defined concept, and we must overcome this hurdle. There are two main
ways to achieve this goal: to describe the motion of quantum systems as
a superposition of all possible paths, or to describe action with the help
of wave functions. Both approaches are

are leptons (i.e., electrons, muons, tauons and neutrinos), quarks, and
intermediate bosons (i.e., photons, W-bosons, Z-bosons and Vol. V, page
161 gluons). More details on these particles will be revealed in the
chapters on the nucleus. Another simple cr

Fink/Wiley VCH). Can two electron beams interfere? Are there coherent
electron beams? Ref. 59 Do coherent electron sources exist? The
question is tricky. Results in the literature, such as the one illustrated in
Figure 54, state that is possible to make h

physicist Hugo Tetrode (18951931). Note that the essential parameter is
the ratio between /, the classical volume per particle, and 3 , the de
Broglie volume of a quantum particle. * Josiah Willard Gibbs (1839
1903), US-American physicist who was, with Ma

lamps (and lasers) can show interference when the beam is split and
recombined with identical path length; this is not a proof of coherence of
the light field. A similar reasoning shows that monochromaticity is not a
proof for coherence either. A state is

and the kinetic energy of the particle. For a system of large number of
particles, the probability is (at most) the product of the probabilities for
the different particles. Let us take the case of a car in a garage, and
assume that the car is made of 102

you may wish to check. The tables in Appendix B in the next volume
make the same point. In short, the second criterion for compositeness is
equivalent to the first. A third criterion for compositeness is more
general: any object larger than its Compton le

function. In summary, the least action principle is also valid in quantum
physics, provided one takes into account that action values below
cannot be found in experiments. The least action principle governs the
evolution of wave function. The least actio

predicted the neutrino. He was admired for his intelligence, and feared
for his biting criticisms, which led to his nickname, conscience of
physics. Despite this trait, he helped many people in their research, such
as Heisenberg with quantum theory, witho

in a system is the integral of the Lagrangian. The Lagrangian operator
is defined in the same way as in classical physics: the Lagrangian =
is the difference between the kinetic energy and the potential
energy operators. The only difference is that, in

experiments and devices. The quantum phase We have seen that the
amplitude of the wave function, the probability amplitude, shows the
same effects as any wave: dispersion and damping. We now return to the
phase of the wave function and explore it in more

acceleration due to rotational motion can do so. In fact, it has been
possible to measure the rotation of the Earth by observing the Ref. 58
change of neutron beam interference patterns. Another important class of
experiments that manipulate the phase of

smaller than the right-hand side of expression (52) is elementary. Again,
only leptons, quarks and intermediate bosons passed the test. (For the
Higgs boson discovered in 2012, the test has yet to be performed, but it
is expected to comply as well.) All o

calculation of the entropy of a simple gas, made of simple particles*
of mass moving in a volume , gives = ln [ 3 ] + 3 2 + ln
. (55) Here, is the Boltzmann constant, ln the natural logarithm, the
temperature, and = 22/ is the thermal wavelength
(approxi

with intereference pattern that depends on wire charge beam splitter
electrically charged wire polarized neutron beam F I G U R E 53 The
AharonovCasher effect: the influence of charge on the phase leads to
interference even for interfering neutrons. the t

62 explained by Llewellyn Thomas as a relativistic effect a few months
after its experimental discovery. By 2004, experimental techniques had
become so sensitive that the magnetic effect of a single electron spin
attached to an impurity (in an otherwise n

for being elementary can thus be reduced to a condition on the value of
the dimensionless number , the so-called -factor. (The expression
/2 is often called the magneton of the particle.) If the -factor differs
from the value predicted by quantum Page 188

remain vertical despite being unstable. Similar situations also occur in
quantum physics. Examples are Paul traps, the helium Ref. 70 atom,
negative ions, Trojan electrons and particle accelerators. Motion
Mountain The Adventure of Physics copyright Chris

whether they have a non-zero height or whether they think that atoms
are round. If they agree, they have admitted that wave functions have
some sort of reality. All everyday objects are made of elementary
particles that are so unmeasurably small that we c

University of Vienna, is well-known for his experiments on quantum
mechanics. Motion Mountain The Adventure of Physics copyright
Christoph Schiller June 1990October 2016 free pdf file available at
www.motionmountain.net 4 the quantum description of matte

space point, bringing the total number of parameters to four real
numbers, or, equivalently, two complex numbers. Nowadays, Paulis
equation for quantum mechanics with spin is mainly of conceptual
interest, because like that of Schrdinger it does not compl

Figure 51. A matter wave of charged particles is split into two by a
cylinder positioned at a right angle to the matters path and the
matter wave recombines behind it. Inside the cylinder there is a
magnetic field; outside, there is none. (A simple way to

solenoid) charged matter beam F I G U R E 51 The AharonovBohm
effect: the influence of the magnetic vector potential on interference
(left) and a measurement confirmation (right), using a microscopic
sample that transports electrons in thin metal wires (

paths. In the second approach to quantum physics, action is defined with
the help of wave functions. In classical physics, we defined the action
(or change) as the integral of the Lagrangian between the initial and final
points in time, and the Lagrangian

free relativistic particle, the classical Hamiltonian function that is, the
energy of the particle is given by = 4 2 + 2 2 with = .
(43) Thus we can ask: what is the corresponding Hamilton operator for
the quantum world? The simplest answer was given, in

solenoid with current screen with intereference pattern that depends on
magnetic field neutral matter beam beam splitter F I G U R E 50
Magnetic fields change the phase of a spinning particle. effect. Let us
explore this point. Page 56 The phase of free m

what Ralph Kronig had also suspected: that electrons rotate around an
axis with a projected component of the angular momentum given by /2.
In fact, this value often called spin 1/2 for short is valid for all
elementary matter particles. (In contrast, all

be turned inside out will be of importance for their description and their
motion. We will also explore the difference between right- and lefthanded particles, though in the next Vol. V, page 244 part of our
adventure. In the present chapter we concentrat

/2( ) . For example, tunnelling of single atoms is observed in
solids at high temperature, but is not important in daily life. For
electrons, the effect is more pronounced: the barrier width Motion
Mountain The Adventure of Physics copyright Christoph Sch

Christoph Schiller June 1990October 2016 free pdf file available at
www.motionmountain.net 5 permutation of particles115 them. The first
step in counting particles is the definition of what is meant by a situation
without any particle at all. This seems a

and is the size of the system. Can you deduce this result, and check it
for the ground state of hydrogen? Challenge 85 s In high school, it
often makes sense to visualize electron wave functions as a special type
of fluid-like matter, called electronium,

allows you to distinguish objects: you will find that it Challenge 87 s
runs into trouble for point-like particles. Therefore, in the quantum
domain something must change about our ability to distinguish particles
and objects. We could argue that differen

followed from the requirement to specify, at each Vol. II, page 72 point
in space, the length of the arrow, its phase, and its plane of rotation.
Earlier on we also found that relativity automatically introduces
antimatter. (We will explore the issue Page

times the variation of the scalar product between initial and final states.
In * We skip the details of notation and mathematics here; in the
simplest description, states are wave functions, operators act on these
functions, and the product of two differe

using their state. The state of a billiard ball is given by its position, its
linear and its angular momentum. We are able to distinguish two
identical billiard balls because the measurement error for the position of
each ball is much smaller than the siz