Quantum physics

What quantum physics is about

Newton's mechanics and Maxwell's theory of electricity and magnetism (together called classical physics) describe everyday objects very well. Around 1900, experiments on atoms and light gave results that classical physics could not explain at all: the radiation given out by hot bodies, the photoelectric effect and the line spectra of gases.

Quantum physics is the branch of physics that studies the behaviour of matter and energy at the atomic and subatomic level. At this scale the rules of classical physics are replaced by new ones that at first look strange. This unit builds four of those ideas.

21.1Electromagnetic waves and the spectrum

In 1864 the British physicist James Clerk Maxwell predicted that an accelerated electric charge produces linked electric and magnetic disturbances that travel through space. A changing electric field produces a magnetic field, and a changing magnetic field produces an electric field, so the two fields keep each other going. This travelling pair of fields is an electromagnetic (EM) wave.

Electromagnetic wave

A wave made of oscillating electric and magnetic fields that are perpendicular to each other and to the direction of travel. It is produced by accelerated charges and travels through vacuum at $c=3\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$.

Properties of electromagnetic waves

• The electric and magnetic fields are perpendicular to each other and to the direction of propagation, so EM waves are transverse waves.
• The direction of propagation is the direction of the vector product $\vec{E}\times \vec{B}$.
• The two fields oscillate in phase and with the same frequency.
• All EM waves travel through vacuum with the same speed $c\approx 3\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$. They need no medium, which is why sunlight reaches Earth through empty space. They can also travel through some materials, more slowly.
• EM waves carry energy and momentum from one point to another.

Frequency $f$, wavelength $\lambda$ and speed $c$ are related by the wave equation:

$c$: speed of light ($3.00\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$), $f$: frequency in hertz (Hz), $\lambda$: wavelength in metres.

The electromagnetic spectrum

Visible light from the glowing filament of a bulb is one example of an EM wave. Radio waves, microwaves, infrared, ultraviolet, X-rays and gamma rays are the same kind of wave. They differ only in wavelength (and so in frequency). Arranged in order of wavelength they form the electromagnetic spectrum.

Inside the visible band the colours in order of decreasing wavelength are red, orange, yellow, green, blue, indigo and violet. Red light has the longest wavelength and the lowest frequency; violet has the shortest wavelength and the highest frequency.

21.1Planck's quantum theory and the photonOften examined

Classical physics treated light energy as a smooth, continuous flow. In 1900 the German physicist Max Planck studied the radiation emitted by hot bodies (from their atoms and molecules). He found that the measured results at all wavelengths could be explained only if energy is emitted or absorbed in small, fixed packets. His idea is called Planck's quantum theory, and it gave birth to quantum physics.

Postulates of Planck's quantum theory

1. Atoms and molecules emit or absorb energy only in discrete amounts (not continuously). The energy is $E=nhf$, where $n=1,2,3,\dots$ is a whole number called the quantum number.
2. The smallest amount of energy that can be emitted or absorbed as electromagnetic radiation is called a quantum (plural quanta). For radiation of frequency $f$ one quantum has energy $hf$.
3. The energies of atoms and molecules are therefore quantized. The allowed energy states are called quantum states. An atom emits or absorbs energy only when it jumps from one quantum state to another.

$n=1,2,3,\dots$ (quantum number), $f$: frequency of the radiation, $h$: Planck's constant.

Planck's constant (h)

The constant that links the energy of a quantum to its frequency, $h=6.626\times 10^{-34}\mathrm{J}\mathrm{s}$ (usually rounded to $6.63\times 10^{-34}\mathrm{J}\mathrm{s}$). Its unit is joule second ($\mathrm{J}\mathrm{s}$).

Quantum

The smallest packet of energy that can be emitted or absorbed as electromagnetic radiation, $E=hf$.

Photon

A quantum of electromagnetic radiation (light). It travels at speed $c$, has zero rest mass, energy $E=hf$ and momentum $p=h/\lambda$.

Energy of a photon

In 1905 Einstein went further: light itself travels as a stream of these packets, now called photons. The energy of one photon depends only on the frequency of the light. Using $c=f\lambda$:

Photon energy is directly proportional to frequency and inversely proportional to wavelength.

The electronvolt

Photon energies in joules are tiny numbers such as $3\times 10^{-19}\mathrm{J}$. Atomic and nuclear physics therefore uses a much smaller unit of energy, the electronvolt.

Electronvolt (eV)

The energy gained (or lost) by an electron when it moves through a potential difference of one volt. $1\mathrm{eV}=1.6\times 10^{-19}\mathrm{J}$.

21.3Momentum of a photonOften examined

A photon has zero rest mass: a photon cannot be at rest, it always moves at speed $c$. Even so it carries momentum. Maxwell's theory already showed that an electromagnetic wave carrying energy $E$ also carries momentum $E/c$. In the photon picture every photon carries its own share of this momentum.

Light can exert a force

When light falls on a surface, the photons are absorbed or reflected, so their momentum changes. By Newton's second law, a rate of change of momentum is a force. So a beam of light pushes on any surface it hits. The push per unit area is called radiation pressure.

21.2Photoelectric effect: experiment and observationsOften examined

The word photoelectric joins two words: photo (light) and electric (electron). The photoelectric effect is an interaction between light (photons) and the electrons of a metal.

Photoelectric effect

The emission of electrons from a metal surface when electromagnetic radiation of sufficiently high frequency falls on it.

Photoelectrons

The electrons emitted from a metal surface because of light falling on it.

Photoelectric current

The current produced when the photoelectrons are collected and flow round a circuit.

Experimental set-up

• An evacuated glass tube contains two metal plates. The emitter E (the metal surface under study) is joined to the negative terminal of a battery, and the collector C (also called the anode or detector) is kept at a positive potential.
• The tube is evacuated so that the emitted electrons do not collide with air molecules and the metal surface stays clean.
• When the tube is kept in the dark the ammeter reads zero: there is no current in the circuit.
• When light of suitable frequency falls on E, the ammeter shows a current at once. Electrons are emitted from the negative plate, cross the gap and are collected by the positive plate, completing the circuit.

Effect of applied voltage on photoelectric current

Keep the frequency of the light fixed and plot the photoelectric current I against the potential V of the collector, for three different intensities. The graph shows these features:

• For positive V the current rises and then becomes constant. This maximum value is the saturation current: every electron emitted per second is now collected by the anode.
• The saturation current is larger for brighter light. Brighter light of the same frequency ejects more electrons per second, but it does not give each electron more energy.
• A small current flows even at V=0, because the electrons leave the metal with some kinetic energy.
• When V is made negative (battery reversed), the collector repels the electrons. Only electrons with kinetic energy greater than e|V| reach it, so the current falls.
• At a certain negative potential -V_0 the current becomes zero: even the fastest electrons are turned back. V_0 is the stopping potential. It is the same for all three intensities.

Stopping potential (V_0)

The minimum negative (retarding) potential of the collector, relative to the emitter, at which the photoelectric current becomes zero. It measures the maximum kinetic energy of the photoelectrons.

Saturation current

The largest photoelectric current, reached when all the photoelectrons emitted per second are collected.

Effect of intensity of incident radiation on photoelectric current

Keep the frequency and the potential difference fixed and change only the intensity. The photoelectric current increases linearly with intensity. The current is proportional to the number of photoelectrons emitted per second, so:

Effect of frequency of incident radiation on the kinetic energy of photoelectrons

Now keep the intensity fixed and change the frequency. The maximum kinetic energy of the photoelectrons depends on the frequency. Even very dim light of high frequency releases electrons with large kinetic energy. For every metal there is a threshold frequency f_0: below it no electrons are emitted, however intense the light is. Classical physics has no explanation for this frequency dependence.

• Below f_0: no emission at all, even with very bright light.
• Above f_0: K.E_{\max} increases linearly with f. The graph is a straight line whose slope is Planck's constant h for every metal.
• Extending the line backwards (dashed) meets the energy axis at -\Phi, where \Phi is the work function of the metal.

Threshold frequency (f_0)

The lowest frequency of electromagnetic radiation that can produce the photoelectric effect in a given metal.

Threshold wavelength (\lambda_0)

The longest wavelength of radiation that can eject electrons from a given metal, \lambda_0=c/f_0. No photoemission occurs if the wavelength is longer than this.

