Introduction to Quantum physics

Black Body Radiation

A black body is an ideal system that absorbs all radiation incident on it.

A hollow object with a small hole in its walls behaves like a black body;

• All hot objects radiate energy;

• The energy radiated by a black body depends on wavelength and temperature;

• The observed spectrum did not coincide with the theoretical spectrum based on Classical Physics, as total energy is not conserved

Max Planck, a German physicist, solved the black body radiation problem in 1900, by introducing the revolutionary concept of “Energy Quanta”;

Thus energy is absorbed or emitted by an object in discrete amounts (quanta).

• Planck developed a theory of blackbody radiation that leads to an equation for the intensity of the radiation;

• This equation is in complete agreement with experimental observations;

• Planck made two assumptions about the nature of the oscillators in the cavity walls

Plank’s Assumptions

• The cavity radiation came from atomic oscillations in the cavity walls

• The energy difference between the initial and final states of these atomic oscillators is emitted or absorbed as a single quantum of radiation.

All oscillators emit or absorb quantised amounts of radiation. That is, radiation comes in discrete packets or quanta. The energy of these quanta is proportional to frequency:

The Electron-Volt

A unit of energy equal to the work done on an electron in moving it through a potential difference of one volt. It is used as a measure of particle energies although it is not an SI unit. (Oxford Dictionary of Physics).

1 eV = 1.60 × 10^-19 J

T he Photoelectric Effect

• When the tube is kept in the dark, the ammeter reads zero

• When the emitter is illuminated by light of an appropriate wavelength, a current is detected by the ammeter.

In a Photocell, a photon is coming in with Energy

E_photon = hf = pc

If the photon has enough energy, it will “kick out” or emit electrons from atoms in metal cathode.

The Photocell is then connected to a circuit, and the freed electrons travel in the circuit and create a photocurrent.

We can find the kinetic energy of these electrons by supplying a very small voltage difference between the anode and cathode, opposing the photocurrent. The smallest voltage that stops the photocurrent is V_s .

• The current arises from photoelectrons emitted from the negative plate and collected at the positive plate;

• For each metal, there is a threshold frequency below which no electrons are emitted;

• The number of electrons emitted is proportional to intensity of radiation;

• Emitted electrons have KE up to a maximum value, which depends on the frequency of radiation (unexplained by classical physics).

• W hen the applied potential difference is equal to or more negative than -DVs, the stopping potential, no electrons reach the anode, and the current is zero.

• The stopping potential is independent of the radiation intensity.

• Some electrons are ejected with slower speeds. The most energetic electrons have KE_MAX = e∆V_S

Albert Einstein and the Photoelectric Effect

E instein extended Planck’s energy quanta concept to electromagnetic waves; Thus one photon is absorbed by one electron, providing it with the necessary energy to escape the atom as a photoelectron;

Einstein suggested:

 is the “Work-Function” i.e. the threshold (minimum) photon energy required to release one electron from the surface of the metal.

Photons with frequency less than f_C (cutoff frequency / threshold frequency) don't have sufficient energy to eject an electron from the metal.

Wave-particle Duality

• In his PhD thesis, Louis de Broglie postulated that all forms of matter have wave and particle properties:

Principle of Complementarity

The principle of complementarity states that the wave and particle models complement each other e.g. sometimes light behaves like a wave or as a particle;

Both models must be used in order to provide a complete description of a phenomenon. However, it is not possible to model both the wave and particle properties simultaneously;

Heisenberg’s Uncertainty Principle

Werner Heisenberg showed that:

Δx is the uncertainty in the position of a particle;

Δp is the uncertainty in its momentum.

Early Models of the Atom – Rutherford

• Planetary model (most of atom is empty)

• Positive charge located at the centre of atom (nucleus)

• Electrons orbit the nucleus (like planets around stars)

Representation of the nucleus

One writes the structure of the nucleus of an atom X as follows:

A (Mass number) = number of protons and neutrons

Z (Atomic number) = number of protons

Atomic spectra – Emission

• Monoatomic gases emit light at discrete wavelengths when electric discharge is applied;

• Each atom has its emission spectrum (series of coloured lines).

