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:

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:

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.

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

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

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

molecular or ionic species from which the electron escapes.

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

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

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

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

angular momentum

angular momentum of an atom depends on its electrons. Electrons can have

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

momentum

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

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

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:

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.

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:

and forth. In such a situation only certain frequencies of vibration are permissible, as the others cancel out, and the

result is a

frequencies occur, these are the

is the

our QM case the waves are solutions of the time-dependent

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

(

energy levels of each wave. There is an infinite number of such states inside the infinite well, called

infinite well and each has its own discrete value for energy (the whole set of energies forming the set of energy

eigenstates is the

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:

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.

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

energy levels in the spectrum. Repeating the measurements on a large set of systems prepared in the same way, called

an

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

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

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.

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

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

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

individual photons -

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 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

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

without destroying its coherence (causing

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

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

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

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.

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.

- 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-.

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.

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

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

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.

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

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

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

intensity (

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:

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+.

(A+ and A-) will detect particles

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!).

*Further reading*

__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.

zero and thus where the wave cuts the horizontal axis) in the (radial) wavefunction (number of nodes =

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.

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

states described by

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

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 (

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.

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!).

statements, real measurements, entanglement and weak values.

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 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.

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