Introduction to Quantum Theory

Previous Knowledge - Quantum Theory

• This section builds on the knowledge from Higher Physics including:
  • Black-body radiation curves
  • Photoelectric effect
  • Rutherford's experiment with alpha radiation
  • Bohr model of the atom
  • Emission and absorption spectra

Introduction to Quantum Theory - Learning Outcomes

• At the end of this section you should be able to:
  • State that the following experimental observations cannot be explained by classical physics, but can be explained using quantum theory.
    • Black-body radiation curves (ultraviolet catastrophe)
    • The formation of emission and absorption line spectra
    • The photoelectric effect
  • Use the relationship E = hf to solve problems involving photon energy and frequency.
  • Describe of the Bohr model of the atom in terms of the quantisation of angular momentum, the principal quantum number n and electron energy states
  • Explain the characteristics of atomic spectra using the Bohr model
  • Use the equation mvr = nh/2π to solve problems involving the angular momentum of an electron and its principal quantum number.
  • Describe experimental evidence for the particle-like behaviour of 'waves' and for the wave-like behaviour of ‘particles'.
  • Use the equation λ = h/p to solve problems involving the de Broglie wavelength of a particle and its momentum.
  • State that it is not possible to know the position and the momentum of a quantum particle simultaneously.
  • State that it is not possible to know the lifetime of a quantum particle and the associated energy change simultaneously.
  • Use the uncertainties relationships to solve problems involving the uncertainties in position, momentum, energy and time.
    • ∆x∆p ≥ h/4π
    • ∆E∆t> h/4π
  • Understand the implications of the Heisenberg uncertainty principle, including the concept of quantum tunnelling.

Introduction to quantum theory

• Quantum Theory was developed to explain the results of experiments that classical physics could not which included:
  • Black-body radiation curves - the ultraviolet catastrophe
  • The photoelectric effect
  • The formation of emission and absorption spectra

Black - body radiation curves and the UV catastrophe

• A black body is a hypothetical perfect absorber and radiator of energy with no reflecting power.
• A black body will absorb 100% of the radiation that hits it, no matter the wavelength or the intensity and will not transmit or reflect any radiation.
• A perfect black body is theoretical, although black holes, planets and stars come close and are referred to as black bodies.
• An item good at absorbing radiation is also a good emitter, so a perfect black body would be the best possible emitter of radiation.
• A perfect black body would emit all wavelengths in the electromagnetic spectrum.
• Black body radiation curves show the intensity of radiation emitted at each wavelength from black bodies.
• All curves have the same characteristic.
• There is a peak wavelength emitted that shifts towards longer wavelengths for cooler items and towards higher wavelengths for hotter bodies.
• Wein's equation could explain what happens with lower frequencies.
• Lord Rayleigh's equation predicted infinity at high frequencies which was known as the 'ultraviolet catastrophe'.
• Planck solved the ultraviolet catastrophe in 1900 and derived an equation which correctly predicted the trends seen in black body graphs.
• Planck postulated that electromagnetic energy was emitted in discrete bundles of energy, where the energy was proportional to the frequency.
• Calculate using: where
  • E = energy of discrete bundle/photon (Joule, J)
  • h = Planck's constant (6.63 x 10-34 Joule seconds, Js)
  • f = frequency of radiation (Hertz, Hz)
• This resembles Einstein's conclusion for the photoelectric effect.
• Planck provided the first evidence for photons at the start of the quantum world.

Photoelectric effect

• The photoelectric effect was explained by Einstein in 1905 and needed a quantum, particle-like photon description of light instead.
• When electromagnetic radiation of high enough frequency shines on a metal surface, electrons are ejected from the metal.
• Lower frequency light does not cause electrons to be emitted as predicted by classical theory.
• Classical theory and considering light as a wave of energy cannot explain the photoelectric effect.
• If considering light as a wave, electrons absorbing light from a wave would be able to release electrons eventually, but there would be a time delay until the electrons absorbed enough energy from lower frequency light.
• Einstein proposed that light must be thought of as small packets of wave energy called photons and the energy of each photon can be calculated using the equation E = hf.

