EM: Classical Electrodynamics
1/ 2
2
Ex
i u
u
tan
()() 2
u
1 1/ 2
1
,
(10.131a)
2
E
y
v ()
so that
v()
Cherenkov
cos
1.
(10.131b) radiation:
u
angle
Remarkably, this direction does not depend on the emission time t ret, so that the radiation of frequency , at each instant, forms a hollow cone led by the particle. This simple result allows an evident interpretation (Fig. 14b): the cone’s interior is just the set of all observation points that have already been reached by the radiation, propagating with the speed v() < u, emitted from all previous points of the particle’s trajectory by the given time t. This phenomenon is an analog of the so-called Mach cone in fluid dynamics,50 besides that in the Cherenkov radiation, there is a separate cone for each frequency (of the range in which v() < u): the smaller is the ()() product, i.e. the higher is the wave velocity v() = 1/[()()]1/2, the broader is the cone, so that the earlier the corresponding
“shock wave” arrives to an observer. Please note that the Cherenkov radiation is a unique radiative phenomenon: it takes place even if a particle moves without acceleration, and (in agreement with our analysis in Sec. 2), is impossible in free space, where v() = c = const is larger than u for any particle.
The Cherenkov radiation’s intensity may be also readily found by plugging the asymptotic expression (127), with imaginary , into Eq. (123). The result is
d E
Z e 2
v 2
Cherenkov
1
d
.
(10.132)
radiation:
dx
2
4 v
( )
u
u
intensity
For non-relativistic particles ( u << c), the Cherenkov radiation condition u > v() is fulfilled only in relatively narrow frequency intervals where the product ()() is very large (usually, due to optical resonance peaks of the electric permittivity – see Fig. 7.5 and its discussion). In this case, the emitted light consists of a few nearly-monochromatic components. On the contrary, if the condition u > v(), i.e.
u 2/()() > 1 is fulfilled in a broad frequency range, as it is for ultra-relativistic particles in condensed media, then the radiated power, according to Eq. (132), is dominated by higher frequencies of the range – hence the famous bluish color of the Cherenkov radiation glow from water-filled nuclear reactors– see Fig. 15.
Fig. 10.15. The Cherenkov
radiation
glow
in
the
Advanced Test Reactor of the
Idaho National Laboratory in
Arco, ID. (Adapted from
http://en.wikipedia.org/wiki/
Cherenkov_radiation
under
the Creative Commons CC-
BY-SA-2.0 license.)
50 Its brief discussion may be found in CM Sec. 8.6.
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Essential Graduate Physics
EM: Classical Electrodynamics
The Cherenkov radiation is broadly used in high-energy experiments for particle identification and speed measurement (since it is easy to pass the particles through layers of different densities and hence with different dielectric constants) – for example, in the so-called Ring Imaging Cherenkov (RICH) detectors that have been designed for the DELPHI experiment51 at the Large Electron-Positron Collider (LEP) in CERN.
A little bit counter-intuitively, the formalism described in this section is also very useful for the description of an apparently rather different effect – the so-called transition radiation that takes place when a charged particle crosses a border between two media.52 The effect may be interpreted as the result of the time dependence of the electric dipole formed by the moving charge q and its mirror image q’ in the counterpart medium – see Fig. 16.
q
q'
Fig. 10.16. The transition radiation’s
d( t)
d( t)
physics.
In the non-relativistic limit, this effect allows a straightforward description combining the electrostatics picture of Sec. 3.4 (see Fig. 3.9 and its discussion), and Eq. (8.27), corrected for the media polarization effects. However, if the particle’s velocity u is comparable with the phase velocity of waves in either medium, the adequate theory of the transition radiation becomes very close to that of the Cherenkov radiation.
In comparison with the Cherenkov radiation, the transition radiation is rather weak, and its practical use (mostly for the measurement of the Lorentz factor , to which the radiation intensity is nearly proportional) requires multi-layered stacks.53 In these systems, the radiation emitted at sequential borders may be coherent, and the system’s physics may become close to that of the free-electron lasers mentioned in Sec. 4.
10.6. Radiation’s back-action
An attentive and critically-minded reader could notice that so far our treatment of charged particle dynamics has never been fully self-consistent. Indeed, in Sec. 9.6 we have analyzed particle’s motion in various external fields, ignoring those radiated by the particle itself, while in Sec. 8.2 and earlier in this chapter these fields have been calculated (admittedly, just for a few simple cases), but, again, their back-action on the emitting particle has been ignored. Only in very few cases we have taken 51 See, e.g., http://delphiwww.cern.ch/offline/physics/delphi-detector.html. For an in-depth review of radiation detectors (including the Cherenkov ones), the reader may be referred, for example, to the classical text by G. F.
Knoll, Radiation Detection and Measurement, 4th ed., Wiley, 2010, and a newer treatment by K. Kleinknecht, Detectors for Particle Radiation, Cambridge U. Press, 1999.
52 The effect was predicted theoretically in 1946 by V. Ginzburg and I. Frank, and only later observed experimentally.
53 See, e.g., Sec. 5.3 in K. Kleinknecht’s monograph cited above.
Chapter 10
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EM: Classical Electrodynamics
the back effects of the radiation implicitly, via the energy conservation arguments. However, even in these cases, the near-field effects, such as the first term in Eq. (19), which affect the moving particle most, have been ignored.
At the same time, it is clear that in sharp contrast with electrostatics, the interaction of a moving point charge with its own field cannot be always ignored. As the simplest example, if an electron is made to fly through a resonant cavity, thus inducing electromagnetic oscillations in it, and then is forced (say, by an appropriate static field) to return into the cavity before the oscillations have decayed, its motion will certainly be affected by the oscillating fields, just as if they had been induced by another source. There is no conceptual problem with applying the Maxwell theory to such “field-particle rendezvous” effects; moreover, it is the basis of the engineering design of such vacuum electron devices as klystrons, magnetrons, and free-electron lasers.
A problem arises only when no clear “rendezvous” points are enforced by boundary conditions, so that the most important self-field effects are at R r – r ’ 0, the most evident example being the charged particle’s radiation into free space, described earlier in this chapter. We already know that such radiation takes away a part of the charge’s kinetic energy, i.e. has to cause its deceleration. One should wonder, however, whether such self-action effects might be described in a more direct, non-perturbative way.
As the first attempt, let us try a phenomenological approach based on the already derived formulas for the radiation power P. For the sake of simplicity, let us consider a non-relativistic point charge q in free space, so that P is described by Eq. (8.27), with the electric dipole moment’s derivative over time equal to qu:
2
2
Z q
2
q
0
2
2
P
u
u .
(10.133)
2
3
6 c
3 c 40
The most naïve approach would be to write the equation of the particle’s motion in the form u
m F F ,
(10.134)
ext
self
and try to calculate the radiation back-action force Fself by requiring its instant power, –Fselfu, to be equal to P. However, this approach (say, for a 1D motion) would give a very unnatural result, u 2
F
,
(10.135)
self
u
that might diverge at some points of the particle’s trajectory. This failure is clearly due to the retardation effect: as the reader may recall, Eq. (133) results from the analysis of radiation fields in the far-field zone, i.e. at large distances R from the particle, e.g., from the second term in Eq. (19), i.e. when the non-radiative first term (which is much larger at small distances, R 0) is ignored.
