# The seven laws of Quantum Mechanics : banishing the bogeys

Urjit A. Yajnik111Email<EMAIL_ADDRESS>
Physics Department, Indian Institute of Technology Bombay, Mumbai 400076

###### Abstract

The laws of quantum mechanics are couched in subtle mathematical language. The
laws are not usually stated in a compact pedagogical form. Here I present a
possible way to correct this. Essential facts can be distilled into seven
statements that are easy to remember and easily referred back. Also, the
current teaching of quantum mechanics is laden with words of negative
connotations, originating as they did during the early decades of the subject
when the subject was intellectually still puzzling. A wide variety of
experiments in the intervening decades, not least those that were awarded the
Nobel Prize of 2022 amply affirm the validity and substantial “reality” of
Quantum Mechanics as a theory. I take a few of the inadequacies of classical
framework to illustrate that some of the complaints against Quantum Mechanics
are patently misplaced. Finally I discuss the bogeys such as “wave article
duality”, “uncertainty”, “indistinguishability” “statistics” and
“entanglement” and advocate adopting better terminology to save new learners
from the old biases.

###### Contents

  1. 1 Why number the laws?
    1. 1.1 Welcoming the new principle
  2. 2 Characteristic experiments
  3. 3 The postulates
    1. 3.1 Remarks on the postulates
  4. 4 Is the Classical all reasonable?
  5. 5 Exorcising the bogeys
  6. 6 Conclusion

##  1 Why number the laws?

We teach three laws of Newton, three laws of Thermodynamics and then include a
zeroth law as well, Four Maxwell’s equations of Electromagnetism etc. This
simple mnemonic pedagogical device is missing from the teaching of Quantum
Mechanics. Stating them as a numbered set of laws would make it easier to
remember them. This can also be used to set logical primacy and separation
between them. Then the critical comments would be easier to make, as in
discussing the subtlety of “Newton’s third law”.

Clearly differentiated laws also expose possible pitfalls. For example
Newton’s first law is essentially due to Galileo, and after Newton’s second
law, the former can be taken to correspond to the special case of no applied
force. Is it logically independent? Or does it still have a logical primacy
due to direct observability? Furthermore, substantial discussion is needed to
explicate whether the second law defines mass or defines force[1]. Thus,
setting apart the First Law asserts its validity independent of the dynamical
framework of Newton. We hope that in the following discussion on Quantum
Mechanics we are able to place logically independent facts and rules under
independent laws. After doing that we will be able to identify the source of
various concerns people have about Quantum Mechanics. It is also hoped that
such laws would be useful to new learners. Considering the strides quantum
science is making, we may soon be teaching such laws in the high school. Clear
laws stated without the negativity imposed by the older generation would
reduce the hesitation and suspicion in accepting and applying the laws. Until
some completely new phenomenon demanding a revision comes to the fore we can
be at peace with the laws of Quantum Mechanics, now known for a century.

This article is mostly motivated by Dirac’s elegant exposition in his famous
textbook[2]. This book is most widely praised but least widely read. Teachers
and even experts are heard saying “.. but that is a difficult book to read”.
It happens that Dirac’s exposition is mathematical, but its style is physics.
It makes physicists feel it is too abstract while the mathematicians need more
sophisticated underpinnings. However the book is really precious for its
clarity and elegance. Among many other points of great clarity in this book is
the introduction to many-body quantum mechanics, laying bare the mistaken
origins of the term “second quantisation”.

###  1.1 Welcoming the new principle

To be specific, our motivation in what follows is to propose that the language
of the “wave function” has been the slow poison of quantum mechanics. At its
inception in 1926 this language was the easiest to follow for physicists.
Further, it serves a very good tool for visualising electronic orbitals and
shall remain a very useful language for many settings. However its overuse,
and debating many points of principle in that language, has lead to
misconceptions and a feeling of incompleteness of the subject and inadequacy
of the framework.

The facts look rather different if we grasp Heisenberg’s seminal contribution.
Heisenberg’s 1925 paper [3] called for complete refurbishing of mechanics :

> “… one realises that … even for the simplest quantum-theoretical problems
> the validity of classical mechanics simply cannot be maintained. In this
> situation it seems sensible to discard all hope of observing hitherto
> unobservable quantities such as the position and the period of the electron,
> … Instead it seems more reasonable to try to establish theoretical quantum
> mechanics … in which only relations between observable quantities occur.”

According to this point of view the notion of classical trajectory $X(t)$
needs to be abandoned. What can be really ascertained physically is only the
transition amplitude $X_{ab}$ for a particle to have been once seen at $a$ and
then at $b$. We must thus forgo the knowledge of what may have happened in
between as inaccessible in principle. What does not exist does not need any
organising principle. No ontology no epistemology.

Heisenberg replaces the trajectory by an array of numbers $X_{ab}$ and
additionally $\dot{X}_{ab}$ for the velocity [4]. He then proceeds to check
the mutual commutability of these arrays of numbers. This is what we call
matrix mechanics now. It was Dirac’s insight[5] to note that instead of
velocity one should use the canonically conjugate $P_{ab}$ and when this is
done we get the very elegant analogy of the commutation relations (CR) needed
in Quantum Mechanics to the Poisson brackets (PB) of Classical Mechanics.
However, unlike the PB which are based on real analysis, the quantum
mechanical brackets CR imply that the dynamical variables cannot be
represented by mere numbers, but need matrices for their representation.