Atomic spectra – Absorption

• Individual atoms can also absorb light at specific wavelengths;

• The absorbed wavelengths of an atom form its absorption spectrum (series of dark lines);

• Example: spectrum of absorption of white light by a hydrogen gas.

Atomic spectroscopy

• Atomic spectral lines (specific wavelengths) observed in emission and absorption spectra

• Each atom has a given set of spectral lines

• Spectroscopy studies allow determination of elemental composition of a source or a sample (e.g. detection of atomic elements in planets’ and stars’ atmospheres)

The Bohr Model of the Atom (3 Postulates)

Postulate 1.

• Only certain electron orbits are stable and allowed where each orbit has a specific energy level;

• The greater the radius, the higher the energy level;

• In these orbits no energy in the form of EM radiation is emitted, so the total energy of the atom remains constant.

Postulate 2.

• Radiation is emitted by the hydrogen atom when the electron “jumps” from a more energetic initial state to a less energetic state. The frequency of the radiation emitted in the jump is related to the change in the atom’s energy, given by

I n a hydrogen atom, when an electron moves from the first excited state (n=2) to the ground state (n=1), a photon of energy:

hf = -3.4 - (- 13.6) eV= 10.2 eV is emitted

For the hydrogen atom, the energy levels are given by:

In general, for a transition from an energy state m to an energy state n (m  n) we write:

if m > n !

Postulate 3.

The circumference of an electron’s orbit must contain an integral number of de Broglie wavelengths:

Electrons’ orbits in Bohr’s atomic model are similar to standing waves fitting into the circumferences of the electrons’ orbits:

The requirement of standing waves along the circumference of stable orbits yields:

the quantization of the electron’s angular momentum requires/defines the principal quantum number n;

Other quantum numbers have been introduced to explain other experimental observations and properties, e.g. orbital quantum number ℓ

The Wave Function ψ

• The wave nature of matter, e.g. electron, can be described by a probability wave function, ψ ;

• ψ(x,t) depends on position and time, and can be used to determine the probability of locating the electron;

• Schrödinger’s wave equation describes matter waves, such as electron waves, and the behaviour of atomic and nuclear systems. It is used to calculate ψ(x,t);

• ψ(x,t) is similar to classical waves, which produce interference patterns.

The electron has “lost” its orbit and could be anywhere;

H owever, its most probable locations correspond to the orbital radii predicted by Bohr’s atomic model.

Atomic Energy Levels

• The principle of quantization of electron angular momentum used to calculate energy levels of the hydrogen atom apply to more complex atoms

• Atoms are characterized by a set of discrete energy levels and quantum numbers

• The most stable configuration for an atom (ground state) has the lowest energy

• Repartition of electrons in atomic shells obeys Pauli exclusion principle

Molecular Bonds

• The bonding mechanisms in a molecule are fundamentally due to electric forces

• The forces between atoms in a molecule are related to a potential energy function

• A stable molecule has a configuration for which the potential energy is minimum

• The force between atoms is repulsive at very small separation distances

  • This repulsion is partially electrostatic and partially due to the exclusion principle

  • Due to the exclusion principle, some electrons in overlapping shells are forced into higher energy states

  • The energy of the system increases as if a repulsive force existed between the atoms

• The force between the atoms is attractive at larger distances

Force-separation graph for matter

R epulsive force due to overlapping electrons;

attractive force due to Van der Waal’s forces.

Imagine neighbouring atoms are connected by a spring.

Energy-separation graph for matter

Note: Because the curve is not symmetrical about r₀ during oscillation, the average separation of the atoms increases. This explains thermal expansion.

Note: The energy needed to break the bond between atoms is U₀ (minimum potential energy). This is the energy needed to turn the matter in a gas.

Equation for energy and atomic separation

In the equation, A and B are positive constants and m and n are integers. Various values are possible for A, B, m and n.

Molecular Bonds – Types

Four main models of molecular bonding:

• Bonding between atoms within molecule

1. Ionic

• An Ionic bond is an interaction between oppositely charged ionized atoms

• Two atoms combine in such a way that one or more outer electrons are transferred from one atom to the other

• Ionic bonds are fundamentally caused by the Coulomb attraction between oppositely charged ions

• Sodium Chloride: NaCl (Na⁺ and Cl^- in solution)