Before exploring the effects of this term, let us, however, make one more attempt at Eq. (133), considering its average effect on some periodic motion of the particle. (A possible argument for this step is that at the periodic motion, the retardation effects should be averaged out – just as at the transfer from Eq. (8.27) to Eq. (8.28).) To calculate the average, let us write the identity
T
1
2
u
uu dt ,
(10.136)
T 0
Chapter 10
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EM: Classical Electrodynamics
and carry out the integration on the right-hand side of Eq. (133) by parts over the motion period T: 2
2
T
T
2
2
q
2
q
1
1
2
q
P
u2
T
u u
u u dt
u u dt . (10.137)
3
3
3 c 4
3 c 4
0
3 c 4
0
0 T
3
0
T 0
0
On the other hand, the back-action force should give
T
1
P
F u dt .
(10.138)
T
self
0
These two averages coincide if54
2
Abraham-
2
q
F
u .
(10.139)
Lorentz
self
3
3 c 4
force
0
This is the so-called Abraham-Lorentz force of back-action. Before going after a more serious derivation of this formula, let us estimate its scale, representing Eq. (139) as
2
2
F
q
m u,
with
,
(10.140)
self
3
3 mc 40
where the constant evidently has the dimension of time. Recalling the definition (8.41) of the classical radius r c of the particle, Eq. (140) for may be rewritten as
2 r
c
.
(10.141)
3 c
For the electron, is of the order of 10-23 s, so that the right-hand side of Eq. (140) is very small. This means that in most cases the Abrahams-Lorentz force is either negligible or leads to the same results as the perturbative treatments of energy loss we have used earlier in this chapter.
However, Eq. (140) brings some unpleasant surprises. For example, let us consider a 1D
oscillator with frequency 0. For it, Eq. (134), with the back-action force given by Eq. (140), takes the form
x
m m 2 x m x .
(10.142)
0
Looking for the solution of this linear differential equation in the usual exponential form, x( t)
exp{ t}, we get the following characteristic equation,
2
2
3
.
(10.143)
0
It may look like that for any “reasonable” value of 0 << 1/ ~ 1023 s-1, the right-hand side of this nonlinear algebraic equation may be treated as a perturbation. Indeed, looking for its solutions in the 54 Just for the reader’s reference, this formula may be readily generalized to the relativistic case, in the 4-form: 2
q 2 2
d p
p
dp
dp
F
,
self
3
2
2
mc
3
4
(
)
0
d
mc
d
d
the so-called Abraham-Lorentz-Dirac force.
Chapter 10
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EM: Classical Electrodynamics
natural form = i0 + ’, with ’ << 0, expanding both parts of Eq. (143) in the Taylor series in the small parameter ’, and keeping only the terms linear in ’, we get
2
0
'
.
(10.144)
2
This means that the energy of free oscillations decreases in time as exp{2 ’t} = exp{- 2
0 t}; this is
exactly the radiative damping analyzed earlier. However, Eq. (143) is deceiving; it has the third root corresponding to unphysical, exponentially growing (so-called run-away) solutions. It is easiest to see this for a free particle, with 0 = 0. Then Eq. (143) becomes very simple,
2
3
,
(10.145)
and it is easy to find all its three roots explicitly: 1 = 2 = 0 and 3 = 1/. While the first two roots correspond to the values found earlier, the last one describes an exponential (and extremely rapid!) acceleration.
In order to remove this artifact, let us try to develop a self-consistent approach to the back-action effects, taking into account the near-field terms of particle fields. For that, we need to somehow overcome the divergence of Eqs. (10) and (19) at R 0. The most reasonable way to do this is to spread the particle’s charge over a ball of radius a, with a spherically symmetric (but not necessarily constant) density ( r), and at the end of the calculations trace the limit a 0.55 Again sticking to the non-relativistic case (so that the magnetic component of the Lorentz force is not important), we should calculate
F
(r)E(r, )
,
(10.146)
self
t d 3 r
V
where the electric field is that of the charge itself, with the field of any elementary charge dq = ( r) d 3 r described by Eq. (19).
To enable an analytical calculation of the force, we need to make the assumption a << r c, treat the ratio R/ r c ~ a/ r c as a small parameter, and expand the resulting right-hand side of Eq. (146) into the Taylor series in small R. This procedure yields
2 1
( )
1 n
n 1
d u
3
3
F
d r d r' ( r) n 1
R ( r' ) .
(10.147)
self
3 4
n
2
n c
n dt n
0
0
!
1
V
V
The distance R cancels only in the term with n = 1,
2
u
2
2
q
F
3
3
d r d r' ( r)( r' )
u
1
,
(10.148)
3
3
3 c 4
3 c 4
0 V
V
0
showing that we have recovered (now in an apparently legitimate fashion) Eq. (139) for the Abrahams-Lorentz force. One could argue that in the limit a 0 the terms higher in R ~ a (with n > 1) could be 55 Note: this operation cannot be interpreted as describing a quantum spread due to the finite extent of the point particle’s wavefunction. In quantum mechanics, different parts of the wavefunction of the same charged particle do not interact with each other!
Chapter 10
Page 36 of 40
Essential Graduate Physics
EM: Classical Electrodynamics
ignored. However, we have to notice that the main contribution to the series (147) is not described by Eq. (148) for n = 1, but is given by the much larger term with n = 0:
2 1
u
3
3
( r) ( r' )
4 u 1 1
3
3
( r) ( r' )
4
F
d r d r'
d r d r'
u , (10.149)
0
2
2
2 U
3 4 c
R
3 c 4 2
R
c
3
0
V
V
0
V
V
where U is the electrostatic energy (1.59) of the static charge’s self–interaction. This term may be interpreted as the inertial “force” 56 (– m efa) with the following effective electromagnetic mass: Electro-4 U
magnetic
m
,
(10.150)
ef
2
mass
3 c
which is a factor of 4/3 larger than it should be according to Einstein’s formula (9.73). This is the famous (or rather infamous :-) 4/3 problem that does not allow one to interpret the electron’s mass as that of its electric field. Some (admittedly, rather formal) resolution of this paradox is possible only in quantum electrodynamics with its renormalization techniques – beyond the framework of this course.
Note, however, that all these issues are only important for motions with frequencies of the order of 1/ ~ 1023 s-1, i.e. at energies E ~ / ~ 108 eV, while other quantum electrodynamics effects may be observed at much lower frequencies, starting from ~1010 s-1. Hence the 4/3 problem is by no means the only motivation for the transfer from classical to quantum electrodynamics. However, the reader should not think that their time spent on this course has been lost: quantum electrodynamics it heavily based on classical electrodynamics, incorporates virtually all its results, and the basic transition between them is surprisingly straightforward.57 So, I look forward to welcoming the reader to the next, quantum-mechanics part of this series.
10.6. Exercise problems
10.1. Derive Eqs. (10) from Eqs. (1) by a direct (but careful!) integration.
10.2. Derive the radiation-related parts of Eqs. (19)-(20) from the Liénard-Wiechert potentials (10) by direct differentiation.
10.3. A point charge q that had been in a stationary position on a circle of
q
radius R is carried over, along the circle, to the opposite position on the same R
diameter (see the figure on the right) as fast as only physically possible, and then is
kept steady at this new position. Calculate and sketch the time dependence of its
electric field E at the center of the circle.
10.4. Express the instantaneous power of electromagnetic radiation by a relativistic particle with electric charge q and rest mass m, moving with velocity u, via the external Lorentz force F exerted on it.
56 See, e.g., CM Sec. 4.6.
57 See, e.g., QM Secs. 9.1-9.4.
Chapter 10
Page 37 of 40
EM: Classical Electrodynamics
10.5. A relativistic particle with rest mass m and electric charge q, initially at rest, is accelerated by a constant force F until it reaches a certain velocity u, and then moves by inertia. Calculate the total energy radiated during the acceleration.
10.6. A charged relativistic particle with initial momentum p0 flies ballistically from a free-space region into a region of a constant, uniform electric field E, whose force is directed opposite to p0.
