Measurements in Quantum Mechanics         Mathematics is undoubtedly a powerful key, essential to unlock our understanding of physics. However, maths alone
would not get us very far, physics is a science and depends ultimately on empirical evidence obtained by observing what
happens when we conduct experiments. Measurements are also an essential key to physics. A good mathematical model
builds on prior observations and makes predictions of what we would observe in certain conditions - that is it predicts the
results of experiments and what values we would actually measure. We rely on meters of various kinds and on making
meters with the required degree of accuracy, however measurement in quantum mechanics takes on new meaning. It
turns out that measurement is a very subtle thing indeed and often the results are far from expected by intuition alone.

In this article we look at some of the features and phenomena measured in quantum mechanics (QM) and delve deeply
into what these measurements really mean. Be warned - Kansas is going bye-bye, to coin a phrase!

We begin with a simple experiment which will illustrate some of the theory we are going to need. The apparatus is
illustrated in the circuit diagram below:   We have a sealed vacuum tube in which we have a single trapped electron that is free to move inside the tube. Electrons
are negatively charged and our electron is indicated by the blue circle in the centre of the tube. At each end of the tube is
a grid, A, held at zero potential (earthed to the equipment chassis). If the electron stays between these grids then it
experiences no force (except for gravity which will be so slight on a particle so light that we will ignore it). However, the
plates at B are negatively charged, so if the electron crosses either grid, A, then it suddenly becomes subject to a
repulsive electric force (like charges repel). We make this force sufficiently strong to decelerate the electron and send it
flying back into the middle of the tube (between A and A). The electron will tend to oscillate backwards and forwards. We
have 'caged' our electron inside a 'box'.

Specifically, we have set-up an electric force-field to confine the electron. This force-field can be modeled as a
one-dimensional (1D) potential energy well; 1D because the electron is only confined along teh x-axis (as indicated in the
figure). We can represent this as follows:
The states for the finite square well can also be obtained by hand, though the calculation is a little more involved. A finite
well has a finite number of eigenstates within it, but even the most shallow well always has at least one. The wave
functions for the finite well also extend part-way through the 'soft' walls of the well before falling to zero (the particle can
tunnel its way through the barrier) though we shall not consider the details of quantum tunneling here. This penetration
of the walls, gives the particles slightly longer wavelengths than in the infinite well case and so slightly lower energies.

Bound and Unbound Solutions and the Continuum

So far we have looked at bound states, that is particles confined inside a force field or potential energy well. The deeper
the well, the more bound states or energy levels it will have, though every well has at least one bound state, no matter
how shallow. As the particle gains energy it jumps up to higher energy levels, until eventually it will jump out of the well
altogether if given sufficient (kinetic) energy (energy at least equal to the depth of the well). If we had an infinite universe
with just one energy well and one particle plus some energy source, then if the particle escapes from the well it would be
in a region of empty space with no confining force fields to perturb it. It would be a
free particle. Such a particle is then
able to possess any amount of energy - it's energy is no longer quantised! Equivalently we can say that above the well,
in free space, there is an infinite
continuum of energy levels. In the real universe, there are always other particles and
forces to interact with and no particle is ever truly free. However, a particle travelling through the 'void' of space would be
relatively free and we still say that it exists in the continuum, except that the energy is still quantised, but such that there
are so many close-spaced energy-states available to the particle that it belongs to a continuum to all intents and
purposes. Such a particle is called a free particle, even though it is never entirely free. Such a continuum would exist, for
example, in a rarified gas of atoms, in an approximate vacuum, and if enough energy is given to one of these atoms for
one of its electrons to escape from the atom (this could be energy from a laser pulse or a collision with an electron beam,
for example) then that electron becomes a free electron. The minimum amount of energy that must be given to the
electron to achieve escape is called the
ionisation energy of the atom. Ionisation energy depends on the atomic,
molecular or ionic species from which the electron escapes.

Strong measurements

When we measure the position of our particle in a box, we find that it occupies any one of the energy level eigenstates,
at 'random' (not strictly random as some states are more likely than others). These eigenstates are stationary and do not
change with time. Once a measurement finds the particle in one state, it will remain in the same state on subsequent
measurements or until the system is perturbed (the disturbance putting the system into a new state).

