Quantum mechanics unit 1

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 www2.le.ac.uk/departments/physics/people/academic-staff/mr6/lectures

2. Workshop questions • 1. Compton effect

3. Workshop questions • 4 & 5. Infinite well, • Ground state • even parity • n=1 Infinite well Å

4. Workshop questions • 4 & 5. Infinite well, • First excited state • odd parity • n=2 Infinite well Å

5. Last time • Finite well – see Rae, Worked Example 2.3 (p29) • Tunnelling through a barrier • Write down the things we know about the solutions to S.E. • Apply the boundary conditions and normalise the wavefunction to find any unknown constants (see notes) www2.le.ac.uk/departments/physics/people/academic-staff/mr6/lectures

6. Tunnelling through a barrier • Transmission probability is • Large barrier, then • General case if

7. T is the Transmission coefficient, R = 1-T is the Reflection coefficient

8. semiconducting nanotube source drain L~1 m SiO2 quantum dot gate Harmonic oscillator • potentials very common in physics! 2R

9. Harmonic oscillator where Time independent Schrödinger equation is Change variable from to , This is often written where is the oscillator length scale.

10. Let then and. Substituting for gives Then, or, where . is a standard equation (Rae, Sec. 2.7) with solutions, whereare Hermite polynomials.

11. Then, if are the harmonic oscillator wavefunctions. The allowed energies are where is the zero point energy. is the normalisation constant. The first 2 normalised wavefunctions are,

13. Correspondence principle - quantum physics reproduces classical results in the limit of large .

14. Cross-sectional Scanning Tunnelling microscopy Mn impurity in GaAs matrix InAs quantum dots in GaAs matrix • A. Yakunin– PhD thesis (2005), TechnischeUniversiteit, Eindhoven. D. M. Bruls– PhD thesis (2003), TechnischeUniversiteit, Eindhoven.