Compton's scattering experiment

• Arthur H. Compton performed an experiment in which he observed the scattering of x-rays from electrons in a carbon target.
• The scattered x-ray had a longer wavelength than the incident x-ray and had a lower energy due to loss in momentum.
• Compton explained this by assuming the light behaved like a particle (photon), and therefore could apply the conservation of energy and the conservation of momentum to the collision between the photon and the electron in the carbon target.
• This gave clear evidence of particle-like behaviour by electromagnetic radiation.

Atomic structure

• Atoms were first considered to consist of a large positively charged mass with negatively charged electrons embedded in it described as a 'plum pudding' model.
• The 'plum pudding' model, suggested by J J Thomson, was then replaced by the Rutherford model of the atom.
• Rutherford interpreted the results of his experiments as evidence that;
  • Most of the atom is empty space.
  • Small dense nucleus holds the neutrons and protons together by means of the Strong Force.
  • Electrons orbit the nucleus at non-relativistic speeds.
  • The atom is electrically neutral.
• There is a problem with this model due to the electron moving with speed v in a circle of radius r, and its centripetal acceleration is equal to v²/r.
• Classical physics theory dictates that an accelerating charge emits electromagnetic radiation.
• A 'classical' electron would therefore be losing energy and would spiral into the nucleus, instead of remaining a circular path.
• Niels Bohr put forward an alternative model for the atom in 1913 which introduced the idea of energy levels where electrons could only occupy discrete orbits, each with its own discrete energy.
• The total energy of the electron is the sum of kinetic energy due to the circular motion and potential energy due to the electrical field in which it was located.
• Each value of energy corresponded to a unique electron orbit, so the orbit radius could only take certain values.
• Emission and absorption spectra were due to electrons moving between allowed orbits.
• Using classical and quantum theory, Bohr calculated the allowed values of energy and radius, matching the gaps between electron energies to the photon energies observed in the line spectrum of hydrogen.
• Bohr suggested that electrons can orbit in stable orbits without emitting radiation and that the angular momentum of the electrons in these orbits is quantised.
• The angular momentum L of any particle of mass m moving with speed v in a circle of radius r is L = mvr.
• The Bohr model proposes that as long as the angular momentum of the electron is a multiple of h/2π, the orbit is stable (h is Planck's constant).
• Thus, the Bohr model proposed the concept of quantisation of angular momentum of the electron in a hydrogen atom, following the condition
  • Angular momentum = mvr = nh/2π
  • where
    • m = mass of electron (9.11 x 10-31 kg)
    • v = linear velocity of electron (metres per second, ms⁻¹)
    • r = radius of electron's orbit (metre, m)
    • n = integer 1, 2, 3, etc (no unit)

Summary of Bohr Model

• Electrons revolve around the nucleus of an atom in certain allowed orbits called energy levels
• Angular momentum of the electron is quantised (L = nh/2π)
• Total energy in each possible energy level is constant.
• Electron does not continuously radiate electromagnetic radiation.
• A single quantum photon of electromagnetic radiation is emitted on transition to a lower energy level

Wave-particle duality

• There is evidence for electromagnetic radiation behaving like both a particle, such as the photoelectric effect, and a wave such as interference and diffraction.
• Electrons also display both particle and wave like properties.
• There is evidence that electrons demonstrate wave like properties where G P Thomason bombarded thin films of metal with an electron beam and diffraction rings were observed.
• Electrons also behave like a wave when passed through a tiny opening, or slit, and produce diffraction fringes.
• Where there is more than one slit for the electrons to pass through, the electrons produce a pattern such as the interference pattern observed when light is passed through a diffraction grating.