Actually, as Schrodinger's equation is linear and these are linear solutions, then just like any linear wave, we can add
more waves together to form a new wave (
superposition). This happens when we pluck a guitar string - the harmonics
are superposed to give a note. We also see this in water waves: when they run into one-another they can cancel out or
reinforce one-another as they add together. This means that our particle can be in a more general state that is a mixture
of stationary eigenstates. Such a state gives us motion - an electron in a mixture of eigenstates can change with time, it
can move and vibrate! This is how we go from standing waves to quantum motion. According to textbook QM the problem
is that we can never actually observe the particle in one of these mixed states. Such states are fragile and as soon as we
take a measurement we disturb the system, causing the mixed wavefunction to
collapse into one of its component
eigenstates. It is generally believed, or at least until recently, that a measurement inevitably collapses the system. This
makes some sense, we are measuring something that is minutely small and we measure it by interacting with it. A good
example is measuring the position of a photon in the
two-slit diffraction experiment - we measure its position by
intercepting it, causing it to collide with a screen! This is clearly quite drastic and is bound to alter the photon in some
way. The result of the measurement then gives us the state of the photon after the measurement, when it has been
brought to rest. These disruptive measurements are called
strong measurements. We can only deduce the original
mixed state in principle by analysing an ensemble of systems in the same initial state and seeing what proportion end up
in each eigenstate, giving us the corresponding mixture.

Similarly, when an electron orbits the proton nucleus in a hydrogen atom, it is in a state of motion by virtue of the fact
that it is in a mixed state, but as soon as we measure its energy or position it collapses into an eigenstate of the
hydrogen atom, such as a 1s or a 2p or a 3d orbital, etc. Once in this state it does not orbit according to the maths, since
it is in a stationary state. If it does orbit, then there are
hidden variables that we can not measure and so know nothing
about. As we shall see, the consensus opinion rules out such hidden variables, though the truth appears somewhat more
subtle, as we shall see. Contradictions and confusions are already plentiful regarding this issue of electron orbits or
trajectories.

Another example of a strong measurement is the
Stern-Gerlach experiment which demonstrated the quantisation of
angular momentum
in atoms. An oven prepares a gas of alkali metal atoms, such as caesium (Cs) atoms. The intrinsic
angular momentum of an atom depends on its electrons. Electrons can have
spin angular momentum, due to their
rotation, which can be pictured (though not literally!) as either clockwise or anticlockwise, or spin-up and spin-down, with
an actual value of 1/2 (in units of Planck's constant over 2 x pi). Additionally, the electron can have
orbital angular
momentum
, which happens when the orbital is not spherically symmetric, as in a p, d or f-orbital, but not the s-orbital
which is spherically symmetric. In complete electron shells, the orbitals combine to give the shell spherical symmetry, so a
complete shell has zero orbital angular momentum (the orbital angular momenta of the electrons in the shell cancel). In
an orbital occupied by the maximum of 2 electrons, the electrons pair with anti-parallel (opposite) spins and so again the
spin-angular momenta cancel. Again this means that in complete shells the net spin-angular momentum is also zero. In
alkali metal atoms, the outer valence shell of electrons, the only incomplete shell, contains a single s-electron. Thus
these atoms have no net orbital-angular momenta, but do possess a net spin-angular momentum of 1/2. When a
measurement is made of the intrinsic angular momentum of these atoms only two possible values are found - either +1/2
or -1/2 (spin-up or spin-down) corresponding to the angular momentum of the single valence electron. These two values
occur with equal probability (p = 0.5). Thus, the angular momentum of the atom is quantised - it can only take one of two
possible values. In classic mechanics we would expect any value to be possible, within reasonable limits, as is the case
for a spinning football, but this is not the case.

Note that, unless we invoke hidden variables, it is NOT generally the case that atoms possess either one or the other
value prior to the measurement! Rather, the atoms usually possess a (linear) mixture or
superposition of the spin-up
and spin-down values, but as soon as the system is disturbed by taking a measurement, then each atom acquires one or
other of the only two possible values. These two values are stable
eigenstates. They are stationary states that do not
change with time, so a repeated measurement on each atom would in this case produce the same result (unless the
system is perturbed or allowed to evolve for sufficient time between measurements in which case it returns to a linear
superposition of the two eigenstates).

In the Stern-Gerlach experiment, the caesium sample is vaporised in the oven and the emerging beam collimated to form
a narrow beam in which all the atoms are travelling more-or-less in one direction (otherwise they do not pass through the
holes in the collimators). This beam then passes through a pair of magnetic poles. These magnets are shaped so that
the magnetic form between them is not uniform and so accelerates the atoms by deflecting them by an amount that is
proportionate to the intrinsic angular momentum of the atoms (and to the strength of the field). Since there are only two
such values, the beam splits into two and then strikes a screen or detector.
These are 1D square wells (of finite depth). V is the voltage (potential difference) which is a function of position x, being
zero in the earthed centre of the box and rising steeply to our set potential (V=V0 at B) at the ends of the tube or walls of
the well. The well at the top represents the force-field in the case when the repelling potential at B is quite weak: the well
is shallow and 'has 'soft walls', that is the walls slope and are not rigid, which means in practice the electron can move
through the walls a short distance (that is it can move past A a short way before being repulsed). The height of the well is
proportional to the potential difference (between a and B) that is the strength of the force-field. If we increase the
potential, that is strengthen the force-field, then the electron will be repulsed more strongly and more sharply: the walls
are steeper and more rigid, and the well is also deeper. The greater the potential or force-field strength, the deeper the
well. The deeper the well, the more vertical and more rigid its walls.