Calculate the energy radiated by the particle during its motion in the field, assuming that it is small in comparison with the particle’s initial kinetic energy.
10.7. Calculate
(i) the instantaneous power, and
(ii) the power spectrum
of the radiation emitted, into a unit solid angle, by a relativistic particle with charge q, performing 1D
harmonic oscillations with frequency 0 and displacement amplitude a.
10.8. Analyze the polarization and the spectral contents of the synchrotron radiation propagating in the direction normal to the particle’s rotation plane. How do the results change if not one, but N > 1
similar particles move around the circle, at equal angular distances?
10.9. Calculate and analyze the time dependence of the energy of a charged relativistic particle performing synchrotron motion in a constant and uniform magnetic field B, and hence emitting the synchrotron radiation. Qualitatively, what is the particle’s trajectory?
Hint: You may assume that the energy loss is relatively slow (– d E/ dt << cE), but should spell out the condition of validity of this assumption.
10.10. Analyze the polarization of the synchrotron radiation propagating within the particle’s rotation plane.
10.11.* The basic quantum theory of radiation shows that the electric dipole radiation by a particle is allowed only if the change of its angular momentum’s magnitude L at the transition is of the order of Planck’s constant .
(i) Estimate the change of L of an ultra-relativistic particle due to its emission of a typical single photon of the synchrotron radiation.
(ii) Do you think quantum mechanics forbid such radiation? If not, why?
10.12. A relativistic particle moves along the z-axis , with velocity uz, through an undulator – a system of permanent magnets providing (in the simplest model) a perpendicular magnetic field, whose distribution near the axis is sinusoidal:58
B n B cos k z .
y
0
0
58 As the Maxwell equation for H shows, this field distribution cannot be created in any non-zero volume of free space. However, it may be created on a line – e.g., on the particle’s trajectory.
Chapter 10
Page 38 of 40
EM: Classical Electrodynamics
Assuming that the field is so weak that it causes negligible deviations of the particle’s trajectory from the straight line, calculate the angular distribution of the resulting radiation. What condition does the above assumption impose on the system’s parameters?
10.13. Discuss possible effects of the interference of the undulator radiation from different periods of its static field distribution. In particular, calculate the angular positions of the power density maxima.
10.14. An electron launched directly toward a plane surface of a perfect conductor is instantly absorbed by it at the collision. Calculate the angular distribution and the frequency spectrum of the electromagnetic waves radiated at this collision if the initial kinetic energy T of the particle is much larger than the conductor’s workfunction .59 Is your result valid near the conductor’s surface?
10.15. A relativistic particle, with rest mass m and electric charge q, flies ballistically, with velocity u, by an immobile point charge q’, with an impact parameter b so large that the deviations of its trajectory from the straight line are negligible. Calculate the total energy loss due to the electromagnetic radiation during the passage. Formulate the conditions of validity of your result.
59 See Sec. 2.9, in particular Fig. 2.27a.
Chapter 10
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EM: Classical Electrodynamics
This page is
intentionally left
blank
Chapter 10
Page 40 of 40
Konstantin K. Likharev
Essential Graduate Physics
Lecture Notes and Problems
Beta version
Open online access at
http://commons.library.stonybrook.edu/egp/
and
https://sites.google.com/site/likharevegp/
Part QM:
Quantum Mechanics
Last corrections: September 2, 2022
A version of this material was published in 2019 under the title
Quantum Mechanics: Lecture notes
IOPP, Essential Advanced Physics – Volume 5, ISBN 978-0-7503-1411-4,
with the model solutions of the exercise problems published under the title
Quantum Mechanics: Problems with solutions
IOPP, Essential Advanced Physics – Volume 6, ISBN 978-0-7503-1414-5
However, by now this online version of the lecture notes, as well as the problem solutions available from the author, have been much better corrected
About the author:
https://you.stonybrook.edu/likharev/
© K. Likharev
QM: Quantum Mechanics
Table of Contents
Chapter 1. Introduction (28 pp.)
1.1. Experimental motivations
1.2. Wave mechanics postulates
1.3. Postulates’ discussion
1.4. Continuity equation
1.5. Eigenstates and eigenvalues
1.6. Time evolution
1.7. Space dependence
1.8. Dimensionality reduction
1.9. Exercise problems (15)
Chapter 2. 1D wave mechanics (76 pp.)
2.1. Basic relations
2.2. Free particle: Wave packets
2.3. Particle reflection and tunneling
2.4. Motion in soft potentials
2.5. Resonant tunneling, and metastable states
2.6. Localized state coupling, and quantum oscillations
2.7. Periodic systems: Energy bands and gaps
2.8. Periodic systems: Particle dynamics
2.9. Harmonic oscillator: Brute force approach
2.10. Exercise problems (43)
Chapter 3. Higher Dimensionality Effects (64 pp.)
3.1. Quantum interference and the AB effect
3.2. Landau levels and the quantum Hall effect
3.3. Scattering and diffraction
3.4. Energy bands in higher dimensions
3.5. Axially-symmetric systems
3.6. Spherically-symmetric systems: Brute force approach
3.7. Atoms
3.8. Spherically-symmetric scatterers
3.9. Exercise problems (40)
Chapter 4. Bra-ket Formalism (52 pp.)
4.1. Motivation
4.2. States, state vectors, and linear operators
4.3. State basis and matrix representation
4.4. Change of basis, and matrix diagonalization
4.5. Observables: Expectation values and uncertainties
4.6. Quantum dynamics: Three pictures
4.7. Coordinate and momentum representations
4.8. Exercise problems (34)
Table of Contents
Page 2 of 4
QM: Quantum Mechanics
Chapter 5. Some Exactly Solvable Problems (48 pp.)
5.1. Two-level systems
5.2. The Ehrenfest theorem
5.3. The Feynman path integral
5.4. Revisiting harmonic oscillator
5.5. Glauber states and squeezed states
5.6. Revisiting spherically-symmetric problems
5.7. Spin and its addition to orbital angular momentum
5.8. Exercise problems (48)
Chapter 6. Perturbative Approaches (36 pp.)
6.1. Time-independent perturbations
6.2. The linear Stark effect
6.3. Fine structure of atomic levels
6.4. The Zeeman effect
6.5. Time-dependent perturbations
6.6. Quantum-mechanical Golden Rule
6.7. Golden Rule for step-like perturbations
6.8. Exercise problems (31)
Chapter 7. Open Quantum Systems (50 pp.)
7.1. Open systems, and the density matrix
7.2. Coordinate representation, and the Wigner function
7.3. Open system dynamics: Dephasing
7.4. Fluctuation-dissipation theorem
7.5. The Heisenberg-Langevin approach
7.6. Density matrix approach
7.7. Exercise problems (14)
Chapter 8. Multiparticle Systems (52 pp.)
8.1. Distinguishable and indistinguishable particles
8.2. Singlets, triplets, and the exchange interaction
8.3. Multiparticle systems
8.4. Perturbative approaches
8.5. Quantum computation and cryptography
8.6. Exercise problems (31)
Chapter 9. Elements of Relativistic Quantum Mechanics (36 pp.)
9.1. Electromagnetic field quantization
9.2. Photon absorption and counting
9.3. Photon emission: spontaneous and stimulated
9.4. Cavity QED
9.5. The Klien-Gordon and relativistic Schrödinger equations
9.6. Dirac’s theory
9.7. Low energy limit
9.8. Exercise problems (21)
Table of Contents
Page 3 of 4
QM: Quantum Mechanics
Chapter 10. Making Sense of Quantum Mechanics (16 pp.)