De Broglie wavelength

• Bohr's quantisation of angular momentum fitted in with the predicted energy levels but left a crucial question unanswered.
• Why were these particular orbits allowed, what made this value of angular momentum so special, and why did angular momentum of nh/2π make the orbit stable?
• The answer came when de Broglie's ideas on wave-particle duality were published a decade later.
• The French physicist Louis de Broglie, suggested in 1923, that since light had particle-like properties, then maybe nature was dualistic and particles like electrons have wave-like properties.
• From relative theory, the energy of a particle with zero rest mass like a photon is given by E = pc and since E = hf, the de Broglie wavelength is
  • λ = h/p
    • where
      • λ = de Broglie wavelength (metre, m)
      • h = Planck's constant (6.63 x 10-34 joule seconds, Js)
      • p = momentum of electron = mv (kilogram metres per second, kg m s-1)
• Therefore, an electron must be treated like a particle with rest mass 9.11 x 10-31 kg or as a wave with a wavelength: λ = h/p
• Treating the electron as a stationary wave allows the smallest allowed circumference of the electron's orbit to corresponds to one wavelength λ.
• The next allowable orbit corresponds to 2λ and so on, and the orbit will be stable, then, if the circumference is equal to nλ.
• De Broglie associated wavelength with the orbiting electron to explain the stability of the orbits in the Bohr atom considering that a stationary wave does not transmit energy.
• If the electron is acting as a stationary wave then all its energy is confined within the atom, and the problem of a classical electron radiating energy does not occur.
• If the circumference of a circle of radius r is 2πr, then the condition for a stable orbit is: nλ = 2πr
• The equation λ = h/p = h/mv where v is the velocity of the electron, can be substituted into nλ = 2πr:
• where nλ = 2πr → nh/mv = 2πr → mvr=nh/2π which states that the angular momentum mvr is a multiple of h/2π, as predicted by Bohr.
• Treating the electron as a stationary wave gives the quantisation of angular momentum predicted by Bohr.
• The unit of allowed angular momentum is sometimes given the symbol ħ ('h bar'), where ħ = h/2π.

Limitations of the Bohr model

• The Bohr model provides a good model for the hydrogen atom but does not give a complete picture.
• The electron is actually moving in three dimensions, whereas the Bohr atom only considers two dimensions.
• Nowadays, a full 3-dimensional wave function is used to describe an electron orbiting in an atom called the Schrodinger wave function which describes the motion in terms of probabilities.
• The wave function is used to determine the probability of finding an electron at a particular location in three dimensional space where the electron cannot be thought of as a point object with a specific position.
• Instead, The probability of finding the electron within a certain region can be calculated within a certain time period.
• Heisenberg was also amongst the first to propose this concept in the Uncertainty Principle.

Heinsenberg Uncertainty principle

• The German physicist Werner Heisenberg produced his Uncertainty Principle in 1927 at 23 years old.
• Heisenberg determined that using an optical microsope to measure the position and momentum of a particle, a light has to be shone on it, and then the reflection detected.
• This method works fine on a macroscopic scale, but on a sub-atomic scale this causes problems where the photons of light will cause the sub-atomic particle to move significantly.
• So, although the position may have been measured accurately, the velocity or momentum of the particle is changed, and by finding out the position any information about the particle's velocity is lost. This is also known as 'The very act of observation affects the observed.'

Position and momentum

• The uncertainty principle states that it is not possible to measure the position and the momentuum of a particle with absolute certainty.
• The more accurately one of these is known, the less accurately the other is known.
• The uncertainty in a quantum particle's position, ∆x, and the uncertainty in the particle's momentum, ∆p, are connected by the relationship:
  • ∆x∆p>=h/4π
    • where
      • ∆x = uncertainty in the quantum particle's position (m)
      • ∆p = uncertainty in the quantum particle's momentum (kg m⋅s-1)
      • h = Planck's constant (6.63 x 10-34 Js)
• ∆p = m∆v where ∆v = uncertainty in the particle’s velocity (m s-1)

Energy and time

• Similar to the uncertainty principle relationships there is an relationship for a quantum particle and the lifetime of the quantum particle, ∆t.
• It is not possible to measure the quantum particle's energy and lifetime with absolute certainty.
• The more accurately one of these is known, the less accurately the other is known:
  • ∆E∆t> h/4π
    • where
      • ∆E = uncertainty in the quantum particle's energy (Joule, J)
      • ∆t = uncertainty in the quantum particle's lifetime (second, s)
      • h = Planck's constant (6.63 x 10-34 Js)
• This discovery meant that particles no longer had well defined positions and velocities or energies and times, but they now exist in a 'quantum state' which is a combination of both their position and momentum or energy and time.
• You cannot know all the properties of the system at the same time, and any you do not know about in detail can be expressed in terms of probabilities.