We may immediately feel very anxious about this radical framework to be used
for something as obvious as a trajectory. But we shall return in a later
section to argue that the real culprit is the Newtonian notion of
instantaneous velocity, which is at least as “unreasonable” as these quantum
hypotheses. However for the moment let us consider the two positive thoughts
put across by Dirac. The existence of $\hbar$ is the natural demarcation
between the “macroscopic” and the “microscopic”. It sets the scale below which
the microscopic world begins. Otherwise the world should remain self-similar
under infinite subdivision. But Dalton, Avogadro, Boltzmann, Thomson,
Rutherford and others had already investigated that the microscopic world is
atomistic and rather different.

Dirac’s second point is that quantum principles need not be viewed negatively,
as a loss of familiar concepts. Rather, there is a new positive principle as a
compensation so to speak. This is the principle of linear superposition.
Agreed, there is an added layer of abstraction. “State of a system” in quantum
mechanics is not given by a list of values that are guaranteed to be the
outcome of measurement. However, this abstract state vector does obey the
Principle of Superposition.

The uncanny nature of the situation is best brought home to students by the
example of the position and velocity of a ball in a playing field. Common
sense says that the possible states of the ball are given by positions
$\vec{R}_{A}$ and corresponding velocities $\vec{v}_{A}$ where $A$ denotes the
point in space and $\vec{R}_{A}$ its position vector. But in quantum mechanics
we get new valid “states” of the ball which are linear “sum”s of states at
$A_{1}$, $A_{2}$, …$A_{n}$. The possible number of states increases manifolds,
as the relative weightages vary from $0$ to $1$ in magnitude, can have
relative complex phases, and can include any number $n$ of classical states.
Thus the Quantum World offers multitudes of possibilities far beyond the
classical imagination. Yet this is not a disaster. Dirac exhorts us to admire
the simplicity of the linear superposition principle rather than be baffled by
it.

Thus the basic attitudes to quantum mechanics need to be changed. Presenting
the set of arguments that substantiate this appeal is the goal of this
article. In the following in Sec. 2 we summarise the classic experiments that
signal the novel behaviour, Sec. 3 presents the statement of the laws, not in
truly pedagogical form but as notes for knowledgeable peers, Sec. 4 contains a
discussion of a few key classical conceptions, where we argue that the latter
are, to wit, as technical as the ones involved in quantum mechanics and indeed
not in accord with observed reality. In Sec. 5 I comment on the prevalent
terminology, identifying the bogeys that need to be banished. Sec. 6 contains
the concluding remarks.

##  2 Characteristic experiments

It will be important to distinguish between what is truly novel in the quantum
phenomena themselves, versus what makes us uneasy about the mathematical
framework. We shall try to show that whatever is novel is indeed rooted in the
phenomena themselves. More interestingly, as argued later, whatever makes us
uneasy about the formalism is perhaps no worse than the state of affairs in
the classical framework.

The enigmas of quantum systems can be summarised as

  1. a)

The observer is free to choose what to measure, though the choices are limited
in any one attempt at measurement.

  2. b)

The system will produce probabilistic outcomes.

  3. c)

Subsequently the system either ceases to exist completely, or the measured
attribute becomes predictable with certainty.

Note the most uncanny feature of the last statement. In macroscopic world we
only see systems change or transform or redistribute. But the quantum world
allows entities to vanish forever, like a photon absorbed by an atom or the
neutrino by inverse beta decay in nuclei. Then the case of measured attributes
becoming certain in values may be understood as one form of extinction of the
other values of the attribute.

These points can be stated succinctly as

> a’) Subjectivity in the choice of measurement,
>
> b’) probabilistic outcomes,
>
> c’) objectivity of the post measurement state

We now summarise the well known phenomena which we may keep in mind as
intrinsic quantum behaviour

  1. 1.

Complementarity of description \- The Davisson-Germer experiment idealised as
double slit experiment.

The electrons although particles can get redistributed to give the
interference pattern displayed by wave phenomena. Extensive work has shown
that one can recover particle like properties or wave like properties but not
both in the same measurement.

  2. 2.

Quantised values in case of some observables \- The Stern-Gerlach experiment

A beam of polarised Silver atoms passes through a region of magnetic field
pointing along $z$-axis. The emerging beam is split into precisely two
streams. The only explanation is that the component of intrinsic spin of the
Silver atoms along the applied field can have only two possible values.
Further only one of these precise values can emerge in any measurement, and
not an averaged value.

  3. 3.

Probabilistic outcomes

Outcomes of measurements on identically prepared systems are not identical. We
can at best associate a probability to any outcome. If the Stern-Gerlach type
experiment is performed with a rarefied beam of atoms so that only one atom is
passing the magnetic field region, we cannot predict which of the two
orientations the atom will finally emerge in, only relative probabilities for
the two outcomes.

  4. 4.

Indeterminate evolution during measurement process

Related to the previous point is the independent fact that there is no theory
for how the quantum state evolves “during” the measurement. The imprint of the
quantum system is recorded and the quantum nature of that attribute then
terminates. However there is no theoretical framework for describing the
evolution of a quantum system into a residual system with specific value as it
leaves its imprint in an apparatus.

More generally the system itself may disappear such as charged particles in a
Geiger counter. The result is a macroscopic current, and we have a quantum
theory of how the single particle cascades into a current, but not a
deterministic theory.

  5. 5.