10.1. Quantum measurements
10.2. QND measurements
10.3. Hidden variables, the Bell theorem, and local reality
10.4. Interpretations of quantum mechanics
* * *
Additional file (available from the author upon request):
Exercise and Test Problems with Model Solutions (277 + 70 = 347 problems; 518 pp.) Table of Contents
Page 4 of 4
QM: Quantum Mechanics
Chapter 1. Introduction
This introductory chapter briefly reviews the major experimental motivations for quantum mechanics, and then discusses its simplest formalism – Schrödinger’s wave mechanics. Much of this material (besides the last section) may be found in undergraduate textbooks, 1 so that the discussion is rather brief, and focused on the most important conceptual issues.
1.1. Experimental motivations
By the beginning of the 1900s, physics (which by that time included what we now call non-relativistic classical mechanics, classical thermodynamics and statistics, and classical electrodynamics including the geometric and wave optics) looked like an almost completed discipline, with most human-scale phenomena reasonably explained, and just a couple of mysterious “dark clouds”2 on the horizon.
However, rapid technological progress and the resulting development of more refined scientific instruments have led to a fast multiplication of observed phenomena that could not be explained on a classical basis. Let me list the most consequential of those experimental findings.
(i) The blackbody radiation measurements, pioneered by G. Kirchhoff in 1859, have shown that in the thermal equilibrium, the power of electromagnetic radiation by a fully absorbing (“black”) surface, per unit frequency interval, drops exponentially at high frequencies. This is not what could be expected from the combination of classical electrodynamics and statistics, which predicted an infinite growth of the radiation density with frequency. Indeed, classical electrodynamics shows3 that electromagnetic field modes evolve in time just as harmonic oscillators, and that the number dN of these modes in a large free-space volume V >> 3, within a small frequency interval d << near some frequency , is
3
d k
4 2
2
k dk
dN 2 V
V
V
d
(1.1)
2
2 3
,
2 3
2 3
c
where c 3×108 m/s is the free-space speed of light, k = / c the free-space wave number, and = 2/ k is the radiation wavelength. On the other hand, classical statistics4 predicts that in the thermal equilibrium at temperature T, the average energy E of each 1D harmonic oscillator should be equal to k B T, where k B is the Boltzmann constant.5
Combining these two results, we readily get the so-called Rayleigh-Jeans formula for the average electromagnetic wave energy per unit volume:
1 See, for example, D. Griffith, Quantum Mechanics, 2nd ed., Cambridge U. Press, 2016.
2 This famous expression was used in a 1900 talk by Lord Kelvin (born William Thomson), in reference to the results of blackbody radiation measurements and the Michelson-Morley experiments, i.e. the precursors of quantum mechanics and relativity theory.
3 See, e.g., EM Sec. 7.8, in particular Eq. (7.211).
4 See, e.g., SM Sec. 2.2.
5 In the SI units, used throughout this series, k B 1.38×10-23 J/K – see Appendix CA: Selected Physical Constants for the exact value.
© K. Likharev
QM: Quantum Mechanics
1 dE
k T dN
2
u
B
k T ,
(1.2)
V d
V d
2
c 3 B
that diverges at (Fig. 1) – the so-called ultraviolet catastrophe. On the other hand, the blackbody radiation measurements, improved by O. Lummer and E. Pringsheim, and also by H. Rubens and F.
Kurlbaum to reach a 1%-scale accuracy, were compatible with the phenomenological law suggested in 1900 by Max Planck:
2
u
.
(1.3a) Planck’s
2 3
c
exp
radiation
/ k T
B
1
law
This law may be reconciled with the fundamental Eq. (1) if the following replacement is made for the average energy of each field oscillator:
k T
,
(1.3b)
B
exp(
/ k T ) 1
B
with a factor
055
.
1
10-34 J
s ,
(1.4)
Planck’s
constant
now called the Planck’s constant.6 At low frequencies ( << k B T), the denominator in Eq. (3) may be approximated as / k B T, so that the average energy (3b) tends to its classical value k B T, and the Planck’s law (3a) reduces to the Rayleigh-Jeans formula (2). However, at higher frequencies ( >> k B T), Eq. (3) describes the experimentally observed rapid decrease of the radiation density – see Fig. 1.
10
1
u
u 0
0.1
Fig. 1.1. The blackbody radiation density u, in units
of u 0 ( k B T)3/22 c 3, as a function of frequency, 0.01
according to the Rayleigh-Jeans formula (blue line)
0.1
1
10
and the Planck’s law (red line).
/ k B T
(ii) The photoelectric effect, discovered in 1887 by H. Hertz, shows a sharp lower boundary for the frequency of the incident light that may kick electrons out from metallic surfaces, independent of the light’s intensity. Albert Einstein, in one of his three famous 1905 papers, noticed that this threshold min could be explained assuming that light consisted of certain particles (now called photons) with energy 6 Max Planck himself wrote as h, where = /2 is the “cyclic” frequency (the number of periods per second) so that in early texts on quantum mechanics the term “Planck’s constant” referred to h 2, while was called “the Dirac constant” for a while. I will use the contemporary terminology, and abstain from using the “old Planck’s constant” h at all, to avoid confusion.
Chapter 1
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Energy
vs
E ,
(1.5)
frequency
with the same Planck’s constant that participates in Eq. (3).7 Indeed, with this assumption, at the photon absorption by an electron, its energy E = is divided between a fixed energy U 0 (nowadays called the workfunction) of electron’s binding inside the metal, and the excess kinetic energy m e v 2/2 > 0 of the freed electron – see Fig. 2. In this picture, the frequency threshold finds a natural explanation as min=
U 0/.8 Moreover, as was shown by Satyendra Nath Bose in 1924, 9 Eq. (5) explains Planck’s law (3).
2
m v
e
E U 0
v
2
E
q e
Fig. 1.2. Einstein’s explanation of the
m m
photoelectric effect’s frequency threshold.
e
(iii) The discrete frequency spectra of the electromagnetic radiation by excited atomic gases could not be explained by classical physics. (Applied to the planetary model of atoms, proposed by Ernst Rutherford, classical electrodynamics predicts the collapse of electrons on nuclei in ~10-10s, due to the electric-dipole radiation of electromagnetic waves.10) Especially challenging was the observation by Johann Jacob Balmer (in 1885) that the radiation frequencies of simple atoms may be well described by simple formulas. For example, for the lightest, hydrogen atom, all radiation frequencies may be numbered with just two positive integers n and n’:
1
1
,
(1.6)
n, n'
0
2
2
n
n'
with 0 1, 2.071016 s-1. This observation, and the experimental value of 0, have found its first explanation in the famous 1913 theory by Niels Henrik David Bohr, which was a phenomenological precursor of the present-day quantum mechanics. In this theory, n,n’ was interpreted as the frequency of a photon that obeys Einstein’s formula (5), with its energy En,n’ = n.n’ being the difference between two quantized (discrete) energy levels of the atom (Fig. 3):
E
E E 0 .
(1.7)
n, n'
n'
n
Bohr showed that Eq. (6) may be obtained from Eqs. (5) and (7), and non-relativistic11 classical mechanics, augmented with just one additional postulate, equivalent to the assumption that the angular 7 As a reminder, A. Einstein received his only Nobel Prize (in 1921) for exactly this work, rather than for his relativity theory, i.e. essentially for jumpstarting the same quantum theory which he later questioned.
8 For most metals, U 0 is between 4 and 5 electron-volts (eV), so that the threshold corresponds to max = 2 c/min
= 2 c/( U 0/) 300 nm – approximately at the border between the visible light and the ultraviolet radiation.
