Introduction to quantum theory
(Concepts of quantum states and
on to the brush, I remove the ladder !"
Introduction and preliminaries
The laws of quantum physics are fundamentally probabilistic. Thus,
to understand them, we need to express probabilistic laws of
evolution. The geometric
expression of Markov processes (the
general case of classical probabilistic law of evolution for
material systems) is a prerequisite here, as it provides the
language of this presentation.
We are going to express the concepts of quantum states and
measurements in a simplified but still mathematically accurate
manner (compared to usual courses of quantum physics) as a variant
of that mathematical description of Markov processes (in other
words, in terms of how quantum probability differs from classical
probability). This will describe and show the coherence of some
famous "paradoxes" of quantum physics, making them almost seem
natural and intuitive. It includes the concepts of indetermination,
the role of the observer, and the description of the quantum system
involved in the double slit experiment.
We shall also explain the details of the measurement process as a
particular case of quantum evolution, by the concept of decoherence
; but that only works "for all practical purposes", not completely
in ontological terms (the question of what is real). The remaining
discrepancy between the role of measurement in the principles of the
theory, and its explanation as a physical process, is the subject of
the debate on the interpretations of quantum physics.
Axioms and fundamental properties of quantum states
We can obtain the basic principles of quantum
theory, from the classical probabilistic
concepts we just developed, by slight modifications and
specifications, as follows (you may ask "Why are things this way ?"
well, we just know that they are this way because experience has
confirmed it countless times) :
Indeed consider the simplest example, n=2 : any point inside the
sphere can be obtained as a barycenter of 2 "pure states", that is,
points one the sphere. These are the 2 intersections of the sphere
with any line going through the point. You can see that any point of
the sphere fits with one of these possible decompositions.
- To every physical system considered within finite limits of
available space and energy, is associated a natural number n
called the "number of possible states" of the system; this
number can become arbitrarily large when more space or energy is
offered. (Of course there are no walls in space, and the concept
of energy remains to be defined, so that these limitation
conditions are not well-defined here, but this strange
claim happens to be true in practice anyway).
- This number n completely determines the geometric shape
of the set B of all states of the system; it is a volume in
an affine space with dimension nČ-1, which we will called a
"quantum n-states shape". (This is the set of positive
semi-definite Hermitian matrices with trace 1, but we
shall not make use of this definition in this
- Some states are "pure" (those of rank 1), others are
composite; the set of pure states forms a (2n-2)-dimensional
surface S enveloping B. In the case n=2, S is a sphere in a
Euclidean 3-dimensional space.
- The whole set B is made of all barycenters of pure states with
positive coefficients; in other words, B is a convex set (thus
for n=2, this is a ball, that is the volume inside the sphere);
- No pure state can be obtained as a barycenter of any list of
several other states (you can check this for the sphere !)
- The natural evolution of a physical system staying "inside its
box" without external interactions, consists in a rotation
of this volume which occurs in a continuous way along time,
letting every pure state remain among pure states. Any pure
state can be sent to any other pure state by such rotations.
- Only if a system interacts with the environment (exchanging
disturbance with it), then the evolution can take the form of
some other affine transformation, shrinking it into itself
(sending pure states into composite states).
- For every state, we can define the number k of states it
is made of (the rank of this Hermitian matrix), that is, the
minimum number of pure states necessary for obtaining it as a
barycenter (with positive coefficients, as always). The list of
these pure states is not unique, but all the pure states from
all possible decompositions of our state as a barycenter of k
pure states, form the (2k-2)-dimensional surface of the
pure states enveloping the only quantum k-states
shape containing our state in its interior. (It can
be obtained by evolution from a system of k possible
Let us focus on this case n=2 to examine how things work there in
The spin 1/2
The most natural case of a physical system with "2 possible states",
is the spin 1/2 of a particle. The simplest and most common example
of a particle with 2 states due to its spin 1/2 is the electron, so
that we will fix the discussion on it, but some other particles
such as the proton (hydrogen kernel), the neutron, and some other
atoms, kernels or ions, have this property too (it does not matter
whether a particle is elementary or not).