Non-locality of states

The quantum state can be spread over a macroscopic region. A state of two
photons can be stretched over many meters. But as soon as the attributes of
one of the photons are measured, the state evolves through that indeterminate
evolution and the attributes of the other photon are instantaneously
determined. One version of it is the famous EPR paradox. However unreasonable
classically, the validity of this outcome has finally been accepted and
recognised by the Nobel Prize of 2022[6].

The non-locality is also manifested over time. We might suspect that once we
have decided on which attribute to measure, the two photons mutually encode
which one is to manifest which value before they are far apart. But since
there is complementarity of which attribute to measure, we need not set up
which attribute we wish to measure until the photons are really far apart.
This is called “delayed choice”. However even under delayed choice, the effect
of measuring an attribute on one photon immediately determines the outcome of
that attribute on the other photon.

  6. 6.

Bose condensation and Pauli Exclusion Principle

Perhaps the most radical departure from classical conceptions arises in the
very notion of identity of the basic entities, particles, or more correctly,
quanta. There are two fundamental aspects to this. One is that there are
identical particles : the primary units are endowed with very few attributes,
and all quanta of a particular species have just a few possibilities for these
attribute. For example all electrons have exactly the same values of charge
and mass, and can have one of two values for the projection of their spin
along any measured direction. This is not encountered classically even in what
we think are identical objects, say balls of same size, color and shine of
polish. Improving the precision of measurement always reveals the differences,
often taking continuum values.

Further the quanta obey peculiar rules for collective states. Atomic physics
verifies most directly that two electrons cannot occupy the exact same state
of energy and spin. Likewise, the spectrum of “light gas” under ideal
conditions of Black Body obeys Planck spectrum, which can be understood only
if photons occupy their available energy states according to the rules
discovered by Planck, Einstein and Bose.

In summary there are identical particles and they obey strange rules for their
collective states. These facts are foundational to the quantum world and the
laws pertaining to them need to be formalised into the core of Quantum
Mechanics. The law should not be postponed as auxiliary rules to be learnt in
more advanced courses.

##  3 The postulates

In this section we enunciate the laws in simple naive language. These are
stated more as pointers to what all practitioners of Quantum Mechanics are
quite familiar with. The idea is to lay out these rules in an order from the
more basic or elementary, moving towards those that build the structure
further. We skirt several subtleties about the Hilbert space and precise
meaning and varieties of measurement at the level of this presentations.
Stating the more detailed version will however not require a change to this
basic list.

Let us make a brief qualitative statement about what is at stake, á la Dirac.
We have a quantum system and we have some apparatus that will record various
clicks and ticks. The requirement is to set up a mathematical framework that
will make predictions about what clicks and ticks can result as outcomes. The
dynamical quantities that can be thus measured will be called observables.
Examples are charge, mass, spin projections, binding energies, lifetimes, etc.
This list needs to be established empirically for every new quantum system one
encounters.

The program of quantum mechanics is to identify a core set of variables which
should satisfy relatively simple kinematic conditions, the CR’s, and all the
other observables should be expressible in terms of them. Fortuitously, but
with no guarantee, this set happens to be the same as the set of classical
canonical variables, satisfying the CR’s that are, upto the fundamental unit
$\hbar$ the same as the classical PB’s. Indeed all the observables, all the
relevant symmetry operators, and the dynamical evolution operator, can be
constructed out of this canonical set. This is not at all obvious, and indeed
it fails in a few major exceptional cases such as spin which has no canonical
representation. A failure also shows up in the occurrence of anomalies in
advanced implementation of quantum principles, such as in Quantum Field Theory
and String Theory. But majority of quantum observables do have simple
classical analogy and that is what has greatly facilitated the prediction and
control of the quantum world.

The rules stated below essentially address the logical structure of these
constructs and the mathematics required to implement them. They are stated in
the simplest context of a single variable and a single observable etc, to keep
the statements compact.

We shall use the convention

$\begin{array}[]{ll}|\Psi\rangle,|\varphi\rangle&\text{A generic state}\\\
|\Psi t\rangle,|\varphi t\rangle&\text{A generic state displaying its time
dependence}\\\ |x\rangle,|p\rangle&\text{A basis state labeled by a canonical
variable}\\\ |\alpha\rangle,|l,m\rangle,|n\rangle&\text{Eigenstates of general
observables labeled by their}\\\ &\text{eigenvalues}\end{array}$

We are dispensing with the convention of putting a hat or a caret above an
operator, so long as there is no ambiguity.

  1. I.

State functions constitute a Hilbert space.

The states of a quantum system obey linear superposition principle and have
the structure of a complex vector space. Further, for physical interpretation
we need to endow this space with a hermitian inner product.

$\langle\psi|\varphi\rangle=\langle\varphi|\psi\rangle^{\ast}$

The framework of Hilbert spaces is applicable with some caveats.

  2. II.

Observables are realised as Hermitian operators.

For an operator $A$, the hermitian conjugate or adjoint operator is defined as
:

$\langle A\psi|\varphi\rangle\equiv\langle\psi|A^{{\dagger}}\varphi\rangle$

Hermitian operators are self-adjoint. All observables are represented by
hermitian operators. Their eigenvalues will be the possible list of answers we
get upon observation. The eigenstates corresponding to these eigenvalues can
be used to construct a basis for the Hilbert space. There can be several
independent choices of bases.