9 See, e.g., SM Sec. 2.5.
10 See, e.g., EM Sec. 8.2.
11 The non-relativistic approach to the problem may be justified a posteriori by the fact the resulting energy scale E H, given by Eq. (13), is much smaller than the electron’s rest energy, m e c 2 0.5 MeV.
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momentum L = m e vr of an electron moving with velocity v on a circular orbit of radius r about the hydrogen’s nucleus (the proton, assumed to be at rest because of its much higher mass), is quantized as Angular
L n ,
(1.8) momentum
quantization
where
is again the same Planck’s constant (4), and n is an integer. (In Bohr’s theory, n could not be equal to zero, though in the genuine quantum mechanics, it can.)
E
n'
E E
n, n'
n'
n
Fig. 1.3. The electromagnetic radiation
of a system at a result of the transition
between its quantized energy levels.
En
Indeed, it is sufficient to solve Eq. (8), m e vr = n, together with the equation 2
2
v
e
m
,
(1.9)
e r
4
2
r
0
which expresses the 2nd Newton’s law for an electron rotating in the Coulomb field of the nucleus. (Here e 1.610-19C is the fundamental electric charge, and m e 0.9110-30 kg is the electron’s rest mass.) The result for r is
2
m
2
/
r n r ,
where
e
r
0529
.
0
nm .
(1.10) Bohr
B
B
2
e / 4
radius
0
The constant r B, called the Bohr radius, is the most important spatial scale of phenomena in atomic, molecular, and condensed-matter physics – and hence in all chemistry and biochemistry.
Now plugging these results into the non-relativistic expression for the full electron energy (with its rest energy taken for reference),
m v 2
e
e 2
E
,
(1.11)
2
4 r
0
we get the following simple expression for the electron’s energy levels:
Hydrogen
E
H
E
0 ,
(1.12) atom’s
n
energy
2 2
n
levels
which, together with Eqs. (5) and (7), immediately gives Eq. (6) for the radiation frequencies. Here E H is called the so-called Hartree energy constant (or just the “Hartree energy”)12
2 e /4
Hartree
0 2
E
360
.
4
10 18
J
eV
21
.
27
.
(1.13a) energy
H
2
/ m
constant
e
(Note the useful relations, which follow from Eqs. (10) and (13a):
12 Unfortunately, another name, the “Rydberg constant”, is sometimes used for either this energy unit or its half, E H/2 13.6 eV. To add to the confusion, the same term “Rydberg constant” is used in some sub-fields of physics for the reciprocal free-space wavelength (1/0 = 0/2 c) corresponding to the frequency 0 = E H/2.
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1/ 2
2
2
2
2
e
e / 4
/ m
0
e
E
,
i.e. r
;
(1.13b)
H
2
B
4
r
m r
E
E
0 B
e B
H
H
the first of them shows, in particular, that r B is the distance at which the natural scales of the electron’s potential and kinetic energies are equal.)
Note also that Eq. (8), in the form pr = n, where p = m e v is the electron momentum’s magnitude, may be rewritten as the condition than an integer number ( n) of wavelengths of certain (before the late 1920s, hypothetic) waves13 fits the circular orbit’s perimeter: 2 r 2 n/ p = n.
Dividing both parts of the last equality by n, we see that for this statement to be true, the wave number k
2/ of the de Broglie waves should be proportional to the electron’s momentum p = mv: Momentum
vs wave
p k
,
(1.14)
number
again with the same Planck’s constant as in Eq. (5).
(iv) The Compton effect14 is the reduction of frequency of X-rays at their scattering on free (or nearly-free) electrons – see Fig. 4.
' / c
/ c
m
e
Fig. 1.4. The Compton effect.
p
The effect may be explained assuming that the X-ray photon also has a momentum that obeys the vector-generalized version of Eq. (14):
p
,
(1.15)
photon
k
n
c
where k is the wavevector (whose magnitude is equal to the wave number k, and whose direction coincides with the unit vector n directed along the wave propagation15), and that the momenta of both the photon and the electron are related to their energies E by the classical relativistic formula16
2
E cp2 mc 2
2
.
(1.16)
(For a photon, the rest energy m is zero, and this relation is reduced to Eq. (5): E = cp = c k = .) Indeed, a straightforward solution of the following system of three equations,
13 This fact was first noticed and discussed in 1924 by Louis Victor Pierre Raymond de Broglie (in his PhD
thesis!), so that instead of speaking of wavefunctions, we are still frequently speaking of the de Broglie waves, especially when free particles are discussed.
14 This effect was observed in 1922, and explained a year later by Arthur Holly Compton, using Eqs. (5) and (15).
15 See, e.g., EM Sec. 7.1.
16 See, e.g., EM Sec. 9.3, in particular Eq. (9.78).
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2
m c '
cp m c
(1.17)
e
( )2 ( 2)2
e
1/2,
'
cos p cos ,
(1.18)
c
c
'
0
sin p sin ,
(1.19)
c
(which express the conservation of, respectively, the full energy of the system and the two relevant Cartesian components of its full momentum, at the scattering event – see Fig. 4), yields the result: 1
1
1
1
( cos ) ,
(1.20a)
2
'
m c
e
which is traditionally represented as the relation between the initial and final values of the photon’s wavelength = 2/ k = 2/(/ c): 17
2
2
'
1
( cos ) 1
( cos ), with
,
(1.20b) Compton
m c
C
C
m c
effect
e
e
and is in agreement with experiment.
(v) De Broglie wave diffraction. In 1927, Clinton Joseph Davisson and Lester Germer, and independently George Paget Thomson succeeded to observe the diffraction of electrons on solid crystals (Fig. 5). Specifically, they have found that the intensity of the elastic reflection of electrons from a crystal increases sharply when the angle between the incident beam of electrons and the crystal’s atomic planes, separated by distance d, satisfies the following relation:
2 d sin n ,
(1.21) Bragg
condition
where = 2/ k = 2/ p is the de Broglie wavelength of the electrons, and n is an integer. As Fig. 5
shows, this is just the well-known condition18 that the path difference l = 2 d sin between the de Broglie waves reflected from two adjacent crystal planes coincides with an integer number of , i.e. of the constructive interference of the waves.19
To summarize, all the listed experimental observations could be explained starting from two very simple (and similarly looking) formulas: Eq. (5) (at that stage, for photons only), and Eq. (15) for both photons and electrons – both relations involving the same Planck’s constant. This fact might give an impression of experimental evidence sufficient to declare the light consisting of discrete particles 17 The constant C that participates in this relation, is close to 2.4610-12 m, and is called the electron’s Compton wavelength. This term is somewhat misleading: as the reader can see from Eqs. (17)-(19), no wave in the Compton problem has such a wavelength – either before or after the scattering.
18 See, e.g., EM Sec. 8.4, in particular Fig. 8.9 and Eq. (8.82). Frequently, Eq. (21) is called the Bragg condition, due to the pioneering experiments by W. Bragg on X-ray scattering from crystals, which were started in 1912.
19 Later, spectacular experiments on diffraction and interference of heavier particles (with the correspondingly smaller de Broglie wavelength), e.g., neutrons and even C60 molecules, have also been performed – see, e.g., the review by A. Zeilinger et al., Rev. Mod. Phys. 60, 1067 (1988) and a later publication by O. Nairz et al., Am. J.
Phys. 71, 319 (2003). Nowadays, such interference of heavy particles is used, for example, for ultrasensitive measurements of gravity – see, e.g., a popular review by M. Arndt, Phys. Today 67, 30 (May 2014), and more recent experiments by S. Abend et al., Phys. Rev. Lett. 117, 203003 (2016).
