.e. with ky = kz = 0, only if the wall’s surface is perfectly plane and exactly normal to the axis x. Any imperfection (and they are so many of them in real physical systems –:) may cause excitation of waves with non-zero values of ky and kz, due to the continuous character of the functions Ey( ky) and Ez( kz).62
There is essentially one, perhaps counter-intuitive way to make the 1D solutions “robust” to small perturbations: it is to provide a rigid lateral confinement 63 in two other directions. As the simplest example, consider a narrow quantum wire (Fig. 9a), described by the following potential:
U ( x
),
0
for y a ,
0
and
z a ,
U (r)
y
z
(1.96)
otherwize.
,
(a)
(b)
y
z
y
z
x
x
Fig. 1.9. Partial confinement in: (a) two dimensions, and (b) one dimension.
Performing the standard variable separation (79), we see that the corresponding stationary Schrödinger equation is satisfied if the partial wavefunction X( x) obeys Eqs. (94)-(95), but now with a discrete energy spectrum in the transverse directions:
2
2
2
2
n
n
U
U
y z .
(1.97)
ef
2
2
2 m a
a
y
z
If the lateral confinement is tight, ay, az 0, then there is a large energy gap, 2
2
U
~
,
(1.98)
2
2 may, z
between the ground-state energy of the lateral motion (with ny = nz = 1) and that for all its excited states.
As a result, if the particle is initially placed into the lateral ground state, and its energy E is much smaller than U, it would stay in such state, i.e. may be described by a 1D Schrödinger equation similar to Eq. (92) – even in the time-dependent case, if the characteristic frequency of energy variations is much smaller than U/. Absolutely similarly, the strong lateral confinement in just one dimension (say, z, see Fig. 9b) enables systems with a robust 2D evolution of the particle’s wavefunction.
62 This problem is not specific to quantum mechanics. The classical motion of a particle in a 1D potential may be also unstable with respect to lateral perturbations, especially if the potential is time-dependent, i.e. capable of exciting low-energy lateral modes.
63 The term “quantum confinement”, sometimes used to describe this phenomenon, is as unfortunate as the
“quantum well” term discussed above, because of the same reason: the confinement is a purely classical effect, and as we will repeatedly see in this course, the quantum-mechanical effects reduce, rather than enable it.
Chapter 1
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The tight lateral confinement may ensure the dimensionality reduction even if the potential well is not exactly rectangular in the lateral direction(s), as described by Eq. (96), but is described by some x-
and t-independent profile, if it still provides a sufficiently large energy gap U. For example, many 2D
quantum phenomena, such as the quantum Hall effect,64 have been studied experimentally using electrons confined at semiconductor heterojunctions (e.g., epitaxial interfaces GaAs/Al x Ga1- x As), where the potential well in the direction perpendicular to the interface has a nearly triangular shape, and provides an energy gap U of the order of 10-2 eV.65 This gap corresponds to k B T with T ~100 K, so that careful experimentation at liquid helium temperatures (4K and below) may keep the electrons performing purely 2D motion in the “lowest subband” ( nz = 1).
Finally, note that in systems with reduced dimensionality, Eq. (90) for the number of states at large k (i.e., for an essentially free particle motion) should be replaced accordingly: in a 2D system of area A >> 1/ k 2,
A
dN
2
Number
d k ,
(1.99)
2
2
of 2D states
while in a 1D system of length l >> 1/ k,
l
dN
dk ,
(1.100) Number
2
of 1D states
with the corresponding changes of the summation rule (91). This change has important implications for the density of states on the energy scale, dN/ dE: it is straightforward (and hence left for the reader :-) to use Eqs. (90), (99), and (100) to show that for free 3D particles the density increases with E
(proportionally to E 1/2), for free 2D particles it does not depend on energy at all, while for free 1D
particles it scales as E-1/2, i.e. decreases with energy.
1.9. Exercise problems
1.1. The actual postulate made by N. Bohr in his original 1913 paper was not directly Eq. (8), but the assumption that at quantum leaps between adjacent large (quasiclassical) orbits with n >> 1, the hydrogen atom either emits or absorbs energy E = , where is its classical radiation frequency –
according to classical electrodynamics, equal to the angular velocity of electron’s rotation.66 Prove that this postulate is indeed compatible with Eqs. (7)-(8).
1.2. Use Eq. (53) to prove that the linear operators of quantum mechanics are commutative:
ˆ
ˆ
ˆ
ˆ
A A A A , and associative: ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
A A A A A A .
1
2
3
1
2 3
2
1
1
2
1.3. Prove that for any time-independent Hamiltonian operator H ând two arbitrary complex functions f(r) and g(r),
f r ˆ g
H r d 3 r
ˆ f
H r gr d 3 r
.