Uncertainty Principle and Quantum tunnelling

• The Uncertainty Principle in terms of energy and time leads to the concept of quantum tunnelling where a particle is repelled by the electromagnetic force of the barrier.
• It would require additional energy to go over the barrier however there is a chance the occurance of appearing on the other side of the barrier can occur.
• This is easiest to visualise with electrons as a wave instead of a particle since a particle clearly cannot pass through the barrier but a wave may slightly overlap the barrier and even though most of the wave is on one side.
• The small part of the wave that does cross the barrier leads to a small chance of the particle that generated the wave appearing on the far side of the barrier.

Examples of quantum tunnelling

• Alpha decay is one example of quantum tunnelling.
• Alpha particles are emitted from a heavy nucleus, such as Uranium 238 (U238).
• Alpha particles would normally be bound inside the nucleus and would need a lot of energy to break the bonds keeping them in place.
• Since an alpha particle inside a nucleus has a very well-defined momentum its position is not so well-defined so there is a chance that is could find itself outside the nucleus even though it does not have enough energy to escape.

Quantum tunnelling composites (QTC)

• This technique is used to make quantum tunnelling composites (QTCs), which are materials that can be used as highly sensitive pressure sensors due to the ability of quantum tunnelling to cross thin barriers.
• QTCs are also used in electronic devices such as modern Tunnel Field Effect Transistors (TFET transistors) replacing the older MOSFET transistors.
• Possible uses of these pressure sensors would include measuring the forces of impacts on fencing helmets or boxing gloves or to accurately read a person's blood pressure.

Previous Knowledge - Particles from Space

• This section builds on the knowledge from from Higher Physics Our Dynamic Universe topic.
  • Moving charges in a magnetic field.
• Aspects of section will use concepts which you will learn in the Advanced Higher Rotational Motion and Electromagnetism topics.
  • Rotational motion - centripetal force
  • Electromagnetism - fields

Particles from Space - Learning Outcomes

• At the end of this section you should be able to:
  • Know the origin and composition of cosmic rays and the interaction of cosmic rays with Earth's atmosphere.
  • Know the composition of the solar wind as charged particles in the form of plasma.
  • Explain the helical motion of charged particles in the Earth's magnetic field.
  • Use appropriate relationships to solve problems involving the force on a charged particle, its charge, its mass, its velocity, the radius of its path, and the magnetic induction of a magnetic field using
    • F = qvB when a particle cuts across a magnetic field at 90°
    • F = mv²/r

Cosmic rays

• There isn't a precise definition for the term 'cosmic ray' but it is accepted to mean ‘high energy particles arriving at Earth, which have originated elsewhere'.
• Lowest energy particles come from the Sun, intermediate energy ones are presumed to be created elsewhere in the Milky Way, often in connection with supernovae.
• The origin of the highest energy particles is uncertain.
• The highest energies produced by particle accelerators on Earth are of the order of 1012 cV which is 1.6 x 10-7 J.
• Cosmic rays have energies in the range of 109 to 1020 eV.
• The highest energy cosmic ray particles have energies of about 50 J.

Composition

• Cosmic rays come in a range of different types, but the most common are protons, as shown in the table below.
  • Approximate % of all cosmic rays:
    • Protons ~ 89%
    • Alpha particles ~ 9%
    • Carbon, nitrogen and oxygen nuclei ~ 1%
    • Electrons ~ Less than 1%
    • Gamma radiation ~ Less than 0.1%

Interaction with the Earth's atmosphere

• When cosmic rays hit the top of the atmosphere, they produce showers of secondary particles which include electrons, photons, neutrinos and muons called a cosmic air shower.
• Cosmic ray observatories are at higher altitudes so that they are closer to the path of the cosmic rays and therefore reduces the interaction of the rays with the atmosphere.
• Observatories are also closer to the poles as the Earth's magnetic field is stronger at the poles and therefore a greater density of cosmic rays are found there.

Movement of charged particles in Earth's magnetic field

• Charged particles approaching Earth can be affected by the Earth's magnetic field.
• A moving charge either:
  • Moves parallel to a magnetic field
  • Crosses a magnetic field at 90°
  • Crosses a magnetic field at an angle

Charged particle moving parallel to a magnetic field

• If a charge is moving parallel or antiparallel to a magnetic field, then no magnetic force acts on the moving charge and its velocity is unchanged.
• In this case, the particle will continue moving in the same direction with the same speed.