  3. III.

Change of basis is implemented by Unitary operators

If we change from a basis constructed using an observable $n$ to a basis
constructed using an observable $\alpha$, then the change is implemented by a
unitary transformation

$|\alpha\rangle=\sum_{n}U_{\alpha n}|n\rangle$
$\sum_{\beta}(U^{{\dagger}})_{m\beta}U_{\beta
n}=\delta_{m,n}\qquad\sum_{m}U_{\alpha
m}(U^{{\dagger}})_{m\beta}=\delta_{\alpha,\beta}\qquad$

where the Kronecker deltas need to be replaced by Dirac delta functions for
continuum eigenvalues.

  4. IV.

Observations and probabilities are described by projection operators.

In an observation process we can only predict the probability for the system
to emerge in a particular eigenvalue of the relevant observable. If the
observable is $\alpha$ and the initial normalised state vector is represented
in the $|\alpha_{i}\rangle$ basis as

$|\Psi\rangle=\sum_{i}C_{i}|\alpha_{i}\rangle,$

then the probability of getting the outcome $\alpha_{r}$ is $|C_{r}|^{2}$. In
Hilbert space this amounts to the state being subjected to a projection
operator.

This leads to a simple rule about the average measured value of that
observable under repeated measurement of identically prepared systems.

$\langle
A\rangle\equiv\langle\Psi|A|\Psi\rangle\equiv\langle\Psi|A\Psi\rangle=\sum_{i}|C_{i}|^{2}\alpha_{i}$

where the second equivalence connects a Physics convention with Hilbert space
operation.

  5. V.

Quantum kinematics.

The observables and other dynamical quantities introduced are operators and
need not commute. With a sufficiently exhaustive set of dynamical quantities
$\mathcal{O}_{i}$ we find that they satisfy a symplectic algebra

$[\mathcal{O}_{i},\mathcal{O}_{j}]\equiv\mathcal{O}_{i}\mathcal{O}_{j}-\mathcal{O}_{j}\mathcal{O}_{i}=\sum_{k}C_{ijk}\mathcal{O}_{k}$

which has closure and obeys the Jacobi identity. This algebra sets up the
quantum kinematics.

Further, a great simplification is afforded by a deep classical analogy.
Corresponding to the classical canonical variables $\\{x_{1}\ldots
x_{N},p_{1}\ldots p_{N}\\}$ there exists a set of quantum variables such that

$[x_{i},p_{j}]\equiv
x_{i}p_{j}-p_{j}x_{i}=i\hbar\\{x_{i},p_{j}\\}_{\operatorname{PB}}=i\hbar\delta_{ij}$

where the $x_{i}$, $p_{j}$ in the third expression are classical variables and
PB denotes the Poisson bracket. Most operators $\mathcal{O}_{i}$, hermitian
and unitary, can be algebraically constructed out of the canonical set, with a
few notable exceptions such as spin.

Another important feature of quantum theory is that in Quantum Field Theory
one also needs anti-commutator kinematic conditions. These have no classical
analogue.

  6. VI.

Quantum dynamics.

a) In analogy with Classical Mechanics, there exists a distinguished hermitian
operator, the Hamiltonian. In the Heisenberg picture dynamical evolution is
expressed in terms of time dependent operators which can be observables or
other operators

$i\hbar\frac{d}{dt}\mathcal{O}(t)=[\mathcal{O}(t),H]$

In particular we have the analogues of Hamilton’s equations of motion for the
canonical variables,

$i\hbar\frac{d}{dt}p(t)=[p(t),H];\qquad i\hbar\frac{d}{dt}x(t)=[x(t),H]$

from which the evolution equation for any operator on the phase space can be
worked out using the canonical commutation rules.

b) A convenient alternative is the Schrödinger picture in which the state
vector is time dependent, $|\Psi t\rangle$. In this case the operators are not
to be evolved in time, and the equation of motion for the state function is

$i\hbar\frac{\partial}{\partial t}|\Psi t\rangle=H|\Psi t\rangle$

In practice we do not work with the abstract state $|\Psi t\rangle$, but the
“wave function” $\Psi(x,t)\equiv\langle x|\Psi t\rangle$ and express
$H(x,p)\equiv H(x,-i\hbar d/dx)$.

c) A third very elegant and fruitful formulation of the dynamics is due to
Dirac and Feynman, the Path Integral version. Usually the basis set
$|x\rangle$ is treated as time independent. Now define a ”moving basis” (see
[2], Sec. 32) which we might call Dirac picture basis,

$|xt\rangle_{D}=e^{iHt/\hbar}|x\rangle$

Then the Path Integral formula gives the amplitude for going to $x_{f}$ at
time $t_{f}$ given that the system was at $x_{i}$ at $t_{i}$ :

${}_{D}\langle
x_{f}t_{f}|x_{i}t_{i}\rangle_{D}=\int\mathcal{D}x(t)\mathcal{D}p(t)\exp\left\\{\frac{i}{\hbar}\int_{t_{i}}^{t_{f}}dt(p\dot{x}-H)\right\\}$

where the action integral in the exponent is on the phase space and the
symbolic integration $\mathcal{D}x(t)\mathcal{D}p(t)$ is over all possible
paths connecting $x_{f}$ and $x_{i}$. Then we can obtain $\Psi(x,t)$ from
$\Psi(x_{i},t_{i})$ for $t>t_{i}$ as

$\Psi(x,t)=\int dx_{i}\ {{}_{D}\langle
xt|x_{i}t_{i}\rangle_{D}}\Psi(x_{i},t_{i})$

  7. VII.

Bosons and fermions.

Another deep and non-classical feature of the quantum world is the existence
of “identical quanta”. These bits of nature have just a few attributes such as
mass, spin and a few charges. In weakly coupled systems, the full multi-quanta
Hilbert space can be constructed out of repeated tensor product of the one-
quantum Hilbert space. This is called the Fock space.

The admissible multi-quanta states are only the symmetrised ones in an
assembly of integer spin quanta, while the admissible states are only the
anti-symmetrised ones for half-integer quanta. For the case of two quanta,
with the states labeled by the values of the observable $\alpha$, these
tensorial constructions are

$|\Psi\rangle_{B}=\frac{1}{\sqrt{2}}\\{|\alpha^{(1)}_{1}\rangle|\alpha_{2}^{(2)}\rangle+|\alpha^{(2)}_{1}\rangle|\alpha_{2}^{(1)}\rangle\\}$
$|\Psi\rangle_{F}=\frac{1}{\sqrt{2}}\\{|\alpha^{(1)}_{1}\rangle|\alpha_{2}^{(2)}\rangle-|\alpha^{(2)}_{1}\rangle|\alpha_{2}^{(1)}\rangle\\}$

where subscripts $B$ and $F$ refer to Bose and Fermi respectively.

This ends the list of the postulates incorporating the most essential rules.
To repeat in a nutshell, the postulates refer to

  1. I.

Superposition principle and Hilbert space,

  2. II.

Observables as hermitian operators,

  3. III.

Change of basis as unitary operators,

  4. IV.

Observation as projection operator,

  5. V.

Kinematics as commutation rules,

  6. VI.

Dynamics via a special unitary operator, and

  7. VII.

Multi-quanta states via Bose-Einstein and Fermi-Dirac rules

###  3.1 Remarks on the postulates

  1. 1.

The Hilbert space postulate captures two important things in one. Firstly
there is superposition principle for the states, the deepest non-classical
feature of the quantum world. Secondly we introduced the inner product on the
space, as required for physical interpretation.

  2. 2.

Many cases of change of basis correspond to classical symmetry operations.
Wigner’s theorem proves that such transformations are indeed represented by
unitary or anti-unitary operators. Thus Postulate III is not entirely an
independent postulate, but it is important enough to be listed here.

  3. 3.

The issue of measurement has been a source of creative proposals and long
standing debate among the finest minds. There are two aspects – the outcome is
probabilistic and there is no satisfactory description of the evolution from
the unmeasured to the measured state. Here my first caveat is that a conscious
observer may not be a key component of the measurement paradox. Scattering
processes and spontaneous decay are directly observed phenomena which capture
most of the unsettling aspect of “collapse of the state”, with purely quantum
evolution.

    1. i.

Intentional measurement has a lot of resemblance to scattering. In Rutherford
type scattering we send in a stream of projectiles, which are momentum
eigenstates. The expected out state obeys the symmetries of the scatterer, for
example the azimuthal rotation symmetry along the direction of the incoming
projectile. However, a particular scattered particle can emerge only in one
fixed direction. Thus while the evolution operator is unitary, the outcome for
a single scattered particle is in a momentum eigenstate projected out from the
evolved state.

Only when a large number of the same projectiles with the same impact
parameter is studied do we recover the azimuthal symmetry. This can be
considered to be a “collapse” into the eigenvector representing that momentum
value. Note that scattering goes on in locations remote from any conscious
observer all over the Universe. The emergence in any one direction of the
scattered particle is a generic event.

    2. ii.

Spontaneous decay is a similar phenomenon. The emitted final state may be
expected to obey the symmetries of the decaying parent ( for example
rotational symmetries of an atom or a molecule). And indeed this is so on the
average. However any particular decay results in the particle emerging only in
a specific direction, and the symmetry can be recovered only through repeated
experiments. Again, this process happens exactly thus, with no observer
needed, though there could be one, light years away.

It is interesting to note that this issue is implicit in Einstein’s 1905 paper
thus[8] :

> According to the assumption considered here, in the propagation of a light
> ray emitted from a point source, the energy is not distributed continuously
> over ever increasing volumes of space, but consists of a finite number of
> energy quanta localised at points of space that move without dividing, and
> can be absorbed or generated only as complete units”.

It is clear that in the emission of any individual quantum, the rotational
symmetries of the source can not be respected. In turn, if the symmetry is to
be recovered over a large number of observations, it should not be a surprise
that the question of which direction is determined only by a probabilistic
law. In hindsight one may wonder why the bearer of so lucid and profound an
insight shied away from the collateral logical consequences.

  4. 4.

Returning to the broader issues of measurement, the problem seems to lie in
the inability to characterise where the validity of classical paradigm ends
and quantum regime begins. In some sense this is due to the fact that $\hbar$
dimensionally involves space, time and mass and it is difficult to demarcate
the transition between the two frameworks purely in terms of length or time or
mass scales. Indeed Bohr’s Correspondence Principle relies on largeness of
quantum numbers to recover a classical description.

In the Schrödinger’s Cat paradox the presumption is that all systems must
rightly be considered as quantum. In other words taking the observer to be
entangled with the system being observed. Further, the cat box plus its
observer can be a combined system being observed by another observer, thus
demanding a combined description with indefinite recursion. This is the
”Wigner’s Brother” paradox.

There are two possibilities for a resolution. Perhaps more delicate
experiments will demand a more sophisticated formalism for their description.
But if not, we only need improved semantics. A characteristic of macroscopic
systems is that they are highly complex, say a Geiger counter or a bubble
chamber, making it clear that they can be simply treated classically, thus at
least avoid the recursion paradox. However the postulate as presented here has
been verified in a variety of experiments directly or indirectly over the past
century. While the paradox may persist, there is no contradiction with the
proposed postulate.

  5. 5.

The Einstein-Podolski-Rosen paradox epitomises another aspect that is counter-
intuitive about measurement. Suppose we have a two-electron state with net
spin $=0$. Suppose an observer Alice at some remote point $\mathbf{x}$ makes
an observation at time $t_{1}^{\prime}$ observing only one of the electrons,
and that in spin-up state, destroying that electron in the process, then
causality demands that at all subsequent times $t>t^{\prime}_{1}$, the other
observer Bob can only find one electron and that in spin-down state. According
to Special Relativity, this latter fact cannot in principle be known to Bob
during the time $t^{\prime}_{1}<t<t^{\prime}_{1}+|\mathbf{x}|/c$. In case Bob
makes an observation during the time
$t^{\prime}_{1}<t<t^{\prime}_{1}+|\mathbf{x}|/c$ they may legitimately ascribe
the outcome to the weightage factor $1/\sqrt{2}$ for the spin-down state, and
nothing will go wrong. The main lesson is that quantum states are
intrinsically non-local and yet consistent with Special Relativity. The
advanced framework of Relativistic Quantum Field Theory does not throw up any
contradictions either. It does require the existence of anti-particles in
order to preserve causality. More on the possible fallacy in EPR in the last
section.

  6. 6.

We make a big leap of faith in assuming that all the observables and other
operators can be expressed in terms of the canonical set. We propose that the
$\mathcal{O}_{i}$ have the same algebraic dependence on the canonical
variables as the corresponding classical variables. This is an immense
simplification. But it requires a price to be paid

     * $-$

It results in operator ambiguity, which however is resolved by simple
prescriptions.

     * $-$

One may encounter observables with no classical analogue, e.g., spin

     * $-$

For fermion fields one needs anti-commutators instead of commutators. There is
no classical limit available for this operation in the normal sense.

Fermion bilinears do have classical limits and one arranges to set up correct
commutation relations between them with desirable classical limits222One might
wonder whether this and the preceding point about spin have any mutual
connection since fermions are prime examples of particles with spin. But
integer spin particles also exist, and therefore the need to insert spin by
hand will persist and is independent from the enigmatic quantisation of
fermions..

Despite all these exceptions, this principle must be viewed as of special
significance. It suggests that the macroscopic canonical structure of
Hamiltonian dynamics is firmly rooted in microscopic principles.

  7. 7.

Quantum kinematics can be equivalently expressed by the overlap between the
bases labeled by canonical observables. This is given by the fundamental
relation

$\langle x|p\rangle=\frac{1}{\sqrt{2\pi\hbar}}e^{ipx/\hbar}$

This is the key building block of the Path Integral formula.

  8. 8.

Postulate VII is most likely not a logically independent one. When Lorentz
invariance is imposed on a quantum system, the “spin-statistics theorem” can
be proved. We state this here as a law to highlight its importance. That is,
that in Quantum Mechanics states are fundamental, not quanta. And to remind
that “quanta are not particles”.

  9. 9.

Quantum Field Theory provides the comprehensive framework of calculation in
Fock Space based on weakly coupled quanta. However, the Quantum Field Theory
framework has a far greater reach, in a wide variety of strongly coupled
systems.

##  4 Is the Classical all reasonable?

The framework of Quantum Mechanics couched in the language of the Hilbert
space is at once forbidding to those unfamiliar with advanced Mathematics.
However there exists several quantum systems where this Hilbert space may be
no more forbidding than a finite dimensional complex vector space for a finite
number of spins. This is very promising because it may become possible to
teach simplified basics to even highschool students who may be able to
understand the technologies that are now emerging.

Despite this, the framework has become notorious for resorting to purely
abstract and therefore perhaps auxiliary constructs, and for being difficult
to grasp instinctively. As a counter to this, I take up here to argue that a
few key concepts of Classical framework are equally counter-intuitive, and it
is more the familiarity developed over several centuries that makes them so
easily acceptable.

  1. 1.

Instantaneous velocity : The most commonly held drawback of Quantum Mechanics
is the loss of the detailed trajectory of the particle, specified by position
and velocity at any given instant of time. It is however worth pondering how
justified this expectation was in the first place. The notion that an object
is at a place and is moving while at that place actually defies common sense.
In a philosophical vain one is led to wonder how something can be at a place
if it is moving. Newton solved this problem through the Calculus notion of
limits. One defines

$\boldsymbol{v}=\lim_{\Delta t\rightarrow 0}\frac{\Delta\boldsymbol{x}}{\Delta
t}$

The conceptual proposal however has never been verified empirically. No one,
in Galileo’s vain, took a meter stick and a stopwatch watch of unlimited
precision and allowed the $\Delta\boldsymbol{x}$ and $\Delta t$ to physically
go to zero. If they did they would discover Quantum phenomena. Thus there was
no loss of paradise of unlimited precision, it was never there in the first
place.

To repeat Dirac’s point of view quoted in the Introduction, $\hbar$ indeed
sets a scale separating the Classical from the Quantum. In the limit the
action function governing our moving particle reaches this level, the
Newtonian paradise withers away and the Quantum emerges.

This does draw our attention further to the carefully constructed notion of
the continuum in Mathematics which buttresses Newton’s conception. Development
of science may have reached a point making it worth revisiting the tenets of
the continuum. And in turn we need to develop mathematics in which quantum
constructs take an intuitive centre stage.

  2. 2.

The abstract nature of the wave function is often contrasted with other
physical quantities of immediate empirical meaning. The notions of electric
and Magnetic fields are however worth re-inspecting. They are only a step away
from being something palpable, since you need to hold a point charge or a
small dipole in that region of space to immediately see their manifestation as
field lines as visualised by Faraday. However when we propose that these
fields oscillate and they carry energy, and they migrate over light years, in
vacuum as Einstein taught us, then we may wonder “what” is actually
propagating in the empirical sense. These fields therefore have a status
comparable to that of the wave function.

What is intriguing then is that according to the Sudarshan-Glauber theory, the
quantum theory of light subsumes all its classical states without any change.
Perhaps the validity of the conception of electromagnetic radiation is
ultimately derived from the more fundamental quantum property.

  3. 3.

Classical conception is willing to live with point particles as well as
continuum fields. However there is an internal contradiction in the
Electrodynamics of a point charge. An oscillating point charge radiates, but
the field of the charge cannot act upon itself. Thus the charge loses energy
without any force slowing it down. Lorentz worked to give a finite size and an
internal structure to the electron without success. If the field of a point
charge is allowed to act upon itself we get an infinite answer. The issue
occupied many stalwarts, Dirac, Wheeler and Feynman, and others. Its futility
became clear after it was realised that the electron can be a point particle,
yet obey quantum rules and never be localised to an ideal point. The questions
in the revised version got transferred to those of a calculable framework of
Quantum Electrodynamics, eventually addressed by renormalisation.

##  5 Exorcising the bogeys

A large number of professionals putting Quantum Mechanics successfully to use
remain unsure about its consistency and validity. To the extent these may only
be anxieties transmitted by the previous generation it is worth examining the
various enigmatic issues, and to finally banish the bogeys which can be
resolved in the light of the postulates formulated here.

To this end I go through a list of terms commonly used which continue to
reinforce a negative perception. It is my appeal that it is appropriate to
replace some of the description with better words and if this is not possible,
at least to stop using the terms in that form.

  1. 1.

The uncertainty principle. This essential novelty of quantum phenomena was
gleaned by Heisenberg before arriving at full quantum mechanics. It holds a
central role in our understanding of quantum systems. It provides a handy tool
for making important estimates, also extended to energy-time uncertainty
relation. However the term “uncertain” brings in serious negative connotation,
which is worth avoiding.

As all textbooks discuss [7], this principle is a direct consequence of the
superposition principle and the kinematics of canonically conjugate variables,
postulates I and V.

  2. 2.

Wave function. The concept of the wave function is of great utility especially
in visualising bound states and the electron probability density distribution
in an atomic orbital. But the concept seems to get confused with waves on
water or on stretched membranes.

In Dirac notation, $\Psi(x)=\langle x|\Psi\rangle$, or it is nothing but the
component of the state along the basis vector $|x\rangle$

$|\Psi\rangle=\sum_{x}|x\rangle\langle
x|\Psi\rangle\equiv\sum_{x}C_{x}|x\rangle\equiv\sum_{x}\Psi(x)|x\rangle$

showing that it has the status of component $C_{x}$. However the wave function
can be obtained in any convenient basis such as the momentum basis
$\Phi(p,t)\equiv\langle p|\Psi t\rangle$, and convenient for bound states in
spherically symmetric potentials, $\Phi(l,m,t)\equiv\langle l,m|\Psi
t\rangle$. These can be interpreted as the corresponding “components” of the
abstract state vector in that basis. Indeed position space has a special
status, more than any other observable. One then accords great primacy to this
description, leading to mental pictures of catastrophic “collapse” of the wave
function all over space.

What about the fact that we have de Broglie wavelength associated with a
particle? A free particle is in a momentum eigenstate, with some value $p$.
Due to the existence of the fundamental constant $\hbar$, we can come up with
a quantity of dimension of length, $\lambda=2\pi\hbar/p$. In hindsight, once
the pilot waves are abandoned, there is nothing more to the “wave picture”
than this convenient re-expression of the momentum eigenvalue. The argument
extends to a “wave packet” which is a linear superposition of momentum
eigenstates.

  3. 3.

Wave particle duality. The term duality immediately suggests ambiguity of
description. The classic experimental situation where wave particle duality
can be demonstrated is the double slit experiment. If we can trace the
electron as streaming through one of the two slits we call it the particle
manifestation, if we forgo the knowledge of which slit it traveled through we
obtain a fringe pattern characteristic of wave phenomena.

Let us be clear that even in the “wave” avatar, the electron is going to be
actually measured as a particle, by a click counter. The number of clicks will
now be distributed in the fringe pattern. Further, the wave is only a mental
construct obtained by relabeling momentum eigenstates in terms of
“wavelengths” $\lambda$ as discussed in Wave Function topic above.

But don’t we still face the paradoxical situation? Not if we properly digest
postulate V, the canonical commutation relation. Effectively we are switching
between position observable ( which slit) and momentum observable (free
streaming to interfere) and the what we observe is easily deduced from
Kinematics. We can summarise the situation by saying that the double slit
experiment will remain a key part of pedagogical discussion as included in
Sec. 2, being the most direct manifestation of quantum phenomena. But it needs
formalisation as Postulate V and accepted as a part of the basic tenets rather
than be mystified.