64 To be discussed in Sec. 3.2.
65 See, e.g., P. Harrison, Quantum Wells, Wires, and Dots, 3rd ed., Wiley, 2010.
66 See, e.g., EM Sec. 8.2.
Chapter 1
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1.4. Prove that the Schrödinger equation (25) with the Hamiltonian operator given by Eq. (41), is Galilean form-invariant, provided that the wavefunction is transformed as
2
v r
'
m
mv t
'
r , t' r, t
exp i
i
,
2
where the prime sign marks the variables measured in the reference frame 0 ’ that moves, without rotation, with a constant velocity v relatively to the “lab” frame 0. Give a physical interpretation of this transformation.
1.5.* Prove the so-called Hellmann-Feynman theorem:67
E
H
n
,
n
where is some c-number parameter, on which the time-independent Hamiltonian H ˆ , and hence its eigenenergies En, depend.
1.6.* Use Eqs. (73) and (74) to analyze the effect of phase locking of Josephson oscillations on the dc current flowing through a weak link between two superconductors (frequently called the Josephson junction), assuming that an external source applies to the junction a sinusoidal ac voltage with frequency and amplitude A.
1.7. Calculate x, px, x, and px for the eigenstate { nx, ny, nz} of a particle in a rectangular hard-wall box described by Eq. (77), and compare the product x px with the Heisenberg’s uncertainty relation.
1.8. Looking at the lower (red) line in Fig. 8, it seems plausible that the 1D ground-state function (84) of the simple potential well (77) may be well approximated with an inverted quadratic parabola: X
,
trial x C x a
x
x
where C is a normalization constant. Explore how good this approximation is.
1.9. A particle placed in a hard-wall rectangular box with sides ax, ay, and az, is in its ground state. Calculate the average force acting on each face of the box. Can the forces be characterized by a certain pressure?
1.10. A 1D quantum particle was initially in the ground state of a very deep, rectangular potential well of width a:
,
0
for a / 2 x a / ,
2
U ( x)
otherwise.
,
67 Despite this common name, H. Hellmann (in 1937) and R. Feynman (in 1939) were not the first ones in the long list of physicists who had (apparently, independently) discovered this equality. Indeed, it has been traced back to a 1922 paper by W. Pauli, and was carefully proved by P. Güttinger in 1931.
Chapter 1
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At some instant, the well’s width is abruptly increased to a new value a’ > a, leaving the potential symmetric with respect to the point x = 0, and then left constant. Calculate the probability that after the change, the particle is still in the ground state of the system.
1.11. At t = 0, a 1D particle of mass m is placed into a hard-wall, flat-bottom potential well
,
0
0
for
x a
,
U ( x)
,
otherwise,
in a 50/50 linear superposition of the lowest (ground) state and the first excited state. Calculate: (i) the normalized wavefunction ( x, t) for arbitrary time t 0, and (ii) the time evolution of the expectation value x of the particle’s coordinate.
1.12. Calculate the potential profiles U( x) for that the following wavefunctions, (i)
c
ax 2
exp
ibt, and
(ii)
c
exp a x ibt
(with real coefficients a > 0 and b), satisfy the 1D Schrödinger equation for a particle with mass m. For each case, calculate x, px, x, and px, and compare the product x px with the Heisenberg’s uncertainty relation.
1.13. A 1D particle of mass m, moving in the field of a stationary potential U( x), has the following eigenfunction
C
x
,
cosh x
where C is the normalization constant, and is a real constant. Calculate the function U( x) and the state’s eigenenergy E.
1.14. Calculate the density dN/ dE of traveling-wave quantum states in large rectangular potential wells of various dimensions: d = 1, 2, and 3.
1.15.* Use the finite-difference method with steps a/2 and a/3 to find as many eigenenergies as possible for a 1D particle in the infinitely deep, hard-wall 1D potential well of width a. Compare the results with each other, and with the exact formula.68
68 You may like to start by reading about the finite-difference method – see, e.g., CM Sec. 8.5 or EM Sec. 2.11.
Chapter 1
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Chapter 2. 1D Wave Mechanics
Even the simplest, 1D version of wave mechanics enables quantitative analysis of many important quantum-mechanical effects. The order of their discussion in this chapter is dictated mostly by mathematical convenience – going from the simplest potential profiles to more complex ones, so that we may build upon the previous results. However, I would advise the reader to focus more not on the math, but rather on the physics of the non-classical phenomena it describes, ranging from particle penetration into classically-forbidden regions, to quantum-mechanical tunneling, to the metastable state decay, to covalent bonding and quantum oscillations, to energy bands and gaps.
2.1. Basic relations
As was discussed at the end of Chapter 1, in several cases (in particular, at strong confinement within the [ y, z] plane), the general (3D) Schrödinger equation may be reduced to its 1D version, similar to Eq. (1.92):
( x, t)
2
2
( x, t)
Schrödinger
i
U ( x, t)( x, t) .