What is a spin ? The first idea for describing a spin, would be that
of a rotating ball that must keep rotating because of the
conservation of angular momentum. However a rotating ball has too
much details that an electron does not have: we can draw a mark on
the ball and see it moving around; the ball may stop spinning and
become at rest, or spin at different speeds.
The electron, on the other hand, has no such details: it cannot
stop spinning, and has no mark on its face that can be seen
moving around. Its spin state only consists in the data of its
angular momentum, and thus remains constant in time as long as it is
not modified by interaction with the environment (namely,
by the magnetic field). For any system, the momentum can only vary
by integers (to multiply by the Planck constant). The electron has
only 2 possible values of the momentum, ± 1/2, thus with a
difference of 1.
In order to measure the spin of an electron and getting one of both
possibilities (clockwise vs. counterclockwise), we first need to
choose the direction of the axis around which this spin will be
measured. And the probabilities of results will of course depend on
the axis chosen (as a continuous change of possible choice ending up
in exchanging both ends, will of course exchange both
Before choosing an axis, any electron's spin is naturally in some
state. Like any angular momentum, it is a pseudo-vector. This means
it belongs to a 3-dimensional vector space, but its representation
as a vector in our space depends on a convention of orientation
of space, and is reversed when we change this convention. For
example, the angular momentum of the Earth can be represented by a
vector towards the North pole, but a representation by a vector
towards the South pole would be an equally possible convention. We
just have to fix the convention once for all.
So, once this space orientation convention is fixed, the ball B of
all spin states of an electron, whose surface is the sphere S of
pure states, is figured by a ball in space.
Let us describe measurements of this spin.
As before, each possible measurement result goes with a probability
calculated as an affine function from B to real numbers, and more
precisely into [0,1]. It can be any such function. So, it can be
represented geometrically by the data of both parallel planes P0
and P1 where this affine function, extended to
the whole space, would take the values 0 and 1 (so, outside B, and
having B between them).
In the case of a binary (yes/no) measurement, the other possible
measurement has the complementary probability (so that the sum is
1), represented with P0 and P1
exchanging their roles.
Now that we have specified "what is to be measured" (the
probabilities of measurement results as depending on the initial
state), what can be the state of the system after the measurement ?
Contrary to classical physics, quantum physics cannot admit a
measurement being done on an elementary system without a physical
interaction with it, that disturbs it (there is no more such a thing
as a non-disturbing measurement). We need a measuring apparatus to
interact with a system, and let the result of the measurement appear
in a macroscopic way, where its description can be summed
up (approximated) in the form of classical probabilities that
we first presented, thanks to the process of decoherence.
We shall precisely describe the form of this necessary disturbance
happening during measurement (effect of the physical processes
making the measurement).
Instead of a non-disturbing measurement, we have the concept of a least-disturbing measurement.
Let us describe its effects geometrically, for the spin of the
The simplest case is the case of a complete measurement, that is
where the probability 0 and 1 planes are tangent to S at two
opposite points. This measurement collapses the spin onto the point
of tangency which is the only pure state having the probability
1 of giving the observed result. As the two possible measurement
results collapse the spin onto 2 opposite points, this is why we say
that the "number of possible states" of the spin is 2.
This collapsing effect works more generally for any case when P0
is tangent to the sphere, disregarding whether P1
is also tangent or not, and so collapses the spin onto the maximum
probability point (opposite to the 0 probability one).
Indeed, we already explained with classical probabilities, that
the meaning of a measurement, and thus its effect on the state of
the system, does not change if the function that gave the
probability of reaching it, was multiplied by a constant.
In this sense, just like in the classical case, the set of all
measurements has the same geometrical shape (a ball) as the set of
all states (and this correspondence also works for any other "number
of states"). To see this, you just need to divide the
probability function of a measurement, by its value at the
center of S, which will thus become 1 (and divide again the
result by 2 if you want it to give a meaningful probability,
with values in [0,1] over the sphere). In a Cartesian coordinates
system (for the 3-dimensional space containing the sphere), you just
need to reinterpret the coefficients (a,b,c) of this function
(x,y,z) -> ax+by+cz+1, as the coordinates of the measurement in a
space of measurements.
In other words, a measurement, as specified by its zero-probability
plane outside the sphere, will be represented by the point inside
the sphere, on the line from the center and orthogonal to
the plane, and at a distance from the center which is the inverse of
the distance of the center to the plane (if the sphere has radius
1), and on the opposite side.
This way, each measurement is represented by the point where it
sends the center of the sphere (the totally undetermined state)
according to its least-disturbing effect.
Each pure measurement is figured as the element of the sphere where
it has its maximal probability, while others are figured inside it.
So, there are many other possible sorts of least-disturbing
measurements: binary measurements where one possible result
collapses the spin while the other doesn't; or where none does;
measurements with arbitrary numbers of possible results, with
arbitrary respective probability functions on B, provided that they
are positive, affine, and that their sum is 1 all over B.
Now let us describe other cases, when the measurement result does
not happen to provide certainty on the state of the system,. i.e.
where P0 is not tangent, but away from S. Then the effect
is that of a projective transformation of the space that sends
P0 to infinity, and globally preserves B and S : each pure
state becomes another pure state.
Only two pure states remain fixed (in the least-disturbing case):
those that were nearest and furthest to P0.
(These projective transformations of the 3-dimensional space that
preserve a sphere, are also those acting on the set of speeds
considered as relatively to different observers according to Special
Relativity theory: the elements of the sphere define the speed
vectors whose length correspond to the speed of light, thus
expressing the fact that going at the speed of light, is a
property that does not depend on the movement of the observer that
measures this speed.)
We can see here that the concept of non-perturbing measurement
cannot make sense in general: not all pure states (points of
the sphere) can be preserved in such a projective transformation.
Only two can, and so must be specified to make sense of the
Popular accounts of quantum physics mention the Heisenberg
inequalities. One of these inequalities say that the position and
the momentum of a particle cannot be both determined, and the more
precisely one of these quantities is known, the less the other is.
What we just explained about the spin, already presents such an
indetermination: it is neither possible to measure nor predict the
spin of the electron along several axis at the same time.
Energy and evolution
The evolution of a physical system is determined by the energy
differences between its possible states.
We will describe the situation in the case of the spin of the
electron, but the same law applies to any other system as well. The
explanation will be based on some concepts of classical mechanics.
Many concepts of classical mechanics are no more valid in quantum
theory, however some properties like those we will mention here,
still apply somehow and can help to understand the situation
The electron has a magnetic moment associated to its spin. This
means that it behaves like a little magnet with the same
orientation as its spin. Like any magnet, its interaction with
an external magnetic field gives it a potential energy that is
minimal when the magnetic moment is aligned with the magnetic field,
and maximal when they are opposite. When the magnetic moment is not
aligned with the magnetic field, the magnetic field exerts a torque
on the magnet, which in the case of ordinary magnets pushes them
towards the minimum energy configuration, aligned with the field.
But the axis of the electron's spin is not like a fixed object that
is turned in the way forces push to turn it. Instead, as it is
defined by the angular momentum, the torque exerted by the magnetic
field produce a gyroscopic
precession of this spin around the direction of the magnetic
Now let us express the situation in the terms of quantum physics.
One of the Heisenberg inequalities says that the energy and the
time cannot be both determined. Thus, whenever the energy of a
system has an exact well-defined value, nothing can happen to it
The spin has two possible states, and thus two possible values of
the energy (when the environment is classically fixed). Each of both
pure states of the spin along the direction of the magnetic field,
has a well-defined value of the energy. For any other state of the
spin, the energy in undetermined.
The measurement of the energy of the electron, coincides with the
measurement of its spin along the direction of the magnetic field.
These two pure states of well-defined energy remain fixed in time,
and give the axis of the rotation of the set B of all spin
states along time.
The frequency of this rotation is proportional to the
difference of energy between both possible values of the energy.
This rotational movement of the spin, being also a rotation of
the magnetic momentum of the electron which affects the
surrounding magnetic field, generates an electromagnetic wave. This
is the frequency of the photon emitted by the electron, by which it
will lose its energy in the long term, and reach its state of lowest
But to say this, means that we don't consider the spin of the
electron as an isolated system anymore.
Other comments on energy in
The quantum theory of electromagnetism is very complex with strange
properties, but here we will focus on the simple case of a single
photon with a well-defined frequency and propagating in
a unique direction,
Like the electron, the photon has a spin, also called polarization, whose number of
possible states is 2, even though the two values of its angular
momentum are no more ±1/2 but ±1. Unlike the electron whose spin
could be mesured along any axis in space, the spin of the photon is
only defined with respect to the axis which is the direction of
propagation. Still, it is possible to measure this spin along any
other direction of its abstract sphere of states, but the
(below described) correspondence between these abstract directions
and our usual space-time differs from the spin 1/2 case; while the
angular momentum that a photon may carry with respect to other
directions, takes the form of the spatial configuration of the wave
(position and direction of propagation) and will not be discussed
We can first understand the polarization in the case of a classical
electromagnetic wave: this is a transverse wave, which means
that the oscillation of the electric field is perpendicular to
the direction of propagation (and the magnetic field too, which at
every point of space-time, coincides with the electric field turned
90° around the direction of propagation).
On the abstract sphere of states of the photon's polarization, let
us mark 6 points, configured like the centers of faces of a cube
containing this abstract sphere; as a cube defines a coordinates
system, so these points are expressed by their 3 coordinates.
Imagine that the photon propagates horizontally, so that the
oscillation of the field happens in a vertical plane.
Let us also represent in the last column of the following table,
another case of a 2-states system: the two possible states of the
electromagnetic field that correspond to the undetermined presence
of a given circularly polarized photon.
||Polarization type for a photon
||Possibly absent circular photon
||Electric field to the left
||Electric field to the right
||Electric field to the top
||Linear, other diagonal
||Electric field to the bottom
||One circular photon
(The situation would be the same for the presence/absence of an
electron as here with a photon, except that there is no direct
measurement possible for this system in any other direction of that
sphere than the presence/absence direction, in contrast with the
case of the photon where such a measurement can be done in terms of
the electric field. In other words, unlike the photon, it is not
possible to "see" any oscillation in the electron, despite the fact
that such an oscillation somehow exists relatively to some contexts
such as the double-slit experiment, see below)
Note that in the case of the possibly absent photon,
the electric field oscillates circularly at the frequency
usually said to be the frequency of the photon, because each of both
poles of the sphere (one photon/zero photon) has a different
well-defined energy, which makes the sphere of states rotate around
this axis at the frequency defined by the energy difference, which
is the energy of the photon.
Also note that we have a nice correspondence between the sphere
of spin states of the electron precessing in the magnetic field, and
the sphere of states for the undetermined presence of a photon:
this is the way the electron comes down to its minimum energy level
by emitting a photon and thus transferring its state to it.
We described the case of the circularly polarized photon. It is what
would be emitted by the spin of the electron in the direction of the
magnetic field, in the case the photon would be detected in this
direction, as the rotation of the electric field follows the
rotation of the spin.
But the photon is emitted in all directions, so that if we only try
to detect it in one direction, we may not get it as it may be going
to another direction instead. In other words, the detection of the
photon in a direction is correlated to its non-detection in another
So, let us consider a photon detector all around the electron, with
a way out in some angular area around the direction of the magnetic
The fact that no photon is detected around, defines a partial
measurement with respect to the initial spin state of the electron:
it is the sure outcome if the electron was already in its minimal
energy level, but it also has a chance to be so if it was in the
maximal energy level, as the photon can go by the exit
(circularly polarized). Thus this case of absence of any
photon emitted in other directions, makes a physical evolution
defined by a projective transformation from the initial spin state
of the electron to the final state of presence/absence of the
circular photon emitted in the direction of the magnetic field;
this transformation maps pure states into pure states.
Or, if we don't wait enough time to let the electron come down to
its minimum energy level for sure, then the presence of an emitted
photon will be correlated with the remaining spin state of the
Next : concept of correlation in quantum theory, or
Back to main page: Foundations of physics (table of contents), with List of
Main site : Set
Theory and foundations of mathematics