  4. 4.

Collapse of the wavefunction. We have discussed above the convenience but also
the secret of the wavefunction. Due to primacy of space variable, our
postulate IV gets construed as “collapse of the wave function”. As we remarked
in 3, Sec 3.1, the enigmatic situation with respect to probabilistic outcomes
will remain or get resolved with more experiments. In the meantime we need to
banish this term and live with this postulate.

  5. 5.

Indistinguishability. We tell students about indistinguishable quanta. The
fallacy is that there was nothing to distinguish to begin with. The term is
only a hang over of the classical ideas about billiard balls. Two of them
machined with great care may be impossible to distinguish. It is with respect
to this classical expectation that we dub the quanta to fall into the same
descriptive category.

More carefully stated, in weakly coupled systems number is an observable
quantity and these are eigenstates of that observable with value 2 or more.
For photons the number is not conserved, but it is a valid observable. Note
that regardless of how many quanta it has, the state is normalised to unity.

We can trace two sources of the misnomer. One is that the many-quanta states
in weakly coupled systems are most conveniently obtained as tensor products of
1-particle or 1-quantum states. This accords an unjustified primacy to the
1-quantum state, and one thinks of the purely mathematical construct as the
physical one of two or more quanta being brought together, whereas it is a
single state in the Hilbert space.

Another is a more empirical source. A state of two quanta reduces to a state
of one quantum if one of them is absorbed or undergoes observation. And
“which” quantum got observed and “which” remained is a meaningless question.
But after the observation of one of them we have two distinct ones. It is with
respect to this putative final state that we call that system to be contain
indistinguishable quanta.

  6. 6.

Statistics. It is common to refer to the Bose-Einstein and Fermi-Dirac
prescriptions to be “statistics”. This may leave the student with the idea
that some randomness or incompleteness is present and we are sampling partial
information, as would be done in macroscopic setting.

The prescriptions are only providing the correct enumeration of states as
tensor products of 1-quantum states, with nothing statistical about the
prescription. The first manifestation of these uncanny rules was deduced in
the context of thermodynamics of quantum gases. It is perhaps a left over
terminology from that.

  7. 7.

Entanglement. A state of several quanta is merely the correctly identified
basis state of the weakly coupled system. There are really no entities
undergoing “tangling” with each other. Only if we refer to a post facto state
of quanta subjected to observation that we can say there were several
different entities which could have been entangled. Per se there are just
many-quanta states as per B-E or F-D rules with nothing tangled up inside
them. Correlations is a less drastic term, though still with respect to post
observation states.

  8. 8.

The EPR paradox. This is now a composite of several of the issues already
described. The conceptual error is to think that there are “two” systems. It
is actually one quantum state normalised to unity, with two quanta, the latter
number being the eigenstate of the number operator. When evolved, it can have
configuration space component (in the sense of point 2, Wavefunction, above)
which is highly non-local. Observation process breaks the quantum system, with
the number operator eigenvalue having reduced to one in the remainder state.
The remaining system needs to be freshly normalised to unity. The uncanny
situation should not be held up as mysterious or unsatisfactory. It only needs
to be referred to postulate IV on Observation.

##  6 Conclusion

We have summarised an essential core of the well known principles of Quantum
Mechanics itemised as seven postulates. They are not logically independent in
the axiomatic sense but are of primary importance to be listed independently.
It is necessary that students develop direct intuition about raw quantum
facts, and understand them as the sources of the mathematical laws. As such we
discussed relevant basic experiments. I have tried to dispel the negative
perception in which a variety of quantum phenomena are held. After one century
of success of the theory one must jettison the baggage of the past and attempt
to come up with better terminology, or at least avoid using its negative
contents. I have tried to argue that if one properly digests the truth of the
postulates, most of the negative connotations and the sense of spooky mystery
should get dispelled. Finally and most importantly, I have pointed out the
fallacy of instantaneous velocity which gives rise to the illusion of perfect
predictive power of classical mechanics. Indeed with hindsight one may note
that the continuum is a purely logical construct, a description not borne out
by the material media once assumed to be continua. Functional analysis
nevertheless continues to be the bulwark of Quantum Mechanics, albeit after
due tweaking. Perhaps there are more subtle formulations of the continuum that
make contact with the reality of fundamental quantum phenomena more easily.

## References

  * [1] T. W. B. Kibble, “Classical Mechanics“, 3rd Edition, Longman 1985
  * [2] P. A. M. Dirac, “Principles of Quantum Mechanics”, 4th Ed., Oxford University Press, 1958
  * [3] Paper 12 in B. L. van der Waerden, ”Source of Quantum Mechanics“ North Holland Pub. Co. 1967
  * [4] I. J. R. Aitchison, D. A. MacManus and T. M. Snyder, Am. J. Phys. 72, no.11, 1370-1379 (2004) [arXiv:quant-ph/0404009 [quant-ph]].
  * [5] K. Gottfried, Am. J. Phys. 79, 261 (2011) [arXiv:1006.4610 [physics.hist-ph]].
  * [6] https://www.nobelprize.org/prizes/physics/2022/summary/
  * [7] See for example L. I. Schiff, “Quantum Mechanics”, 3rd Ed., McGraw Hill Book Co., 1968; E. Merzbacher, “Quantum Mechanics”, 3rd Ed. John Wiley and Sons Inc., 1999
  * [8] John Stachel, Ed., “Einstein’s Miraculous Year”, Princeton University Press, 1998