Note: our 1D finite square well potential is a simplified model of the force-field containing our electron. The same potential
well can be used to model a range of situations, such as an electron bound to an atom. In this case, however, the square
well is very inaccurate and is replaced by a Coulomb potential well, for the electrostatic force of attraction between the
electron and the proton in say a hydrogen atom. More complexly shaped wells are harder to solve mathematically, and
usually require a computer to approximate the mathematical solution to the required degree of accuracy. The square
well, however, is a good example used in foundation QM courses, since it can be solved relatively easily by hand.
Another elementary energy well is a parabolic well, which gives us the QM simple harmonic oscillator, which can be used
to represent vibrating bonds in molecules.

If we increase our potential indefinitely, then our well will become infinitely deep and then its walls will be perfectly vertical
and impenetrable. This is an ideal that can only approximate our experiment, but the so-called infinite square well is
easier to handle mathematically than the finite square well and introduces many of the key features of quantum force
fields. We can then calculate what possible energy values our electron might have. Consider many electrons in our box,
or even better many boxes, each with one electron it, then it stands to reason that measuring the energy, velocity or
momentum of the electrons will give us a well-defined average for a given potential, but what sorts of values would we
obtain for the energy of an individual electron? Clearly this will vary statistically - not all the electrons will have exactly the
same value. What is found is illustrated below:
To see the full calculation for the 1D
infinite well, click the sequence of four
thumbnails, running from left to right.
Recall that a particle is also a wave, as a result of wave-particle duality. Our electron is a confined wave bouncing back
and forth. In such a situation only certain frequencies of vibration are permissible, as the others cancel out, and the
result is a
standing wave. A similar result is obtained when a guitar string vibrates - only standing waves of certain
frequencies occur, these are the
harmonics, and the wave of lowest frequency (lowest energy and longest wavelength)
is the
fundamental. The frequencies of the other higher-energy harmonics are definite multiples of the fundamental. In
our QM case the waves are solutions of the time-dependent
Schrodinger wave equation (TDSWE) and are called
wave functions. The square of the wave function corresponds to the wave pattern actually observed and the particle
will be positioned somewhere on this wave. In fact we find that the time-dependent part disappears and the wave
functions are solutions of the time-independent Schrodinger wave equation (TIDSWE) - like the guitar-string harmonics
they are standing waves that do not change with time, they are
stationary states or eigenstates. The wave functions
(
eigenfunctions) are shown on the diagram above by the red curves. The blue horizontal lines correspond to the
energy levels of each wave. There is an infinite number of such states inside the infinite well, called
eigenstates, for the
infinite well and each has its own discrete value for energy (the whole set of energies forming the set of energy
eigenvalues). The fact that only certain discrete energies are allowed is the quantisation of energy. The total set of
eigenstates is the
spectrum of states. The lowest energy wave is the one nearest the 'bottom' of the well and this has
the longest wavelength (it is the fundamental). Recall that for a photon in a light wave frequency and energy are linked
by: Where E is particle/wave energy, h is Planck's constant, and v (Greek nu) is frequency. This means that the lower the
energy of the photon, the lower its frequency and the longer its wavelength (it is toward the red-end of the spectrum). A
similar condition applies to our particle-waves - the lower the energy of the particle/wave, the lower its frequency and the
longer its wavelength. Also for a light wave: This applies to other waves, including particles, so long as we replace c by the speed of the wave.

Thus, the lowest energy state has the smallest frequency and the longest wavelength - it is the one at the bottom of the
well with energy E = En = E1, the first eigenvalue (n = 1). Note that as we move to higher energy states (E2, E3, ...) the
frequency increases by one half-cycle, the wavelength shortens and an extra half a wavelength occurs within the well. E1
is the
ground state, and the other states are higher-energy or excited states.

Where is our particle?

A measurement of a particle in a potential well will reveal the particle to be in any one of the possible energy states or
energy levels in the spectrum. Repeating the measurements on a large set of systems prepared in the same way, called
an
ensemble of states, will reveal that each state occurs with a given probability. Some will be more probable than
others, some will be highly unlikely, but we can never determine beforehand which energy level (energy eigenstate) the
particle will actually be found in. As for its position, it will be found somewhere within the space given by the square of the
wavefunction. Where this squared wavefunction is zero (a node) the particle can never be found, and it will be most likely
found where the square wavefunction is highest in amplitude. Again, we can never tell exactly where it will be until we
measure it - there appears to be inherent
indeterminacy in the system. Our particle seems elusive, having become
more like a probability cloud. The best we can do, in principle as this is not apparently a practical limitation, is to state the
probability that a particle in a given system will be found in a certain region of space. We can also calculate average or
expectation values for the various properties of the particle, such as its energy.