(2.1)
equation
t
2
2
m
x
It is important, however, to remember that according to the discussion in Sec, 1.8, U( x, t) in this equation is generally effective potential energy, which may include the energy of the lateral motion, while ( x, t) may be just one factor in the complete wavefunction ( x, t)( y, z). If the transverse factor
( y, z) is normalized to 1, then the integration of Eq. (1.22a) over the 3D space within a segment [ x 1, x 2]
gives the following probability to find the particle on this segment:
x 2
Probability
W t
( ) Ψ( x, t Ψ
) *( x, t) dx
.
(2.2)
x 1
If the particle under analysis is definitely somewhere inside the system, the normalization of its 1D
wavefunction ( x, t) is provided by extending integral (2) to the whole axis x:
Normalization
(
w x, t) dx ,
1
where (
w x, t) Ψ( x, t)Ψ*( x, t)
.
(2.3)
A similar integration of Eq. (1.23) shows that the expectation value of any observable depending only on the coordinate x (and possibly time), may be expressed as
Expectation
A t
( )
value
Ψ*( x, t A ˆ
) Ψ( x, t) dx .
(2.4)
It is also useful to introduce the notion of the probability current along the x-axis (a scalar): Probability
*
2
I ( x, t) j dydz
current
Im Ψ
Ψ
Ψ( x, t
)
,
(2.5)
x
m
x
m
x
where jx is the x-component of the current density vector j(r, t). Then the continuity equation (1.48) for any segment [ x 1, x 2] takes the form
© K. Likharev
QM: Quantum Mechanics
dW
Continuity
I( x ) I( x ) 0 .
(2.6)
2
1
equation
dt
The above formulas are sufficient for analysis of 1D problems of wave mechanics, but before proceeding to particular cases, let me deliver on my earlier promise to prove that Heisenberg’s uncertainty relation (1.35) is indeed valid for any wavefunction ( x, t). For that, let us consider the following positive (or at least non-negative) integral
2
J
x
dx 0
,
(2.7)
x
where is an arbitrary real constant, and assume that at x the wavefunction vanishes, together with its first derivative – as we will see below, a very common case. Then the left-hand side of Eq. (7) may be recast as
2
*
J
x
dx
x
x
dx
x
x
x
(2.8)
2
*
*
*
*
2
x dx x
dx
dx.
x
x
x x
According to Eq. (4), the first term in the last form of Eq. (8) is just x 2, while the second and the third integrals may be worked out by parts:
*
x
x
*
x
dx
x *
dx xd *
*
*
x
dx 1
, (2.9)
x
x
x
x
x
2
*
x
2
p
*
* x
*
1
* 2
dx
d
dx
ˆ
x
p dx
. (2.10)
2
2
x
2
x
x
x
x
x
x
x
As a result, Eq. (7) takes the following form:
2
2
2
2
p
x
J
2
2
x
2
x
,
0
i.e. a b ,
0
with a
, b
.
(2.11)
2
2
2
p
p
x
x
This inequality should be valid for any real , so that the corresponding quadratic equation, 2 + a + b
= 0, can have either one (degenerate) real root – or no real roots at all. This is only possible if its determinant, Det = a 2 – 4 b, is non-positive, leading to the following requirement: 2
2
2
x
p
.
(2.12)
x
4
In particular, if x = 0 and px = 0,1 then according to Eq. (1.33), Eq. (12) takes the form 1 Eq. (13) may be proved even if x and px are not equal to zero, by making the following replacements: x x –
x and / x / x + i p/ in Eq. (7), and then repeating all the calculations – which in this case become somewhat bulky. In Chapter 4, equipped with the bra-ket formalism, we will derive a more general uncertainty relation, which includes the Heisenberg’s relation (13) as a particular case, in a more efficient way.
Chapter 2
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Heisenberg’s
2
uncertainty
~2 ~2
x
p
,
(2.13)
x
relation
4
which, according to the definition (1.34) of the r.m.s. uncertainties, is equivalent to Eq. (1.35).
Now let us notice that the Heisenberg’s uncertainty relation looks very similar to the commutation relation between the corresponding operators:
x ˆ, p ˆ
ˆˆ
ˆ ˆ
.
(2.14a)
x p
x
p x
x
x
x i i
x i
x
x
Since this relation is valid for any wavefunction ( x, t), it may be represented as an operator equality: Coordinate/
momentum
operators’
ˆ x, ˆ p
.
(2.14b)
x i 0
commutator
In Sec. 4.5 we will see that the relation between Eqs. (13) and (14) is just a particular case of a general relation between the expectation values of non-commuting operators, and their commutators.
