PHY202 Quantum Mechanics

Topic 1

Introduction to Quantum Physics

Outline of Topic 1

1. Dark clouds over classical physics

2. Brief chronology of quantum mechanics

4. The photoelectric effect

5. Wave-particle duality

6. Waves as particles: Compton scattering

3. Black body radiation

7. Particles as waves I: Double-slit interference

8. Particles as waves II: Electron diffraction

9. The Bohr atom

10. The Heisenberg Uncertainty Principle

Dark Clouds over Classical Physics

and the failure of the

“Maxwell-Boltzmann doctrine regarding the equipartition of energy.”

(i.e. black body radiation and the heat capacity of solids)

In his address to the British Association for the Advancement of Science in 1900 Lord Kelvin said,

“The beauty and clearness of the dynamical theory, which asserts light and heat to be modes of motion, is at present obscured by two clouds.”

“... how the earth can move through an elastic solid such as essentially is the luminiferous ether.”

Brief Chronology Between 1900 and 1925 Quantum Physics was developed by a number of physicists, including Planck, Einstein, Bohr and de Broglie.

From 1925 onwards a more mathematical approach was developed by Schrödinger (wave mechanics), Heisenberg (matrix mechanics) and Dirac (who developed a more general formulation).

Quantum mechanics underpins all of the physics of elementary particles, nuclei, atoms, molecules and solids.

Quantum mechanics does not explain how a quantum particle behaves. Instead, it gives a recipe for determining the probability of the measurement of the value of a physical variable (e.g. energy, position or momentum).

This information enables us to calculate the average value of the measurement of a physical variable.

This indeterminism in the measurement of a physical variable is profoundly different from classical physics.

Einstein, who never accepted this interpretation of quantum mechanics, declared that “God does not play dice”.

We will discuss some experiments (e.g. double slit diffraction or tunnelling) that

cannot be “explained” classically.

Bohr said, “Anyone who is not shocked by quantum theory has not understood it”.

Nevertheless, for all its philosophical difficulties, no prediction of quantum theory has ever been disproved.

Black Body Radiation

The energy density per unit wavelength, w(λ,T), emitted by the surface of a black body is a universal function of wavelength and temperature.

A black body is by definition a perfect absorber.

Planck (1900) obtained the correct expression for w(λ,T) by assuming that the energy emitted and absorbed by the surface of a black body is quantised in units of hν, where h is Planck’s constant (= 6.63×10−34 J s) and ν is frequency.

Black Body Radiation

The assumption from classical physics that the energy density satisfies equipartition (i.e. ) implies that

This is known as the ultraviolet catastrophe.

The Photoelectric Effect

In 1905 Einstein used the idea of quantised energy to explain the photoelectric effect.

The kinetic energy of electrons emitted from the surface of a metal when light of frequency ν is radiant upon it is Kinetic

energy

hν

Both black body radiation and the photoelectric effect show that energy is quantised with the quantum of energy being Planck’s constant.

Note, the KE is not proportional to the light intensity, although the number of electrons ejected is.

Wave-Particle Duality

In 1916 Einstein suggested that light carries quanta of momentum as well as energy. Light particles are called photons.

In 1923 de Broglie suggested a particle with a momentum p has an associated wavelength, λ:

Waves as Particles: Compton Scattering

Compton (1923) found that the change of wavelength of X-rays scattered from electrons in aluminium foil satisfies

Using E = hν = hc/λ = pc (as p = h/λ), and conservation of energy and momentum this expression is easily derived.

Particles as Waves I: Double Slit Diffraction

Each particle is detected at a definite place on the screen

Particles interfere with themselves !

Particles incident on a parallel pair of slits

The probability of a particle arriving at a particular place on the screen is determined by the diffraction pattern:

Video from http://www.hqrd.hitachi.co.jp/em/index.cfm

Electron diffraction Note that the interference pattern builds up even though the electrons are detected singly (and presumably therefore pass through the slits singly).

Particles as Waves II: Electron Diffraction in Solids

Example: Use the de Broglie relationship to calculate the wavelength of an electron whose energy is 6 eV.

How does this compare to the lattice spacing in a typical solid?

Davisson and Germer (1925) and Thompson (1927) diffracted electrons from crystalline structures according to Bragg’s Law of diffraction,

nλ = 2d sin θ

Using the non-relativistic expression = 5 Å

Interference patterns demonstrate the fundamental concept of superposition.

We will see later that we can describe a quantum particle by a wavefunction.

The particle’s wavefunction is a superposition of all the “waves” for all the possible paths taken by it.

1.  Atomic system exists in a discrete set of stationary states.

The Bohr Atom

2.  Radiation is absorbed or emitted in discrete quanta during a transition between states:

Ef – Ei = hν

3. Quantisation of angular momentum:

Classically, atoms are unstable!

1. Energy is quantised: E = hν. (How energy is quantised will be explained later.)

2. Wave-particle duality: λ = h/p (or p = h/λ ).

3. Interference phenomena imply superposition of waves.

We will see later that we can describe a quantum particle by a wavefunction.

The particle’s wavefunction is a superposition of all the “waves” for all the possible paths taken by it.

The Heisenberg Uncertainty Principle

Measurement of physical variables is different in the quantum world from the classical world.

In particular, for some pairs of variables it is impossible to know their exact values simultaneously.

The microscope’s resolving power is

Thus, the position of the particle is uncertain to within Δx.

For light to enter the microscope lens its x-component of momentum

must satisfy (where p is its total momentum).

x

y

Heisenberg’s Miscroscope

This change of momentum is imparted to the particle.

So the uncertainty in the x-component of the particle’s momentum is

The product of the uncertainties is

As we will see in Topic 8, the exact statement is that

where (hbar) is

Summary of Topic 1

1. Energy is quantised: E = hν.

2. Wave-particle duality: p = h/λ.

4. Interference phenomena imply superposition of waves.

5. Particles exist in stationary states (where energy is conserved).

6. Discrete transitions occur between stationary states.

3. Angular momentum is quantised:

7. The Heisenberg Uncertainty Principle: