AHL Quantum

  2. Atomic, nuclear and particles
  3. AHL Quantum
    4. 10'

Quantum physics is the study of the smallest objects in the universe: photons of electromagnetic radiation, the wavelengths of electrons and the energy levels of an atom. 

Key Concepts

Photon interactions with matter

Photoelectric effect

The photoelectric effect provides evidence for the quantum nature of light. Individual photons of light interact with individual electrons to provide energy that may release an electron from a metal surface. Check out the page on the Photoelectric effect for a full description of the observations and explanations.

Photons have energy proportional to their frequency: 

\(E=hf\)

  • \(E\) is photon energy (J)
  • \(h\) is Planck's constant (6.63×10−34 Js)
  • \(f\) is frequency (Hz)

Conservation of energy can be applied to the photoelectric effect:

\(hf=hf_0+E_\text{max}\)

\(hf=\Phi+E_\text{max}\)

  • \(f_0\) is threshold frequency for a given material (Hz)
  • \(\Phi\) is work function for a given material (J)
  • \(E_\text{max}\) is maximum kinetic energy

High energy interactions

Photons with more energy than is required for the photoelectric effect can transfer energy and momentum to electrons through Compton scattering. Conservation laws suggest that the photons have momentum.

The highest energy photons can convert their energy into mass, with the mass-energy equivalence calculated through \(E=mc^2\). A particle-antiparticle pair is produced to conserve charge and other properties. A particle and its antiparticle may conversely annihilate to produce a pair of photons (in opposite directions to conserve momentum).

 

Electrons as waves

The quantum nature of light indicates that there must be a wave nature of matter. Indeed, electrons are diffracted and produce interference patterns when passed through a sheet of graphite or double slits.

Electrons, as with all other particles, can behave as a wave function rather than a discrete sphere. The de Broglie wavelength of matter can be calculated as follows:

\(\lambda={h\over p}\)

  • \(\lambda\) is the de Broglie wavelength (m)
  • \(h\) is Planck's constant (Js)
  • \(p\) is the momentum of the particle (\(=mv\) in kgms-1)

 

 

 

Essentials

Schrödingers model

The probability of an electron being in a particular position at a particular time is defined by Schrödinger as the square of the wave function, where the wave function varies with position and time. The wave function is denoted as \(\Psi\).

\(P(r)=|\Psi|^2\Delta V\)

  • \(P(r)\) is the probability of the electron at a given radius
  • \(|\Psi|\) is the magnitude of the wave equation
  • \(\Delta V\) is the volume

 

Heisenberg's uncertainty principle

To identify the onward path of a particle, we must know its position and momentum. The Heisenberg uncertainty principle describes the inherent impossibility of measuring both momentum and position with certainty as equations:

\(\Delta p \Delta x \geq{h\over 4\pi}\)

  • \(\Delta p\) is uncertainty in momentum (kgms-1)
  • \(\Delta x\) is uncertainty in position (m)
  • \(h\) is Planck's constant (Js)

\(\Delta E\Delta t\geq{h\over 4\pi}\)

  • \(\Delta E\) is uncertainty in energy (J)
  • \(\Delta t\) is uncertainty in time (s)

 

Bohr model

The Bohr model predicts the energy of an electron in an energy level as being discrete:

\(E=-{13.6\over n^2}\)

  • \(E\) is the energy associated with the \(n^{th}\) energy level (eV)
  • \(n\) is the number of the energy level

Angular momentum is also quantised:

\(mvr={nh\over 2\pi}\)

  • \(mvr\) is angular momentum (kgm2s-2)
  • \(n\) is an integer equal to 1 minus the quantum number
  • \(h\) is Planck's constant (Js)

Observations from the Stern-Gerlach experiment demonstrate that electron spin is another quantised property.

 

Electron in a box model

A helpful model for understanding the probability of an electron's position is a standing wave. Just as an electron cannot move outside an atom, a wave cannot move from a string clamped at both ends. A standing wave may only oscillate at particular harmonic frequencies; an electron may only have particular discrete energies.

The kinetic energy of an electron is given by:

\(E_k={h^2\over 2\lambda^2m}\)

For the \(n^{th}\) harmonic:

\(E_k={n^2h^2\over 8L^2m}\)

 

Electron tunnelling

Tunnelling is the term given to a particle passing through a potential energy barrier without having sufficient energy to surmount the barrier. The continuous nature wave function of the particle means that there is a finite probability of the particle's position being on the far side of the potential barrier.

To reduce the likelihood of tunnelling we could increase:

  • the width of the barrier
  • the mass of the particle
  • the energy deficit between the particle and the barrier energy potential

 

 

 

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Just for Fun

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