Charged particle crossing a magnetic field at 90°

• When a charged particle cuts across a magnetic field at 90°, the magnetic force experienced by the charge is at a maximum.
• The magnitude of the force acting on a charged particle moving at right angles to a magnetic field can be calculated using the equation:
  • F = qvB
    • where
      • F = force on charged particle (Newtons, N)
      • q = charge of particle (Coulombs, C)
      • v = speed of charged particle perpendicular to direction of field (meter/second, ms −1)
      • B = magnetic induction of the field (Tesla, T)
• The direction in which the force acts on the particle can be found out using the right hand motor rule: remember this rule applies to a negative charge, for a positive charge reverse the direction of the force.
• If the charged particle is moving at right angles to the magnetic field, the force acting on it will always be perpendicular to its direction of travel.
• The force will cause the particle to change its direction but not its speed, as the velocity will change because of the change in direction as it orbits in a circular path

Circular Motion

• The force of the magnetic field provides the centripetal force which is:
  • F = mv²/r and F = qvB
  • Therefore mv²/r = qvB which can be rearranged to r = mv/qB
    • where
      • r = radius of circular motion (metre, m)
      • m = mass of charged particles (kilogram, kg)
      • v = speed of particle (m s⁻¹)
      • q = charge of charged particle (Coulomb, C)
      • B = magnetic induction of field (Tesla, T)

Charged particle crossing a magnetic field at an angle

• When a charged particle crosses the magnetic field at an angle the velocity can be resolved into two perpendicular components.
• Circular motion is caused by the perpendicular component of the velocity (vsinθ), and linear motion is caused by the component of the velocity (vcosθ) parallel to the magnetic field.
• This combination of circular and linear motion causes the particle to follow a helical path with the axis of the helix along the direction of the Earth's magnetic field lines.

Solar Wind

• Solar wind is a stream of charged particles in the form of plasma, escaping from the upper atmosphere of the Sun at speeds between 3 x 105 m⋅s-1 and 8 x 105 m⋅s-1.
• High temperature of the corona(over 1,000,000 K) gives the the particles sufficient kinetic energy to escape the sun.
• Plasma contains:
  • Equal numbers of protons and electrons in ionized hydrogen
  • Alpha particles are approximately 8%
  • Trace amounts of heavy ions and nuclei (Carbon, Nitrogen, Oxygen, Neon, Magnesium, Silicon, Sulphur and Iron)
• Solar wind reaches Earth around 3 days after leaving the Sun.
• Solar flares are explosive and release energy across the whole electromagnetic spectrum.
• High energy solar flare particles arrive at Earth within half an hour of the flare maximum can disrupt power generation and communications.
• The magnetosphere protects Earth from having its atmosphere ripped away by the solar wind, as has happened to Mercury and Mars.

Earth's Magnetosphere

• The magnetosphere is the part of the Earth's atmosphere dominated by the Earth's magnetic field.
• Particles in the solar wind become trapped in the Van Allen belts.
• The magnetic field is strongest near the North and South poles in these regions and particles penetrate the upper atmosphere more strongly.
• These particles that strike nitrogen molecules and oxygen atoms cause excitation, which in turn causes light to be emitted known as the Aurorae.
• Green Aurorae occurs where oxygen is produced and red or violet from nitrogen.

Simple Harmonic Motion - Learning Outcomes

• At the end of this section you should be able to:
• Define simple harmonic motion (SHM) in terms of the restoring force and acceleration proportional to, and in the opposite direction to, the displacement from the rest position
• Use calculus to show that expressions in the form of y = Asinwt and y = A coswt are consistent with the definition of SHM (a = -w²y)
• Derive the relationships v = ±w√(A2 − y²) and Ek = mw²(A² – y²)
• Use appropriate relationships to solve problems involving the displacement, velocity, acceleration, force, mass, spring constant k, angular frequency, period and energy of an object executing SHM where:
  • F = -ky
  • ω = 2πf = 2π/T
  • a = d²y/dt² = -w²y
  • y = A sin wt or y = A cos wt
  • v = ±w√(A² − y²)
  • Ek = ½mw²(A² – y²)
  • Ep = ½mw²y²
  • Effects of damping in SHM include underdamping, critical damping and overdamping.