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(photons), and, on the contrary, electrons being some “matter waves” rather than particles. However, by that time (the mid-1920s), physics has accumulated overwhelming evidence of wave properties of light, such as interference and diffraction.20 In addition, there was also strong evidence for the lumped-particle (“corpuscular”) behavior of electrons. It is sufficient to mention the famous oil-drop experiments by Robert Andrew Millikan and Harvey Fletcher (1909-1913), in which only single (and whole!) electrons could be added to an oil drop, changing its total electric charge by multiples of electron’s charge (- e) –
and never its fraction. It was apparently impossible to reconcile these observations with a purely wave picture, in which an electron and hence its charge need to be spread over the wave’s extension, so that its arbitrary part of it could be cut off using an appropriate experimental setup.
d
Fig. 1.5. The De Broglie wave interference
d sin
d sin
at electron scattering from a crystal lattice.
Thus the founding fathers of quantum mechanics faced a formidable task of reconciling the wave and corpuscular properties of electrons and photons – and other particles. The decisive breakthrough in that task has been achieved in 1926 by Ervin Schrödinger and Max Born, who formulated what is now known either formally as the Schrödinger picture of non-relativistic quantum mechanics of the orbital motion 21 in the coordinate representation (this term will be explained later in the course), or informally just as the wave mechanics. I will now formulate the main postulates of this theory.
1.2. Wave mechanics postulates
Let us consider a spinless,22 non-relativistic point-like particle, whose classical dynamics may be described by a certain Hamiltonian function H(r, p, t),23 where r is the particle’s radius-vector and p is its momentum. (This condition is important because it excludes from our current discussion the systems whose interaction with their environment results in irreversible effects, in particular the friction leading to particle energy’s decay. Such “open” systems need a more general description, which will be discussed in Chapter 7.) Wave mechanics of such Hamiltonian particles may be based on the following set of postulates that are comfortingly elegant – though their final justification is given only by the agreement of all their corollaries with experiment.24
20 See, e.g., EM Sec. 8.4.
21 The orbital motion is the historic (and rather misleading) term used for any motion of the particle as a whole.
22 Actually, in wave mechanics, the spin of the described particle has not to be equal to zero. Rather, it is assumed that the particle spin’s effects on its orbital motion are negligible.
23 As a reminder, for many systems (including those whose kinetic energy is a quadratic-homogeneous function of generalized velocities, like mv 2/2), H coincides with the total energy E – see, e.g., CM Sec. 2.3. In what follows, I will assume that H = E.
24 Quantum mechanics, like any theory, may be built on different sets of postulates/axioms leading to the same final conclusions. In this text, I will not try to beat down the number of postulates to the absolute possible Chapter 1
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(i) Wavefunction and probability. Such variables as r or p cannot be always measured exactly, even at “perfect conditions” when all external uncertainties, including measurement instrument imperfection, varieties of the initial state preparation, and unintended particle interactions with its environment, have been removed.25 Moreover, r and p of the same particle can never be measured exactly simultaneously. Instead, the most detailed description of the particle’s state allowed by Nature, is given by a certain complex function (r, t), called the wavefunction (or “wave function”), which generally enables only probabilistic predictions of the measured values of r, p, and other directly measurable variables – in quantum mechanics, usually called observables.
Specifically, the probability dW of finding a particle inside an elementary volume dV d 3 r is proportional to this volume, and hence may be characterized by a volume-independent probability density w dW/ d 3 r, which in turn is related to the wavefunction as w (r, t) 2
*
(r, t)(r, t) ,
(1.22a)
Probability
where the sign * denotes the usual complex conjugation. As a result, the total probability of finding the via particle somewhere inside a volume V may be calculated as
wavefunction
W wd 3 r * d 3 r .
(1.22b)
V
V
In particular, if volume V contains the particle definitely (i.e. with the 100% probability, W = 1), Eq.
(22b) is reduced to the so-called wavefunction normalization condition
*
3
d r 1
.
(1.22c) Normalization
condition
V
(ii) Observables and operators. With each observable A, quantum mechanics associates a certain linear operator A ˆ , such that (in the perfect conditions mentioned above) the average measured value of A (usually called the expectation value) is expressed as26
Observable’s
A *
A d 3
ˆ
r ,
(1.23) expectation
V
value
where … means the statistical average, i.e. the result of averaging the measurement results over a large ensemble (set) of macroscopically similar experiments, and is the normalized wavefunction that obeys Eq. (22c). Note immediately that for Eqs. (22) and (23) to be compatible, the identity (or “unit”) operator defined by the relation
I
ˆ ,
(1.24) Identity
operator
has to be associated with a particular type of measurement, namely with the particle’s detection.
minimum, not only because that would require longer argumentation, but chiefly because such attempts typically result in making certain implicit assumptions hidden from the reader – the practice as common as regrettable.
25 I will imply such perfect conditions in the further narrative, until the discussion of the system’s interaction with its environment in Chapter 7.
26 This key measurement postulate is sometimes called the Born rule, though sometimes this term is used for the (less general) Eqs. (22).
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(iii) The Hamiltonian operator and the Schrödinger equation. Another particular operator, Hamiltonian H ˆ , whose observable is the particle’s energy E, also plays in wave mechanics a very special role, because it participates in the Schrödinger equation,
Schrödinger
equation
i
ˆ
H ,
(1.25)
t
that determines the wavefunction’s dynamics, i.e. its time evolution.
(iv) The radius-vector and momentum operators. In wave mechanics (in the “coordinate representation”), the vector operator of particle’s radius-vector r just multiples the wavefunction by this vector, while the operator of the particle’s momentum is proportional to the spatial derivative: rˆ ,
r
pˆ i
,
(1.26a)
Operators of
coordinate and where is the del (or “nabla”) vector operator.27 Thus in the Cartesian coordinates, momentum
ˆr r x, y, z ,
ˆp i , , .
(1.26b)
x y z
(v) The correspondence principle. In the limit when quantum effects are insignificant, e.g., when the characteristic scale of action28 (i.e. the product of the relevant energy and time scales of the problem) is much larger than Planck’s constant , all wave mechanics results have to tend to those given by classical mechanics. Mathematically, this correspondence is achieved by duplicating the classical relations between various observables by similar relations between the corresponding operators. For example, for a free particle, the Hamiltonian (which in this particular case corresponds to the kinetic energy T = p 2/2 m alone) has the form
2
2
Free
ˆ p
particle’s
2
ˆ
ˆ
H T
.
(1.27)
Hamiltonian
2 m
2 m
Now, even before a deeper discussion of the postulates’ physics (offered in the next section), we may immediately see that they indeed provide a formal way toward resolution of the apparent contradiction between the wave and corpuscular properties of particles. Indeed, for a free particle, the Schrödinger equation (25), with the substitution of Eq. (27), takes the form
Free
2
particle’s
i
Schrödinger
2 ,
(1.28)
t
2 m
equation
whose particular, but the most important solution is a plane, single-frequency (“monochromatic”) traveling wave,29
Plane
wave
i(kr
t)
(r, t) ae
,
(1.29)
27 If you need, see, e.g., Secs. 8-10 of the Selected Mathematical Formulas appendix – below, referred to as MA.
Note that according to those formulas, the del operator follows all the rules of the usual (geometric) vectors. This is, by definition, true for other quantum-mechanical vector operators to be discussed below.
28 See, e.g., CM Sec. 10.3.
29 See, e.g., CM Sec. 6.4 and/or EM Sec. 7.1.
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where a, k and are constants. Indeed, plugging Eq. (29) into Eq. (28), we immediately see that such plane wave, with an arbitrary complex amplitude a, is indeed a solution of this Schrödinger equation, provided a specific dispersion relation between the wave number k k and the frequency : (
Free
k)2
.