Work function (\Phi)

The minimum energy needed to remove an electron from the surface of a metal (the book also calls it the threshold energy). It is measured in joules or electronvolts, and \Phi=hf_0=hc/\lambda_0.

Why classical wave theory fails

21.2Photon theory of the photoelectric effectOften examined

In 1905 Albert Einstein reinterpreted Planck's theory: light itself exists as discrete packets of energy called photons, each of energy E=hf. Photoelectric emission is the result of an interaction between one photon of the incident radiation and one electron in the metal. The photon gives all its energy to the electron and disappears.

A part of this energy (the work function \Phi) is used by the electron to break away from the metal. The rest appears as the kinetic energy of the electron. Electrons from just below the surface lose a little extra energy on the way out, so only electrons from the surface leave with the maximum kinetic energy.

The equation gives three cases:

• If hf<\Phi: there is no photoelectric effect.
• If hf=\Phi: the photoelectric effect just occurs, but the ejected electron has zero kinetic energy.
• If hf>\Phi: electrons are emitted with kinetic energy up to K.E_{\max}=hf-\Phi.

How the photon theory explains the observations

Stopping potential against frequency

Since V_0=\dfrac{h}{e}f-\dfrac{\Phi}{e}, a graph of stopping potential against frequency is a straight line. It lets us find both the work function and Planck's constant.

• Slope =h/e, so h=e\times\text{slope}. The slope is the same for every metal.
• Frequency intercept =f_0, so \Phi=hf_0.
• Intercept on the stopping potential axis (line extended back to f=0) =-\Phi/e, so \Phi=e\times|\text{intercept}|.

Uses of the photoelectric effect

• Photocells (photoelectric cells) convert light energy into electric current. They switch on street lights at dusk, open automatic doors, operate burglar alarms and count objects on a conveyor belt.
• Photoconductive devices, such as light dependent resistors, whose resistance falls when light releases charge carriers.
• Solar cells, which convert sunlight directly into electrical energy.
• Light meters in cameras and the image sensors of digital cameras.

21.3Compton's effectOften examined

In 1923 the American physicist Arthur H. Compton directed a beam of X-rays of wavelength \lambda at a block of graphite. He found that the scattered X-rays had a slightly longer wavelength \lambda' than the incident X-rays. The increase in wavelength (the loss of energy) depended on the angle through which the X-rays were scattered.

Compton effect

The scattering of X-ray (or gamma-ray) photons by electrons, in which the photons transfer energy and momentum to the electrons, so the wavelength of the scattered radiation increases.

Compton shift (\Delta\lambda)

The change in wavelength between the scattered and the incident X-ray, \Delta\lambda=\lambda'-\lambda.

Classical wave theory cannot explain this. A wave makes an electron oscillate at the wave's own frequency, so the electron should re-radiate at the same frequency and wavelength. Compton explained the result by treating the X-ray as a stream of photons.

Explanation with photons

Compton assumed that if a photon behaves like a particle, its collision with an electron is like a collision between two billiard balls: both energy and momentum must be conserved. The incident photon (energy hf, momentum h/\lambda) hits a free electron at rest and gives part of its energy and momentum to it. The electron recoils. The scattered photon is left with less energy, so its frequency is lower and its wavelength is longer.

Conservation of energy:

hf and hf': energies of the incident and scattered photons, K.E: kinetic energy given to the recoiling electron.

Conservation of momentum (a vector equation): the momentum \vec p of the incident photon equals the vector sum of the momenta of the scattered photon \vec p\,' and the recoil electron \vec p_e. The momentum of each photon is p=E/c=h/\lambda (Eq. 21.6).

Take the incident direction as the x-axis. Resolving momenta into components:

From these equations Compton derived the shift in wavelength:

m_0: rest mass of the electron, c: speed of light, \theta: scattering angle of the photon.

Compton wavelength (\lambda_c)

The constant \lambda_c=\dfrac{h}{m_0c}=2.43\times10^{-12}\,\mathrm{m}=0.00243\,\mathrm{nm}. It has the dimension of a wavelength and equals the Compton shift for scattering at 90^\circ.

Features of the Compton shift

• The shift depends only on the scattering angle \theta: \Delta\lambda=0 at \theta=0^\circ, \Delta\lambda=\lambda_c at 90^\circ and \Delta\lambda=2\lambda_c=4.86\times10^{-12}\,\mathrm{m} (the maximum) at 180^\circ.
• It does not depend on the wavelength of the incident photon or on the nature of the scattering material.
• Since (1-\cos\theta) is never negative, \lambda'\ge\lambda: the scattered photon never gains energy from an electron at rest. (The book says (1-\cos\theta) is always positive; it is zero only at \theta=0, where there is no scattering.)

Physical significance of the Compton effect

• It is strong evidence for the particle nature of electromagnetic radiation: an X-ray photon collides with an electron like a particle.
• It confirms that a photon carries momentum p=h/\lambda as well as energy E=hf.
• It shows that the laws of conservation of energy and momentum hold in each individual photon-electron collision.
• Together with the photoelectric effect it describes how photons interact with electrons: in the photoelectric effect the photon is completely absorbed; in the Compton effect it is scattered with less energy.

21.4Pair productionOften examined

Every particle has an antiparticle with the same mass but opposite charge. The antiparticle of the electron is the positron (e^+): mass m_0=9.11\times10^{-31}\,\mathrm{kg}, charge +e.

Pair production

The creation of a particle and its antiparticle from energy. When a high-energy photon passes near a heavy nucleus, it disappears and an electron-positron pair is created. The photon is totally absorbed.

Pair production is direct evidence that energy can be converted into mass, as Einstein's relation E=mc^{2} says. Three quantities are conserved in the process: charge, momentum and energy.

Conservation laws in pair production

Conservation of charge. The photon has zero charge. The electron (-e) and positron (+e) together also have zero charge. So a photon can never produce a single electron, a single positron, two electrons or two positrons: charge would not be conserved.

Conservation of momentum. The photon's momentum hf/c must equal the total momentum of the products. Here m v_{e^+} and m v_{e^-} are the momenta of the positron and electron (vector sum), and the nucleus takes a small recoil:

Conservation of energy. The photon energy becomes the total energy of the electron and positron:

The total energy of a particle is its kinetic energy plus its rest mass energy m_0c^{2}, so

Conditions for pair production

• The photon energy must be at least 2m_0c^{2}=1.02\,\mathrm{MeV} (a gamma-ray photon).
• A heavy nucleus (or another particle) must be nearby to take up momentum.
• Charge, energy and momentum must all be conserved, so a particle and its antiparticle (electron and positron) are always created together.

21.5Pair annihilationOften examined

Pair annihilation is the reverse of pair production. In pair production energy (a photon) is converted into mass (an electron-positron pair). In annihilation mass (an electron-positron pair) is converted into energy (photons).

Pair annihilation

The process in which an electron and a positron combine, both disappear, and their mass is converted into two gamma-ray photons.

Why two photons and not one

Just before annihilation the electron and positron are nearly at rest, so their total momentum is (almost) zero. A single photon always moves at speed c and carries momentum hf/c, which is never zero. One photon therefore cannot conserve momentum. Two photons of equal energy moving in opposite directions have equal and opposite momenta, so their total momentum is zero, just as before. Charge is also conserved: (+e)+(-e)=0, and photons carry no charge.

Another example of annihilation is proton-antiproton annihilation. Besides confirming the photon model of electromagnetic radiation, pair production and pair annihilation clearly illustrate Einstein's idea of mass and rest energy.

21.6Wave-particle duality and de Broglie wavesOften examined

Interference, diffraction and polarization are the basic properties of all electromagnetic waves. Light shows all three, so light can be considered a wave. But the photoelectric effect can be explained only if light is made of small energy packets (particles) called photons. So light displays a dual nature: it has particle characteristics as well as wave characteristics.

Wave-particle duality

The property of light (and of matter) of behaving as a wave in some experiments and as a stream of particles in others.

The two pictures are joined by two equations. The particle properties (energy E and momentum p) are fixed by the wave properties (frequency f and wavelength \lambda):

de Broglie's hypothesis

In 1924 the French physicist Louis de Broglie argued that nature should be symmetric. If light, which shows pure wave properties, can also have particle-like properties, then particles such as electrons can also behave as waves. Particles of matter obey a wave equation just as photons do. Compton's work showed that a photon of wavelength \lambda has momentum p=h/\lambda; de Broglie proposed the same relation for every moving particle.

de Broglie wavelength

The wavelength of the matter wave associated with a moving particle, \lambda=h/p=h/mv.

Matter waves

Waves associated with moving material particles such as electrons, protons and neutrons.

Macroscopic objects and electrons compared

Wavelength of an electron accelerated through a potential difference

21.6Electron diffraction: evidence for matter waves

If electrons are waves they must show diffraction, just like light passing through a grating. A diffraction grating works best when its spacing is comparable to the wavelength. The atoms in a crystal are about 10^{-10}\,\mathrm{m} apart, and electrons accelerated through a few hundred to a few thousand volts have wavelengths of about 10^{-10} to 10^{-11}\,\mathrm{m}, comparable to this spacing. So a thin crystal acts as a natural grating for electrons.

What the experiment shows

• A pattern of bright and dark concentric rings appears on the screen, similar to the pattern made when X-rays pass through a powdered crystal. Rings of this kind are produced by constructive and destructive interference, which only waves can do.
• When the accelerating voltage is increased, the electrons move faster, their wavelength \lambda=h/\sqrt{2meV} becomes shorter, and the rings become smaller (less diffraction).
• The wavelength calculated from the size of the rings agrees with de Broglie's formula \lambda=h/mv.
• A magnet brought near the tube shifts the whole pattern. So the pattern is made by charged electrons, not by X-rays or light produced in the tube.

21.7Electron microscopeOften examined

The electron microscope is a device based on the wave nature of electrons. Instead of light, it uses a beam of accelerated electrons as its source of illumination. It lets us observe objects in detail down to the nanometre scale, with remarkably high resolution. It is commonly used to study microscopic organisms such as viruses and bacteria, and the structure of metals and crystals.

Why the wavelength decides the resolution

Resolution is the ability of a microscope to show two very close points as separate. Because of diffraction, any microscope can only show details that are about as large as, or larger than, the wavelength of the radiation it uses. Shorter wavelength means finer detail and higher resolution.

• Visible light has wavelengths from 4000\,\text{\AA} to 7000\,\text{\AA} (400 to 700\,\mathrm{nm}). The best optical microscope cannot resolve details smaller than about 200\,\mathrm{nm}.
• Electrons accelerated through a p.d. V have wavelength \lambda=h/\sqrt{2meV}. Through 10\,000\,\mathrm{V} (10\,\mathrm{keV}) the wavelength is about 0.12\,\text{\AA} (0.012\,\mathrm{nm}), tens of thousands of times shorter than visible light. At 100\,\mathrm{kV} it is about 0.004\,\mathrm{nm}.
• In practice the magnetic lenses are not perfect, so the resolution is limited to about 0.1 to 0.2\,\mathrm{nm}. This is still about a thousand times finer than an optical microscope, enough to see individual large molecules and rows of atoms.

Construction and working

1. Electron gun: a tungsten filament is heated and emits electrons. A high potential on the anode (tens of kilovolts) accelerates them into a narrow, fast beam, like a beam of light.
2. Condenser lens: a magnetic coil that focuses the electron beam onto the specimen.
3. Specimen (sample) stage: the sample must be very thin, so that electrons can pass through it, and carefully prepared. It is placed in the vacuum chamber.
4. Objective lens: forms the first magnified image of the specimen.
5. Intermediate and projector lenses: magnify the image further and project it downwards.
6. Fluorescent screen or detector: at the very bottom. Electrons striking it make it glow, forming a visible image. A photograph of it is called an electron micrograph.
7. Vacuum system: the whole column is evacuated so that the electrons are not scattered or stopped by air molecules.

The electron beam passes through a series of coil-shaped electromagnets that replace the glass lenses of an optical microscope. The magnetic field of a coil bends the paths of the electrons and brings them to a focus. Increasing the current in a coil makes its field stronger, so the magnetic lens becomes stronger (shorter focal length). The electron gun and condenser lens are in the upper part; the sample stage and objective lens are in the middle; the intermediate and projector lenses are in the lower part; the screen is at the bottom.

21.8Heisenberg's uncertainty principleOften examined

On the microscopic level we cannot measure any property of a particle without interacting with it in some way, for example by bouncing a photon off it. This interaction introduces an unavoidable uncertainty into the result. From Planck's point of view, E=hc/\lambda: a photon of short wavelength has a large energy and a large momentum, so it disturbs whatever it hits.

The German physicist and Nobel laureate Werner Heisenberg formulated the uncertainty principle in 1927. It is a consequence of the dual (wave and particle) nature of matter.

Heisenberg's uncertainty principle

It is impossible to find both the position and the momentum of a particle accurately at the same time. The product of the uncertainties in position and momentum, measured at the same instant, is approximately equal to \hbar (or greater).

\Delta x: uncertainty in position, \Delta p: uncertainty in momentum, \hbar (read as h-cut or h-bar): the reduced Planck constant.

If the position were known exactly (\Delta x=0), the momentum would be completely unknown (\Delta p=\infty), and the other way round. The more precisely we fix one, the less precisely we can know the other.

The wave picture

In quantum mechanics a particle is described by a wave. Position means where the wave is concentrated; the uncertainty in position is how far the wave is spread out. Momentum is fixed by the wavelength (p=h/\lambda); the uncertainty in momentum is how unclear the wavelength is. A wave cannot be both short (definite position) and have one exact wavelength (definite momentum).

Why measuring position disturbs momentum

If we want accuracy in position we must use photons of shorter wavelength, because the best resolution we can get is about the wavelength of the radiation used. But short-wavelength radiation means high-frequency, high-energy photons with large momentum h/\lambda. When these collide with the electron they give it a large and unpredictable kick. If we use longer wavelength (less energetic) photons, the kick is gentler but the position becomes blurred.

Uncertainty in energy and time

Another kind of uncertainty concerns the energy of a quantum state and the time for which it lasts. If \Delta E is the uncertainty in energy and \Delta t is the uncertainty in the time to which that energy refers, then:

Why every measurement has an uncertainty

• These uncertainties are built into the physical world. They have nothing to do with poor instruments or the skill of the observer, and better apparatus cannot remove them.
• Every measurement needs an interaction (for example a photon hitting the object), and every interaction disturbs the object being measured.
• Because Planck's constant is so small, the uncertainties are far too small to notice for everyday objects. For electrons inside atoms they are very important.

21.9Discrete energy levels in an atom and its spectraOften examined

The electrons in an atom can occupy only certain allowed orbits (also called shells). Each orbit has a definite energy, called an energy level: E_1, E_2, E_3 and so on. The energy of the atom is therefore quantized, as Planck's theory requires. An electron can move from one allowed level to another only by gaining or losing a definite amount of energy.

Energy level

One of the definite, allowed energies that an electron in an atom can have.

Ground state

The lowest energy level of an atom, E_1. Higher levels are called excited states.

Excitation

The transition of an electron from a lower to a higher energy level. It needs energy, for example the absorption of a photon.

De-excitation

The transition of an electron from a higher to a lower energy level. Energy is released as a photon.

An electron moves to a higher or lower level by absorbing or emitting electromagnetic radiation of frequency f. The frequency depends only on the energy difference between the two levels:

E_n: higher energy level, E_p: lower energy level, f: frequency and \lambda: wavelength of the photon absorbed or emitted.

Obtaining a spectrum

A spectrum is produced by using a prism or a diffraction grating to separate the different wavelengths in a beam of light. Photons emitted in such transitions are passed through a narrow slit and a prism (or grating); each wavelength is bent through a different angle and forms a separate line on the detector.

A line spectrum contains a discrete set of wavelengths. It is a characteristic feature of an element, unique to it.

Emission line spectrum

An emission spectrum of an element consists of a series of separate bright lines of definite frequencies (or wavelengths) on a dark background. It is produced by the excited atoms of the element in a vapour or gas at low pressure, for example in a discharge tube. The photons are emitted as electrons fall from higher to lower excited states or to the ground state. Each line corresponds to one particular transition.

Since each element has a unique set of orbital electrons and energy levels, its emission line spectrum is also unique. The spectrum can therefore be used to identify the element, like a fingerprint. Different coloured lines show which elements a substance is made of, because each element radiates its own particular energies.

Absorption line spectrum

An absorption spectrum of an element consists of a series of separate dark lines of definite frequencies (or wavelengths) on a continuous coloured background. White light from a source such as a tungsten filament lamp (containing all wavelengths) is passed through the cool vapour of the element and then through the slit and grating. The atoms absorb only those photons whose energy exactly matches a jump between their levels. These wavelengths are removed from the beam, leaving dark lines. The excited atoms soon return to the ground state and re-emit the same photons, but in all directions, so very few of them continue in the original direction.

Formula sheet

Key terms

Quantum physics

The branch of physics that studies the behaviour of matter and energy at the atomic and subatomic level.

Electromagnetic wave

A transverse wave of oscillating electric and magnetic fields, perpendicular to each other and to the direction of travel, moving through vacuum at c=3\times10^{8}\,\mathrm{m\,s^{-1}}.

Quantum

The smallest amount of energy that can be emitted or absorbed as electromagnetic radiation, E=hf.

Photon

A quantum (packet) of electromagnetic radiation with energy E=hf, momentum p=h/\lambda and zero rest mass.

Planck's constant

h=6.63\times10^{-34}\,\mathrm{J\,s}, the constant linking photon energy and frequency.

Quantum number

The whole number n in E=nhf that labels the allowed (quantized) energy states.

Electronvolt

The energy gained by an electron moving through a potential difference of one volt; 1\,\mathrm{eV}=1.6\times10^{-19}\,\mathrm{J}.

Radiation pressure

The force per unit area exerted by light on a surface because the photons' momentum changes.

Photoelectric effect

The emission of electrons from a metal surface when electromagnetic radiation of sufficiently high frequency falls on it.

Photoelectrons

Electrons emitted from a metal surface by the action of light.

Saturation current

The maximum photoelectric current, reached when all emitted photoelectrons are collected.

Stopping potential

The minimum retarding potential at which the photoelectric current becomes zero; eV_0=K.E_{\max}.

Threshold frequency

The lowest frequency of radiation that can eject electrons from a given metal, f_0=\Phi/h.

Threshold wavelength

The longest wavelength of radiation that can eject electrons from a given metal, \lambda_0=c/f_0.

Work function (threshold energy)

The minimum energy needed by an electron to break free from a metal surface, \Phi=hf_0.

Einstein's photoelectric equation

hf=\Phi+K.E_{\max}: the photon energy equals the work function plus the maximum kinetic energy of the photoelectron.

Photocell

A device based on the photoelectric effect that converts light energy into an electric current.

Compton effect

The increase in wavelength of X-rays or gamma rays scattered by electrons, because the photons give some energy and momentum to the electrons.

Compton shift

The change in wavelength \Delta\lambda=\lambda'-\lambda=\dfrac{h}{m_0c}(1-\cos\theta) of a scattered photon.

Compton wavelength

h/m_0c=2.43\times10^{-12}\,\mathrm{m}, the Compton shift at a scattering angle of 90^\circ.

Antiparticle

A particle with the same mass as another particle but opposite charge; the positron is the antiparticle of the electron.

Positron

The antiparticle of the electron: mass 9.11\times10^{-31}\,\mathrm{kg}, charge +e.

Rest mass energy

The energy m_0c^{2} equivalent to the rest mass of a particle; 0.511\,\mathrm{MeV} for an electron.

Pair production

The creation of an electron and a positron from a photon of energy at least 1.02\,\mathrm{MeV} near a heavy nucleus (energy converted into mass).

Pair annihilation

The combination of an electron and a positron into two gamma-ray photons moving in opposite directions (mass converted into energy).

Wave-particle duality

The property of light and matter of behaving as waves in some experiments and as particles in others.

de Broglie wavelength

The wavelength of the wave associated with a moving particle, \lambda=h/p=h/mv.

Matter waves

Waves associated with moving material particles such as electrons.

Electron diffraction

The spreading of an electron beam into a ring or spot pattern when it passes through a thin crystal; evidence that electrons behave as waves.

Electron microscope

A microscope that uses a beam of fast electrons, focused by magnetic lenses, instead of light; its very short electron wavelength gives very high resolution.

Resolution

The ability of an instrument to show two close points as separate; it is limited by the wavelength of the radiation used.

Electron micrograph

A photograph of the image formed in an electron microscope.

Heisenberg's uncertainty principle

The position and momentum of a particle cannot both be measured exactly at the same time; \Delta x\,\Delta p\ge\hbar.

Reduced Planck constant

\hbar=h/2\pi=1.054\times10^{-34}\,\mathrm{J\,s}, read as h-cut or h-bar.

Metastable state

An excited state in which an atom stays for a comparatively long time, so its energy is well defined.

Energy level

One of the definite allowed energies of an electron in an atom.

Ground state

The lowest energy level of an atom.

Excitation

The jump of an electron to a higher energy level by absorbing energy.

De-excitation

The fall of an electron to a lower energy level, emitting a photon of energy hf=E_n-E_p.

Line spectrum

A spectrum containing only a discrete set of wavelengths, characteristic of an element.

Emission line spectrum

Bright lines of definite wavelengths on a dark background, produced by excited atoms of a gas at low pressure.

Absorption line spectrum

Dark lines of definite wavelengths on a continuous background, produced when white light passes through a cool gas.

FBISE 2026 questions

Questions from the FBISE HSSC-II 2026 (1st Annual) papers that belong to this unit, answered the way an examiner expects.

Exam focus

1. Photoelectric effect: definition, observations with graphs, failure of classical physics and derivation of Einstein's equation hf=\Phi+K.E_{\max}. (long)
   The most examined topic of the unit: a Section C question in all three 2026 papers (Set B Q.5 OR, Set D Q.5 OR, old syllabus Q.4).
   Go to answer
2. Draw and explain the graphs: current against collector potential for different intensities, current against intensity, and K.E_{\max} against frequency. (long)
   Graphs earn marks in Section C (Set B Q.5 OR asked for the effect of intensity with a graph; Set D asked for the effect of frequency).
   Go to answer
3. Stopping potential and K.E_{\max}=eV_0; how to find \Phi and h from a V_0 against f graph. (short)
   Old-syllabus 2026 Q.2 (i) OR asked how kinetic energy of photoelectrons is calculated; book short question 5 asks about the graph.
   Go to answer
4. Photoelectric numericals: photon energy in eV, K.E_{\max}, stopping potential and wavelength. (numerical)
   Direct use of hf=\Phi+eV_0; book numericals 1 and 2 (note the misprinted answer of numerical 1).
   Go to answer
5. Pair production: definition, conservation laws and minimum energy 2m_0c^{2}=1.02\,\mathrm{MeV}. (short)
   Asked in 2026 Set B Q.2 (vii) (1 + 2 marks); also a comprehensive question in the book.
   Go to answer
6. Pair annihilation: why two photons, conservation of energy and momentum, 0.511\,\mathrm{MeV} per photon. (short)
   Asked in 2026 Set D Q.2 (vii) (1 + 2 marks).
   Go to answer
7. Electron microscope: why electron wavelength gives very high resolution; comparison with an optical microscope. (short)
   Asked in both 2026 new-syllabus sets (Set B and Set D Q.2 (viii) OR).
   Go to answer
8. de Broglie wavelength \lambda=h/mv and its numericals (convert km/h and mg to SI units). (mcq)
   Two 2026 MCQs (Set B MCQ 12 and Set D MCQ 13) and book examples 21.4 and assignment 21.4.
   Go to answer
9. Compton effect: explanation with photon collisions, Compton shift \Delta\lambda=\dfrac{h}{m_0c}(1-\cos\theta) and its significance. (long)
   Book comprehensive question 2 and several short questions (8, 9, 10) and numericals 4 and 6 come from this section.
   Go to answer
10. Heisenberg's uncertainty principle: statement, \Delta x\,\Delta p\ge\hbar, \Delta E\,\Delta t\ge\hbar and simple numericals. (long)
    Book comprehensive question 3, MCQ 8 and numerical 5; state which form of the principle you use.
    Go to answer
11. Electronvolt: definition and derivation of 1\,\mathrm{eV}=1.6\times10^{-19}\,\mathrm{J}; converting MeV to joules. (short)
    Old-syllabus 2026 Q.2 (i) (1 + 2 marks) and book short question 2.
    Go to answer
12. Discrete energy levels and the formation of emission and absorption line spectra, with hf=E_n-E_p. (long)
    Book comprehensive question 9 and two learning outcomes; energy level diagrams earn marks.
    Go to answer