321 Quantum Mechanics Unit 1

Quantum mechanics unit 1• Foundations of QM• Photoelectric effect, Compton effect, Matter waves

• The uncertainty principle

• The Schrödinger eqn. in 1D

• Square well potentials and 1D tunnelling

• The harmonic oscillator

321 Quantum Mechanics Unit 1

Last time• Photons• Photoelectric effect• Compton effect

• Matter waves• de Broglie relations• Electron diffraction• Waves related to probability

www.le.ac.uk/physics -> people -> Mervyn Roy

321 Quantum Mechanics Unit 1

Heisenberg uncertainty principle• Fundamental property of quantum systems• Course 3210 Unit 3, Rae Chapter 4

• Inherent uncertainties in predictions – or average spread of a set of repeated measurements

∆ 𝑥 ∆𝑝𝑥≥12ℏ

∆𝐸 ∆ 𝑡≥12ℏ

Heisenberg - formulation of QM 1927, Nobel prize 1932

321 Quantum Mechanics Unit 1

Example• The position of a proton is known to within 10-11 m.

Calculate the subsequent uncertainty in its position 1 second later.

km

321 Quantum Mechanics Unit 1

321 Quantum Mechanics Unit 1

321 Quantum Mechanics Unit 1

Example• An atom in an excited state emits a photon of

characteristic frequency, . n The average time between excitation and emission is 10-8 s. Calculate the irreducible linewidth of the transition.

Hz

321 Quantum Mechanics Unit 1

Schrödinger equation

• Developed by induction

• Tested against experiment for over 80 years

• is the wavefunction. This is essentially complex – cannot be identified with any one physical property of the system

Schrödinger - formulation of QM 1925-26, Nobel prize 1933

321 Quantum Mechanics Unit 1

Schrödinger equation • related to probability

dx

• Wavefunction must be normalised

= 1

321 Quantum Mechanics Unit 1

Time independent Schrödinger equation

eigenfunction

eigenvalue

321 Quantum Mechanics Unit 1

Constraints• The wavefunction and its first derivative must be:

• Single valued

• Finite

• Continuous