(1.30) particle’s
2 m
dispersion
relation
The constant a may be calculated, for example, assuming that the wave (29) is extended over a certain volume V, while beyond it, = 0. Then from the normalization condition (22c) and Eq. (29), we get30
2
a V 1.
(1.31)
Let us use Eqs. (23), (26), and (27) to calculate the expectation values of the particle’s momentum p and energy E = H in the state (29). The result is
( k)2
p k,
E H
;
(1.32)
2 m
according to Eq. (30), the last equality may be rewritten as E = .
Next, Eq. (23) enables calculation of not only the average (in the math speak, the first moment) of an observable but also its higher moments, notably the second moment – in physics, usually called variance:
~
2
A
A A 2
2
2
A A ,
(1.33) Observable’s
variance
and hence its uncertainty, alternatively called the “root-mean-square (r.m.s.) fluctuation”, 1/ 2
~
Observable’s
2
A
A
.
(1.34) uncertainty
~
The uncertainty is a scale of deviations A A A of measurement results from their average. In the particular case when the uncertainty A equals zero, every measurement of the observable A will give the same value A; such a state is said to have a definite value of the variable. For example, in application to the state with wavefunction (29), these relations yield E = 0, p = 0. This means that in this plane-wave, monochromatic state, the energy and momentum of the particle have definite values, so that the statistical average signs in Eqs. (32) might be removed. Thus, these relations are reduced to the experimentally-inferred Eqs. (5) and (15).
Hence the wave mechanics postulates indeed may describe the observed wave properties of non-relativistic particles. (For photons, we would need its relativistic generalization – see Chapter 9 below.) On the other hand, due to the linearity of the Schrödinger equation (25), any sum of its solutions is also a solution – the so-called linear superposition principle. For a free particle, this means that any set of plane waves (29) is also a solution to this equation. Such sets, with close values of k and hence p = k (and, according to Eq. (30), of as well), may be used to describe spatially localized “pulses”, called wave packets – see Fig. 6. In Sec. 2.1, I will prove (or rather reproduce H. Weyl’s proof :-) that the wave packet’s extension x in any direction (say, x) is related to the width kx of the distribution of the 30 For infinite space ( V ), Eq. (31) yields a 0, i.e. wavefunction (29) vanishes. This formal problem may be readily resolved considering sufficiently long wave packets – see Sec. 2.2 below.
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corresponding component of its wave vector as x kx ½, and hence, according to Eq. (15), to the width
px of the momentum component distribution as
Heisenberg’s
uncertainty
x
p
.
(1.35)
x
relation
2
(a)
(b)
a
kx
Re
x
k
Im
0
0
k
x
0
k p /
x
x
the particle is
Fig. 1.6. (a) A snapshot of a typical wave packet
(somewhere :-)
propagating along axis x, and (b) the corresponding
here!
distribution of the wave numbers kx, i.e. the momenta px.
This is the famous Heisenberg’s uncertainty principle, which quantifies the first postulate’s point that the coordinate and the momentum cannot be defined exactly simultaneously. However, since the Planck’s constant, ~ 10-34 Js, is extremely small on the human scale of things, it still allows for particle localization in a very small volume even if the momentum spread in a wave packet is also small on that scale. For example, according to Eq. (35), a 0.1% spread of momentum of a 1 keV electron ( p ~
1.710-24 kgm/s) allows its wave packet to be as small as ~310-10 m. (For a heavier particle such as a proton, the packet would be even tighter.) As a result, wave packets may be used to describe the particles that are quite point-like from the macroscopic point of view.
In a nutshell, this is the main idea of wave mechanics, and the first part of this course (Chapters 1-3) will be essentially a discussion of various effects described by this approach. During this discussion, however, we will not only witness wave mechanics’ many triumphs within its applicability domain but also gradually accumulate evidence for its handicaps, which will force an eventual transfer to a more general formalism – to be discussed in Chapter 4 and beyond.
1.3. Postulates’ discussion
The wave mechanics’ postulates listed in the previous section (hopefully, familiar to the reader from their undergraduate studies) may look very simple. However, the physics of these axioms is very deep, leading to some counter-intuitive conclusions, and their in-depth discussion requires solutions of several key problems of wave mechanics. This is why in this section I will give only an initial, admittedly superficial discussion of the postulates, and will be repeatedly returning to the conceptual foundations of quantum mechanics throughout the course, especially in Chapter 10.
First of all, the fundamental uncertainty of observables, which is in the core of the first postulate, is very foreign to the basic ideas of classical mechanics, and historically has made quantum mechanics so hard to swallow for many star physicists, notably including Albert Einstein – despite his 1905 work, which essentially launched the whole field! However, this fact has been confirmed by numerous Chapter 1
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experiments, and (more importantly) there has not been a single confirmed experiment that would contradict this postulate, so that quantum mechanics was long ago promoted from a theoretical hypothesis to the rank of a reliable scientific theory.
One more remark in this context is that Eq. (25) itself is deterministic, i.e. conceptually enables an exact calculation of the wavefunction’s distribution in space at any instant t, provided that its initial distribution, and the particle’s Hamiltonian, are known exactly. Note that in the classical statistical mechanics, the probability density distribution w(r, t) may be also calculated from deterministic differential equations, for example, the Liouville equation.31 The quantum-mechanical description differs from that situation in two important aspects. First, in the perfect conditions outlined above (the best possible initial state preparation and measurements), the Liouville equation is reduced to the 2nd Newton law of classical mechanics, i.e. the statistical uncertainty of its results disappears. In quantum mechanics this is not true: the quantum uncertainly, such as described by Eq. (35), persists even in this limit. Second, the wavefunction (r, t) gives more information than just w(r, t), because besides the modulus of , involved in Eq. (22), this complex function also has the phase arg, which may affect some observables, describing, in particular, interference of the de Broglie waves.
Next, it is very important to understand that the relation between the quantum mechanics and experiment, given by the second postulate, necessarily involves another key notion: that of the corresponding statistical ensemble, in this case, a set of many experiments carried out at apparently ( macroscopically) similar conditions, including the initial conditions – which nevertheless may lead to different measurement results ( outcomes). Indeed, the probability of a certain ( n th) outcome of an experiment may be only defined for a certain statistical ensemble, as the limit
N
M
W
n
lim
,
M
with
M ,
(1.36) Probability:
n
M
n
M
definition
n1
where M is the total number of experiments, Mn is the number of outcomes of the n th type, and N is the number of different outcomes.
Note that a particular choice of statistical ensemble may affect probabilities Wn very significantly. For example, if we pull out playing cards at random from a standard pack of 52 different cards of 4 suits, the probability Wn of getting a certain card (e.g., the queen of spades) is 1/52. However, if the cards of a certain suit (say, hearts) had been taken out from the pack in advance, the probability of getting the queen of spades is higher, 1/39. It is important that we would also get the last number for the probability even if we had used the full 52-card pack, but by some reason discarded results of all experiments giving us any rank of hearts. Hence, the ensemble definition (or its redefinition in the middle of the game) may change outcome probabilities.
In wave mechanics, with its fundamental relation (22) between w and , this means that not only the outcome probabilities, but the wavefunction itself also may depend on the statistical ensemble we are using, i.e. not only on the preparation of the system and the experimental setup, but also on the subset of outcomes taken into account. The sometimes accounted attribution of the wavefunction to a single experiment, both before and after the measurement, may lead to very unphysical interpretations of the results, including some wavefunction’s evolution which is not described by the Schrödinger equation (the so-called wave packet reducti on), subluminal action on distance, etc. Later in the course, we will see that minding the fundamentally statistical nature of quantum mechanics, and in particular the 31 See, e.g., SM Sec. 6.1.