Simple Harmonic Motion (SHM)

• Simple harmonic motion is motion in where an object oscillates around a fixed equilibrium position, the acceleration of the object is proportional to its displacement, and is always directed towards the equilibrium point.
• If an object (e.g. a mass on a spring) is displaced from its rest position, the restoring force will be in the opposition direction to the displacement.
• The force is also proportional to the acceleration of the mass.

Mathematical description of SHM

• The restoring force always acts in the opposite direction and in proportion to the displacement, therefore F = -ky
  • F = restoring force (Newton - N)
  • k = force constant (Newton per metre- Nm⁻¹)
  • y = displacement (metre - m)
• Where d²y/dt² = k/m.y

SHM Formulas

• The graph would normally be labelled x on the horizontal axis.
• The graph would be labelled y to correlate with the equations given.
• The period of the motion is the time taken for one complete oscillation to occur
• In addition
• f = 1/T and ω = 2π/T and ω = 2πf where for a stable, simple pendulum T =
• If we differentiate displacement we obtain velocity and it we differentiate velocity we obtain acceleration
• Expressions in the form of y = A sinωt and y = A cos ωt are consistent with the definition of SHM (a = −ω²y)
  • Assume an object is moving anti-clockwise in a circle of radius a at a constant angular velocity ω, with the origin of x-y axis at the centre of the circle, and the object is at y=0 at t=0
  • Differentiating this equation, the velocity in the y-direction is dy/dt = v = ωa cos ωt
  • The maximum velocity will equal v = ±ωa when y = 0.
  • If we differentiate again to get the acceleration in the y-direction a = d²y/dt² = −ω²a sin ωt and a = -ω²y
  • The maximum acceleration will equal a = ±ω²a when y =±a.
  • In the x- direction the displacement at time t is equal to x = a cos wt, dv/dt = ωa sin ωt, dv/dt = a = -ω² a cos ωt, and a = d²x/dt²== -ω²x
• Remeber when calculating problems involving the sine and cosine of ω make sure your calculator is in radians mode.

Derivation of v = ±ω√(A² – y²)

• This expression shows how velocity of a vibrating object (e.g. a tuning fork) varies with displacement.
• Substitute y = a sin ωt and v = ωa cos ωt into the trig identity = 1 to give:
  • v = ±ω√A² – y²
• This shows that there are two possible solutions for the equation for values other than y = a, and is usually written v = +ω√A² – y²

Energy changes in an oscillating system

• During simple harmonic motion an object will oscillate between maximum potential energy (PE) when it is at its maximum displacement and maximum kinetic energy (KE) at the equilibrium position, when velocity is at a maximum following PE + KE = Emax.
• A particle with maximum amplitude A and period T = 2π/ω has:
  • KE = ½ mv²
    • Substitute for v = ±ω√A² – y² to get KE = = mw²(A² – y²)
  • When at position O the potential energy is zero so the kinetic energy is a maximum.
  • If KE is a maximum, when y = 0: Ek max = =½mw²A² so the total energy Etot Ek + PE = ½mw²A²
  • Since Etot = ½mw²A² Or Etot ½kA² since, ω² can be swapped with k/m, the total energy E the same all points in the motion.
  • Thus for any point on the swing Etot = KE + PE following Etot = ½mw²(A² – y²) + PE, thich can be rearranged to PE= ½mw²y²

Damping of Oscillations

• Oscillating systems come to rest eventually because their motion is damped where energy is transformed from the system due to frictional forces such as air resistance.

Damping

• Undamped - If no energy is lost a mass would oscillate indefinitely with each oscillation reaching the same height as the previous one (ideal condition)
• Underdamped - If a mass oscillates on the end of a spring, it tends to overshoot its starting position, and then return, overshooting again as energy is dissipated resulting in oscillations towards zero,
• Critical Damping - the frictional resistance is just sufficient to prevent any oscillation past the rest position.
  • the system will come to rest in the minimum possible time.
• Overdamped - no complete oscillations are seen therefore the object does not travel past the equilibrium point

How are Systems Damped

• Increase the surface area of the mass.
• closed air chamber
• light vane swinging in chamber with
• small clearance
• The mass oscillates in a viscous fluid e.g. oil or water
• A car shock absorber has very thick oil in the dampers to ensure over bumps will not be a long period.