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• Solving the Schrödinger equation (for given )

1. Examine the physics

2. Write down the solution to the S.E. (which will generally contain some arbitrary constants)

3. Apply the boundary conditions and normalise the wavefunction to find the unknown constants

4. Find the allowed energies and the probability density

321 Quantum Mechanics Unit 1

Finite square well

𝑉 0

−𝑎 𝑎𝑉=0

www2.le.ac.uk/departments/physics/people/mervynroy/lectures

321 Quantum Mechanics Unit 1

𝑉 0=25ℏ2

2𝑚𝑎2

Graphical solution:

𝑘0𝑎=5

Even parity states

𝑥=𝑘𝑎=1.3 𝑥=3 .8

321 Quantum Mechanics Unit 1

𝑉 0=25ℏ2

2𝑚𝑎2

𝑘0𝑎=5

Odd parity states

𝑥=𝑘𝑎=2.6 𝑥=4.9

Graphical solution:

321 Quantum Mechanics Unit 1

Compare infinite to finite wellWell half width, Å, Finite well depth, eV

Infinite well eV eV eV…

Finite well eV eV eV eV

321 Quantum Mechanics Unit 1

Quantum tunnelling

321 Quantum Mechanics Unit 1

Quantum tunnelling

321 Quantum Mechanics Unit 1

Tunnelling of classical waves

Tippler – 35.4 Reflection and transmission of water waves: Barrier penetration

321 Quantum Mechanics Unit 1

Tunnelling of classical waves

321 Quantum Mechanics Unit 1

Tunnelling through a barrier

321 Quantum Mechanics Unit 1

Tunnelling through a barrier• Write down solutions to S.E.

• Apply boundary conditions at ,

• Eliminate coefficients - see notes at www2.le.ac.uk/departments/physics/people/mervynroy/lectures

• Find

321 Quantum Mechanics Unit 1

Å eV

321 Quantum Mechanics Unit 1

Tunnelling through a barrier• Transmission probability is

• Large barrier, then

• General case if

321 Quantum Mechanics Unit 1