Chapter 1
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dependence of wavefunctions on the statistical ensembles’ definition (or redefinition), readily resolves some, though not all, paradoxes of quantum measurements.
Note, however, again that the standard quantum mechanics, as discussed in Chapters 1-6 of this course, is limited to statistical ensembles with the least possible uncertainty of the considered systems, i.e. with the best possible knowledge of their state.32 This condition requires, first, the least uncertain initial preparation of the system, and second, its total isolation from the rest of the world, or at least from its disordered part (the “environment”), in the course of its evolution in time. Only such ensembles may be described by certain wavefunctions. A detailed discussion of more general ensembles, which are necessary if these conditions are not satisfied, will be given in Chapters 7, 8, and 10.
Finally, regarding Eq. (23): a better feeling of this definition may be obtained by its comparison with the general definition of the expectation value (i.e. the statistical average) in the probability theory.
Namely, let each of N possible outcomes in a set of M experiments give a certain value An of observable A; then
N
N
Definition
1
A lim
.
(1.37)
of statistical
A M A W
M
n
n
n
n
average
M n1
n1
Taking into account Eq. (22), which relates W and , the structures of Eq. (23) and the final form of Eq.
(37) are similar. Their exact relation will be further discussed in Sec. 4.1.
1.4. Continuity equation
The wave mechanics postulates survive one more sanity check: they satisfy the natural requirement that the particle does not appear or vanish in the course of the quantum evolution.33 Indeed, let us use Eq. (22b) to calculate the rate of change of the probability W to find a particle within a certain volume V:
dW
d
*
d 3 r
.
(1.38)
dt
dt V
Assuming for simplicity that the boundaries of the volume V do not move, it is sufficient to carry out the partial differentiation of the product * inside the integral. Using the Schrödinger equation (25), together with its complex conjugate,
*
ˆ
*
i
( H) ,
(1.39)
t
we readily get
*
dW
*
3
*
3
1 *
*
d r
d r
(1.40)
ˆ H ˆ H 3
d r.
dt
t
t
t
i
V
V
V
32 The reader should not be surprised by the use of the notion of “knowledge” (or “information”) in this context.
Indeed, due to the statistical character of experiment outcomes, quantum mechanics (or at least its relation to experiment) is intimately related to information theory. In contrast to much of classical physics, which may be discussed without any reference to information, in quantum mechanics, as in classical statistical physics, such abstraction is possible only in some very special (and not the most interesting) cases.
33 Note that this requirement may be violated in the relativistic quantum theory – see Chapter 9.
Chapter 1
Page 13 of 28
QM: Quantum Mechanics
Let the particle move in a field of external forces (not necessarily constant in time), so that its classical Hamiltonian function H is the sum of the particle’s kinetic energy T = p 2/2 m and its potential energy U(r, t).34 According to the correspondence principle, and Eq. (27), the Hamiltonian operator may be represented as the sum35,
ˆ 2
2
p
ˆ
ˆ
ˆ
H T U
U ( ,t
r )
2
U (r, t) .
(1.41) Potential field:
2 m
2 m
Hamiltonian
At this stage, we should notice that this operator, when acting on a real function, returns a real function.36 Hence, the result of its action on an arbitrary complex function = a + ib (where a and b are real) is
H ˆ H ˆ ( a ib
ˆ
) a
H i ˆ b
H ,
(1.42)
where a
H ând b
H âre also real, while
*
*
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
( H) ( a
H iHb)* Ha i b
H H ( a ib) H .
(1.43)
This means that Eq. (40) may be rewritten as
dW
2
1
*
1
H ˆ H ˆ *
d 3 r
* 2
2
*
d 3 r
.
(1.44)
dt
i
2 m i
V
V
Now let us use general rules of vector calculus37 to write the following identity:
*
*
* 2
2
*
Ψ Ψ ΨΨ Ψ Ψ Ψ Ψ ,
(1.45)
A comparison of Eqs. (44) and (45) shows that we may write
dW
( ) 3
j d r,
(1.46)
dt
V
where the vector j is defined as
i
Probability
*
j
ΨΨ c.c.
Im Ψ*Ψ
,
(1.47) current
2 m
m
density
where c.c. means the complex conjugate of the previous expression – in this case, (*)*, i.e. *.
Now using the well-known divergence theorem,38 Eq. (46) may be rewritten as the continuity equation Continuity
dW
I ,
0
I j d 2
with
r ,
(1.48) equation:
n
dt
integral
S
form
34 As a reminder, such description is valid not only for conservative forces (in that case U has to be time-independent), but also for any force F(r, t) that may be expressed via the gradient of U(r, t) – see, e.g., CM
Chapters 2 and 10. (A good example when such a description is impossible is given by the magnetic component of the Lorentz force – see, e.g., EM Sec. 9.7, and also Sec. 3.1 below.)
35 Historically, this was the main step made (in 1926) by E. Schrödinger on the background of L. de Broglie’s idea. The probabilistic interpretation of the wavefunction was put forward, almost simultaneously, by M. Born.
36 In Chapter 4, we will discuss a more general family of Hermitian operators, which have this property.
37 See, e.g., MA Eq. (11.4a), combined with the del operator’s definition 2 .
38 See, e.g., MA Eq. (12.2).
Chapter 1
Page 14 of 28
QM: Quantum Mechanics
where jn is the component of the vector j, along the outwardly directed normal to the closed surface S
that limits the volume V, i.e. the scalar product jꞏn, where n is the unit vector along this normal.
Equalities (47) and (48) show that if the wavefunction on the surface vanishes, the total probability W of finding the particle within the volume does not change, providing the intended sanity check. In the general case, Eq. (48) says that dW/ dt equals the flux I of the vector j through the surface, with the minus sign. It is clear that this vector may be interpreted as the probability current density –
and I, as the total probability current through the surface S. This interpretation may be further supported by rewriting Eq. (47) for the wavefunction represented in the polar form = aei, with real a and : j a 2 .
(1.49)
m
Note that for a real wavefunction, or even for a wavefunction with an arbitrary but space-constant phase
, the probability current density vanishes. On the contrary, for the traveling wave (29), with a constant probability density w = a 2, Eq. (49) yields a non-zero (and physically very transparent) result:
p
j w k w
wv ,
(1.50)
m
m
where v = p/ m is particle’s velocity. If multiplied by the particle’s mass m, the probability density w turns into the (average) mass density , and the probability current density, into the mass flux density v.
Similarly, if multiplied by the total electric charge q of the particle, with w turning into the charge density , j becomes the electric current density. As the reader (hopefully :-) knows, both these currents satisfy classical continuity equations similar to Eq. (48).39
Finally, let us recast the continuity equation, rewriting Eq. (46) as
w
3
j d r 0
.
(1.51)
t
V
Now we may argue that this equality may be true for any choice of the volume V only if the expression under the integral vanishes everywhere, i.e. if
Continuity
equation:
w
j .
0
(1.52)
differential
t
form
This differential form of the continuity equation may be more convenient than its integral form (48).
1.5. Eigenstates and eigenvalues
Now let us discuss the most important corollaries of wave mechanics’ linearity. First of all, it uses only linear operators. This term means that the operators must obey the following two rules:40
ˆ
ˆ
A A A A
(1.53)
1
2
ˆ
ˆ
,
1
2
39 See, e.g., respectively, CM 8.3 and EM Sec. 4.1.
40 By the way, if any equality involving operators is valid for an arbitrary wavefunction, the latter is frequently dropped from notation, resulting in an operator equality. In particular, Eq. (53) may be readily used to prove that the linear operators 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