For those who are interested, the solution of Schrodinger's wave equation for the infinite well, including the derivation of
the eigenfunctions and eigenvalues is shown in the series of thumbnails below.
• The Stern-Gerlach experiment demonstrates that the angular momentum of atoms is quantised.

• It also measures the spin angular momentum of a single electron when alkali metals are used.

Weak measurements

Strong measurements, as we have seen, disrupt the system being measured to such an extent that the system collapses
into one of the possible eigenstates at 'random' (that is stochastically, some states may be more probable than others
and so this is not strictly random, but which state results seems to be probablistic rather than deterministic). In a sense,
the eigenstates or stationary states are the most stable and so when the atom system is strongly disturbed, or
perturbed, it falls into one of these states. By this approach it is not possible to measure the original superposition of
eigenstates directly. This is the state of affairs described in textbooks on quantum mechanics. However, recent
experiments have shown that another type of measurement is possible - so-called
weak measurements. In a weak
measurement, the measurement perturbs the system so slightly that the particles do not collapse into an eigenstate,
rather any change that occurs to them is reversible. However, this does not allow us to measure the superposed state of
a single particle, such as an atom, because the weakness of the measurement means that very little information about
the system is extracted. To counter this a large number or
ensemble (population) of particles prepared in an identical
manner must be used. This does not necessarily mean that we have to take an average of many experiments involving
one particle, though we could do it that way we can also use many particles and perform a single measurement to
measure the so-called weak values to arbitrary accuracy. Thus, weak measurements describe the behaviour of
ensembles rather than any particular particle.

For example, weak measurements to try and pin-point which path a photon follows in a
double-slit diffraction experiment
have yielded average trajectories along which the
ensembles of photons travel, but not the trajectories travelled by
individual photons -
Heisenberg's Uncertainty Principle still forbids an individual particle from travelling along a
precise trajectory since its energy and momentum can not, in principle, be both measured simultaneously with arbitrary
position so we can not plot a graph of momentum versus time for a single particle! This would require hidden variables
again - if the particles do indeed travel along precise trajectories, as some scientists think, then we still can not measure
them, they remain hidden. As we shall see later the evidence is strongly against the idea of hidden variables, but does
not rule it out completely. As we will explain later, I do not consider it more 'natural' to expect particles to follow precise
trajectories and I do not think that intuition should be used to expect any particular behaviour. Those that believe in
definite particle trajectories adopt the idea is that since a football follows a definite trajectory then so should an electron
or atom, but I see no reason why we should expect this and indeed I consider this viewpoint at least as paradoxical as
the more widely accepted interpretation given here as I will explain under the section 'Hidden variables'. Intuition simply
fails us - the atomic and subatomic worlds are 'simply different'!

Several theoretical and practical methods have been devised for taking weak measurements. For example, to measure
the movement (momentum) of an ensemble of charged particles, one could use a large charge attached to a sensor and
measure the deflection of that charge (the momentum transferred to it by the electrostatic interaction). In the case of the
double-slit experiment, scientists measured the polarisation of the photons, which weakly couples (is weakly dependent
on) the momentum of the photons and so gives an imprecise measurement of the photon's momentum without changing
its momentum irreversibly to any appreciable degree.

Another example, of relevance to quantum computing, uses a
quantum point contact to measure electric charge. A
quantum point contact is a narrow channel in a semiconductor between two metal electrodes (such as gold electrodes
condensed onto the semiconductor base) indeed it is very narrow, typically of the order of one micrometre (one
thousandth of a mm) or less. Such a narrow channel can make a very sensitive charge sensor, so that not much charge
is needed to produce a reading. This is valuable when coupled to systems of
qubits (quantum bits) used in a quantum
computer. For example, a pair of qubits may consist of a pair of electron spins in a superposition state, in which the
spins of the two electrons are
entangled in a coherent state. A weak measurement is needed to read the qubit
without destroying its coherence (causing
decoherence) as a strong measurement would by forcing the coherent state
or wavefunction to collapse into an eigenstate. The process is a little more subtle in practise (see our introduction to
quantum computing for more details). In reality, rather than using a single pair of electrons, quantum dots are used. A
quantum dot is a group of about one million semiconductor atoms, larger than a molecule, but much smaller than a
normal solid-state crystal or visible lump of matter.