2.2. Free particle: Wave packets
Let us start our discussion of particular problems with the free 1D motion, i.e. with U( x, t) = 0.
From Eq. (1.29), it is evident that in the 1D case, a similar “fundamental” (i.e. a particular but the most important) solution of the Schrödinger equation (1) is a sinusoidal (“monochromatic”) wave
( x, t) const
i k x t .
(2.15)
0
exp ( 0
0
)
According to Eqs. (1.32), it describes a particle with definite momentum2 p 0 = k 0 and energy E 0 = 0
= 2 k 2
0 /2 m. However, for this wavefunction, the product * does not depend on either x or t, so that the particle is completely delocalized, i.e. the probability to find it the same along all axis x, at all times.
In order to describe a space-localized state, let us form, at the initial moment of time ( t = 0), a wave packet of the type shown in Fig. 1.6, by multiplying the sinusoidal waveform (15) by some smooth envelope function A( x). As the most important particular example, consider the Gaussian wave packet Gaussian
ik x
1
2
x
wave
( x,0)
A x 0
e
,
with
A x
exp -
.
(2.16)
packet:
1/4
1/2
2
(2 ) ( x)
t = 0
(2 x)
(By the way, Fig. 1.6a shows exactly such a packet.) The pre-exponential factor in this envelope function has been selected in the way to have the initial probability density,
*
*
1
2
x
(
w x,0) ( x,0)( x,0) A ( x) (
A x)
,
(2.17)
2
exp
1/ 2
2
x
2( x)
normalized as in Eq. (3), for any parameters x and k 0.3
2 From this point on to the end of this chapter, I will drop the index x in the x-components of the vectors k and p.
3 This fact may be readily proven using the well-known integral of the Gaussian function (17), in infinite limits –
see, e.g., MA Eq. (6.9b). It is also straightforward to use MA Eq. (6.9c) to prove that for the wave packet (16), the parameter x is indeed the r.m.s. uncertainty (1.34) of the coordinate x, thus justifying its notation.
Chapter 2
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To explore the evolution of this wave packet in time, we could try to solve Eq. (1) with the initial condition (16) directly, but in the spirit of the discussion in Sec. 1.5, it is easier to proceed differently.
Let us first represent the initial wavefunction (16) as a sum (1.67) of the eigenfunctions k( x) of the corresponding stationary 1D Schrödinger equation (1.60), in our current case
2
2
2
2
d k
k
E ,
with E
,
(2.18)
2
2
m dx
k
k
k
2 m
which are simply monochromatic waves,
ikx
a e .
(2.19)
k
k
Since (as was discussed in Sec. 1.7) at the unconstrained motion the spectrum of possible wave numbers k is continuous, the sum (1.67) should be replaced with an integral:4
( x,0) a eikxdk
.
(2.20)
k
Now let us notice that from the point of view of mathematics, Eq. (20) is just the usual Fourier transform from the variable k to the “conjugate” variable x, and we can use the well-known formula of the reciprocal Fourier transform to write
2
1
ikx
1
1
x
~
~
a
( x,0) e
dx
exp -
ikx dx,
where k k k . (2.21)
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 by dx’ d[ x + 2 i( x)2 k ]. As a result, we get:5
1
1
x'
a
k x
k
dx
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 that 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
4 For the notation brevity, from this point on the infinite limit signs will be dropped in all 1D integrals.
5 The fact that the argument’s shift is imaginary is not important. Indeed, since the function under the integral tends to zero at Re x’ Re x , the difference between infinite integrals of this function along axes of x and x’ is equal to its contour integral around the rectangular area x Imz x’. Since the function is also analytical, it obeys the Cauchy theorem MA Eq. (15.1), which says that this contour integral equals zero.
Chapter 2
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k 1
2 x .
(2.24)
Thus we may represent the initial wave packet (16) as
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 case6
Gaussian
wave
1 1/ 2
1
( k k )2
2
0
ikx
k
packet:
( 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 2.2
t 0
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
t 20
v
v
Re
0
0
0
1
10
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
x / x
6 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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The plots clearly show the following effects:
(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, which 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.7 Indeed, let us consider a 1D
wave packet of the type (26), but more general:
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 both derivatives have to be evaluated at the point k = k 0. In this approximation,8 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
dk
2
2
dk
(2.29)
d
d
k x t k x
t
k t
0
0
~
1 2 ~2 ,
dk 2
2
dk
so that 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 neglect the last term in square brackets (which is much smaller than the first term if the dispersion relation is smooth enough and/or the time interval t is sufficiently small), 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 e kixdk
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:
7 See, e.g., brief discussions in CM Sec. 6.3 and EM Sec. 7.2.
8 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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Group
d
and phase
v
,
v
.
(2.33a)
gr
velocities
ph
k k
k k
dk
0
k
0
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 :-) 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