A quantum dot can be used instead of a single electron or photon as a qubit. In a typical quantum dot, gold electrodes
generate an electric field to confine a small number of electrons inside the quantum dot. A pair of such dots side-by-side
can form a
double quantum dot. Electrons within each dot interact weakly with the million or so nuclei (via the
hyperfine interaction) and can interact with one-another to form a coherent state. Such a state can have a definite value
of spin, since a coherent state behaves as a single quantum state. The two dots in a pair can then also be entangled
into a single coherent state. Either the spin or the charge distribution in this dot-pair can be measured (for example if
one excess electron is shared between the pair then different states exist depending which dot has the electron). The
system must be kept very cool, since heat energy disrupts coherent states. Coherence is necessary for the
computations to occur. One way to measure the spins of the two dots without destroying coherence is to measure it
indirectly, via a third qubit linked to the other two. This third quantum dot is allowed to get warm, so that it acts as a sink
for the entropy (disorder) caused by thermal effects. Electrons will slowly leak from the quantum dots by quantum
tunnelling. By using a sensitive detector, such as a quantum point contact, it is possible to measure this leaked charge
whilst maintaining the state of the dots.

Optical systems of qubits are also possible, in which we deal with photon polarisation rather than electron spin. Such a
quantum dot was used as a source of single photons of well-defined wavelength in the double-slit diffraction experiment
involving weak measurements.

The objective of a weak measurement is to transfer very little momentum from the system under study to the measuring
apparatus, causing a minimum and reversible perturbation. This means that there are large uncertainties in the
measurements, which can be reduced by studying a large number of identically prepared particles (an ensemble).

In contrast, strong measurements are more accurate (though limited in principle by the uncertainty principle) but do so
by changing the momentum and state of the system irreversibly, in particular they remove any superposition (including
entanglement and coherence) by placing the system in a stationary eigenstate. This also means that the state of the
system prior to the measurement can not be measured in this way, only the final eigenstate. This is sometimes called
post-selection, as the measurement returns the value the system has
after the measurement! Atoms and subatomic
particles are so minute and thus easily disturbed, that most measurements are of the strong type and weak
measurements are a relatively recent realisation.

The advantage of weak measurements is that they can yield information about a system prior to a strong measurement. Above: the apparatus used in the Stern-Gerlach experiment. In
reality, since the magnetic field weakens  on either side of the
beam, two arcs are formed, resulting in an approximate
ellipse-shape as shown on the left-hand side of the diagram on
the left. Classically we would expect any value within the range to
occur, if the angular momentum of atoms was not quantised, in
which case a solid ellipse would result as shown on the
right-hand side of the diagram on the left.
What we would expect to happen?

• We might expect the positron to always arrive at the B+ detector whenever the electron arrives at the A-
detector.

• We might expect the positron to arrive at the A+ detector whenever the electron arrives at the B- detector.

• We might not expect the positron to arrive at detector A+ whenever the electron arrives at A- as then the two
should have crossed and annihilated instead!

• Suppose the electron arrives at B- and the positron at B+ ( a B+B- coincidence detection). We might suppose
that the electron must have been in the overlapping arm, otherwise there would be no interference to offset the
self-destructive interference of the positron. If the electron was not in the overlapping arm then the positron can
not reach detector B+ (as was the case when the interferometers never overlapped).

• Similarly, when the electron arrives at B- and the positron at B+, we suppose that the positron must have been
in the overlapping arm, otherwise it could not have interfered with the electron so as to prevent its destructive
self-interference and so allow it to reach B-.

This then is the paradox: How can we have a B+B- coincidence detection in the first place, as both particles would
have to have been in the overlapping arms and so should have annihilated one-another resulting in no detection at
all.

Many would simply dismiss this and say that it's all theoretical, or that clearly the particles would annihilate and so a
B+B- event could never be recorded. (We have assumed that the particles always annihilate when they meet).
However, think about the original set-up. In the single interferometer, a single positron (or electron) could interfere
with itself, preventing itself from reaching the B+ (or B-) detector. This is because of the wave nature of the particles -
they can be in both paths at the same time!

For this reason, it is not so obvious that annihilation and hence no B+B- event could occur. We could attempt to
resolve the paradox by making weak measurements on each path to determine whether or not the particles are in one
or both paths. We couldn't use strong measurements as this would collapse the system and force the particles to be
in one or other of the paths but never both.

What we could measure, by weak measurements, are the numbers of particles in each path (we have to use an
ensemble of particle-antiparticle pairs to make weak measurements rather than a single particle-antiparticle pair).
This situation can be analysed quantitatively, using quantum theory with
number operators - mathematical
operators that act on the mathematical representation of our system to extract the (average) numbers of electrons
and positrons in each path.

Thus,
N(O+) gives us the number of positrons in the overlapping path;
N(NO+) the number of positrons in the non-overlapping path;
N(O-) the number of electrons in the overlapping path;
N(NO-) the number of electrons in the non-overlapping path;

In addition to these single-occupancy number operators, when the electron and positron are entangled in a coherent
superposition, then we can also use the following
paired occupancy operators:

N((NO+,NO-) gives the number of positrons and electrons in the non-overlapping paths;
N(O+,O-) gives the number of positrons and electrons in the overlapping paths;
N(NO+,O-) gives the number of positrons in the non-overlapping path and the number of electrons in the overlapping
path;
N(O+,NO-) the number of positrons in the overlapping path and the number of electrons in the non-overlapping path.

I wont give the calculations here, but the results for an observed B+B- coincidence detection are:

N(O+) = N(O-) = 1, so the electron and positron can indeed interfere with one-another;
N(NO+) = N(NO-) = 0;

N(O+,O-) = 0, so both were not in the overlapping arm and so did not annihilate;
N(NO+,O-) = 1 (pair), so the electron was in the overlapping arm, allowing the positron to reach B+;
N(O+,NO-) = 1 (pair), so the positron was in the overlapping arm, allowing the electron to reach B-;

N(NO+,NO-) = -1 (pair); this is very interesting since it means that the total number of electrons in the non-overlapping
path is:

N(NO+,NO-) + N(O+,NO-) = -1 + 1 = 0;

Without this -1 value the number of particle-antiparticle pairs would be:
N(NO+,O-) + N(O+,NO-) = 1 + 1 = 2, which can not be since we are putting in one pair at a time!

Thus, the N(NO+,NO-) = -1 operator ensures conservation of total particle number, whilst the electron is allowed to
reach B- at the same time the positron reaches B+ without annihilation!

Perhaps we should not dismiss Hardy's paradox after all. Perhaps it has given us an insight into how bizarre quantum
mechanics can be! It also illustrates that weak measurements likely will not remove the 'oddities' of quantum
mechanics, as some who adhere to the hidden variable theory might hope.

The full implications of weak measurements are not yet understood.

Atomic and molecular orbitals - are they real?

The wavefunctions (eigenfunctions) of the hydrogen atom represent the solutions to Schrodinger's wave equation for
a Coulomb (electrostatic) potential or force field. We envisage the negatively charged electron as trapped inside the
force-field of electrostatic attraction with the positively charged nucleus. These wavefunctions are the energy levels or
orbitals of the electron in the atom, that is they describe the possible states of the electron. The energy of the
electron depends primarily on how close to the nucleus it is and is given by the
principal quantum number, n = 1,
2, 3, ... (a higher energy electron climbs up the well and so is further from the nucleus). The eigenvalues
corresponding to these solutions represent the energy of an electron in the corresponding state described by the
wavefunction and depend mainly on
n. This is illustrated below:
Hardy interferometers, the Hardy Paradox and more weak measurements

To further understand the issues of quantum measurements, we shall look at one particular gedanken (thought
experiment - an 'experiment' carried out in the mind by application of theory, typically prior to the techniques being
available to actually perform the measurement). We can use photons of opposite phase or electron-positron pairs for
this experiment, we chose the latter. Electrons and their positively charged anti-matter counterparts, positrons, can be
generated as electron-positron pairs. electrons and positrons, like photons, exhibit wave-like behaviour and so can be
used in an interferometer. An interferometer splits a beam of particles from a common source (which produces particles
with a well-defined energy and wavelength) into two and directs these two beams along different paths before
recombining them and measuring the resultant intensity. When two waves combine, they interfere with one-another by
superposition (see our introduction to
waves for a description of this). If the waves of the particles are exactly out of
phase when they meet and combine then, just like water waves, they cancel out to produce nothing, a process called
destructive interference. If, however, they are exactly in-phase then they recombine to give a wave of higher
intensity (
constructive interference). If the two path-lengths of the beams are identical then the two will generally
meet in-phase and undergo constructive interference. (However, it should be noted that reflecting a wave, such as by
bouncing it off a mirror, causes the phase to invert - crests become troughs and vice-versa). If the two path-lengths
differ by exactly half a wavelength then the two destructively interfere. Differences of fractions of a half-wavelength will
produce other results. In this way an interferometer can be used to measure distances with extreme accuracy.

In our experiment, we shall use a pair of interferometers, one for the electrons (e-) that are produced, another (which is
identical to the first) for the positrons (e+). One such interferometer is shown below:
This is the one we shall use for the positron, as indicated by the + symbols. The positron beam is split into two by a
beam splitter (BS) labelled BS1+. In the case of light this would consist of a half-silvered mirror that reflects half of the
light whilst allowing the rest through. For charged particles we can use electrostatic deflectors. Each beam then
bounces off a 'mirror' (such as a positively charged sheet) and the two sub-beams are recombined at a second
beam-splitter (BS2+) before arriving at two detectors (A+ and B+). The arrows indicate the beam directions. The
interferometer used for the electrons is identical, but with negative labels. Both are shown below:
It is possible to adjust the path-lengths of the two sub-beams (between BS1 and BS2) such that only the A-detectors
(A+ and A-) will detect particles
- particles headed for the B-detectors cancel by destructive interference (even a
single particle, acting like a wave, can interfere with itself!). We shall fire a large number of electron-positron pairs into
this system, one at a time. Whether each particle takes the inner or outer path is determined probabilistically - that is
there is an equal chance that each particle will follow the inner path as the outer path and there is no way to predict
this, other than to say that each path will be taken 50% of the time on average.

What happens if we overlap the two interferometers such that their paths cross and interfere? Electrons and positrons
will annihilate when they meet. As shown in the diagram below, we have overlapped the inner arms or paths of the two
interferometers, leaving the outer paths as non-overlapping. This has two effects:

1) if the electron and positron both take the overlapping paths, then they will meet and destroy one-another.

2) Additionally, we can arrange things so that the paths interfere with one-another, whenever one of the particles is
one of the overlapping paths, such that the self-interference of each particle is removed, allowing each particle, in
principle, to reach the B-detector. E.g., if the electron is in the overlapping arm, it can interfere with the positron in
such a way as to stop the positron interfering with itself, allowing the positron to reach detector B+.
Notice that n also corresponds to the number of nodes (points at which the displacement along the vertical axis is
zero and thus where the wave cuts the horizontal axis) in the (radial) wavefunction (number of nodes =
n - 1) - higher
energy wavefunctions vibrate at a higher frequency and so have more nodes. The combination of nodes as
determined by n and also the angular momentum of the electron determine the various shapes of the atomic orbitals
(such as spherical s-orbitals, dumbbell-shaped  p-orbitals, etc.). Some of these shapes can be seen
here. The
pertinent question now is: Can we see atomic orbitals?

First of all, the eigenvalues can be verified by experiment as they account for atomic spectra which are very well
understood. Schrodinger's model does have some simplifying assumptions, such as its failure to account for
relativistic effects, but is still very accurate in predicting the eigenfunctions, and is extremely accurate when several
corrections are made (such as incorporating relativistic effects).

Can we observe the wavefunctions? First of all, to be precise it is the square of the wavefunction we observe, since
this corresponds to the charge density which is what we observe when we observe an electron. Of course these
structures are too small to be 'seen' in the normal sense using light, but nevertheless are they real? Charge density
can be measured in many ways and agrees with the predictions of quantum mechanics, but observing the radial
structure and the nodes is a different matter. This would be like viewing an atom in cross-section.  It has been
argued that wavefunctions are not real but are rather simply mathematical constructs.

The wavefunctions actually form an eigenbasis, essentially building blocks from which other states are made, by
superposition. The eigenbasis depends on the choice of mathematical system used to describe the atom (position
space or momentum space for example). Thus, we may end up building the same composite states from a different
set of wavefunctions. Chemists will be familiar with the construction of d-orbitals from several eigenfunctions.
However, some actual states are indeed described by single wavefunctions and so we must conclude that at least
some of the wavefunctions correspond to real atomic shapes. However, eigenfunctions correspond to stable
stationary states into which an initial wavefunction collapses after a typical strong measurement. The initial state is
often a superposition of stationary states (a wave packet or mixture of wavefunctions in various proportions) so the
initial mixed state usually elludes us (weak measurements possibly provide an exception to this).

This has been confirmed by recent experiments which use a photoionisation microscope to view the 'nodal structure'
of hydrogen atom wavefunctions almost directly. Not every state of an atom can be observed so directly, but
hydrogen atoms have been prepared in Rydberg states, that is in high energy states by applying lasers and then
placed in an electric field.
Rydberg atoms are large distended atoms (up to one hundredth of a millimetre in
diameter, or about the size of a 'typical' animal cell!) and so easily distorted by relatively weak magnetic fields. This
distorts the atom in a direction determined by the field, such that instead of states described by
n, we now have
states described by
n1 and n2, in which n2 can be large (e.g. n2 = 28, a Rydberg state) whilst n1 can be small (n1 =
0, 1, 2, 3, 4, etc.). An atom distorted by an electric field in this manner is in a Stark state and the change to its
wavefunctions and eigenvalues is the
Stark effect. Now, Rydberg atoms are unstable, lasting about one second
before shrinking by losing an electron (causing 'n' to reduce) but the electric field can be arranged to act as a barrier
to the electron's escape. However, electrons will sometimes quantum tunnel through this barrier and escape, as the
atom becomes ionised (the electron can be described as quasi-bound). As it happens, calculations show that ,
remarkably, the tunneling electrons in this case, tunneling from an ensemble of similar atoms, carry information
about the nodal structure of the wavefunctions (
n1 specifically) with them, as predicted by calculation. These
patterns are very similar to the 1s, 2s, 3s and 4s orbitals in form (though not quite identical as these are Stark
states). More precisely, these experiments prepared the atoms in a superposition of stationary states, but the wave
packet collapses into one eigenfunction following (strong) measurement. The results of these experiments are
discussed here:
http://physicsworld.com/cws/article/news/2013/may/23/quantum-microscope-peers-into-the-
hydrogen-atom

It is worth mentioning that a number of other experiments have given insight into the structure of wavefunctions in
both individual atoms and molecules. In molecules, molecular orbitals form as the result of combination of certain
wavefunctions in the constituent atoms (at least approximately). A recent experiment with an atomic-force
microscope, a device which passes a very tiny probe over a surface and measures the force acting on the probe
due to interactions such as covalent bonding or quantum tunneling, gave a good visualisation of what appears to be
a hybrid orbital between a 3s atomic orbital and two 3p atomic orbitals consisting of two hemispherical lobes.

Those with a chemistry background may be familiar with the shapes of the p, d and f orbitals presented in chemistry
texts, for example the three dumbbell-shaped p-orbitals which are identical to one-another in shape but lie along
different axes (they are at right-angles to one-another). However, these do not correspond to the shapes of the
eigenfunctions (stationary states) obtained by solving Schrodinger's equation for the hydrogen atom (see:
mathematical plots of the atomic orbitals). Why the discrepancy? First of all, mathematically speaking, any linearly
weighted sum of eigenfunctions is also a solution to Schrodinger's equation. Thus, we can, for example, add one-
quarter of an s-orbital to 3 quarters of a p-orbital and obtain an acceptable solution. However, such superpositions
are no-longer stationary states (the electron can undergo wave-motion as an oscillating wave packet) and any
strong measurement performed on our hybrid orbital will cause it to collapse into either the s or p state: it will not
remain in a superposition after a strong measurement. What is done to construct the standard orbitals shown in
chemistry books is to apply superposition, but only to the
spherical harmonics. The spherical harmonics are
combined in such a way that the imaginary component disappears to obtain real (i.e. non-complex) spherical
harmonics (it is not clear why this is done, since squaring the wavefunction to obtain the probability distribution
abolishes the imaginary part anyway). In this way a new set of basis states (eigenstates) are obtained.

The end result, for the p-orbitals at least, is that each p-orbital becomes equivalent: they each have the same shape
but in a different orientation. This perhaps makes sense when you consider that chemists are concerned with atoms
surrounded by other atoms in molecules. Consider an atom in a solid crystal, for example, it is surrounded by
magnetic fields on all sides and if the crystal is isotropic (uniform in each direction) then there is no reason to
suppose the orbital;s will have a preferred shape along any one axis, as is the case with the H atom eigenfunctions
given in physics texts. We might expect the p-orbitals to combine in some way to equalise themselves. Some authors
dismiss the whole manoeuvre as inappropriate. Personally, I also have my doubts, but I am not prepared to dismiss
this approach in the absence of empirical data. It is important, therefore, to carry out measurements on the shapes
of atomic orbitals, wherever possible, to verify which solutions of Schrodinger's equation give the correct eigenstates.

Hidden variables - what are they and do they exist?

under construction ...

Bell's inequality

Uncertainty Principles

This article is still under construction (it is a large and complex topic requiring lots of research, so bear with us!).

Square wells

French, A.P. and taylor, E.F. 1998. An introduction to quantum physics. M.I.T. Introductory physics series.

Weak measurements and Hardy's Paradox

Aharonov, Y., Botero, A., Popescu, S., Reznik B. and J. Tollaksen, 2002. Revisiting Hardy’s paradox: counterfactual
statements, real measurements, entanglement and weak values.
Physics Letters A 301: 130–138.

Neben, A. 2011. Weak measurements in quantum mechanics. http://hep.uchicago.edu/~rosner/p342/projs/neben.pdf

Kocsis, S., Braverman, B., Ravets, S., Stevens, M.J., Mirin, R.P., Shalm,L.K., and A.M. Steinberg, 2011. Observing
the Average Trajectories of Single Photons in a Two-Slit Interferometer.
Science 332: 1170.

Observing atomic and molecular orbitals

Giessibl, FJ; Hembacher, S; Bielefeldt, H; and J. Mannhart, 2000. Surface Observed by Atomic Force Microscopy
Science 289: 422-425.

Stodolna,AS; Rouze´e, A.; Le´pine, F; Cohen, S; Robicheaux, F; Gijsbertsen, A; Jungmann, JH; Bordas, C and M. J.
J. Vrakking, 2013. Hydrogen Atoms under Magnification: Direct Observation of the Nodal Structure of Stark States.
PRL 110: 213001.  