1. PHYS 30101 Quantum Mechanics PHYS 30101 Quantum Mechanics Dr Jon Billowes Nuclear Physics Group (Schuster Building, room 4.10) j.billowes@manchester.ac.uk These slides at: www.man.ac.uk/dalton/phys30101 Lecture 8

2. Plan of action 1.Basics of QM 2.1D QM Will be covered in the following order: 1.1 Some light revision and reminders. Infinite well 1.2 TISE applied to finite wells 1.3 TISE applied to barriers – tunnelling phenomena 1.4 Postulates of QM (i) What Ψ represents (ii) Hermitian operators for dynamical variables (iii) Operators for position, momentum, ang. mom. (iv) Result of measurement 1.5 Commutators, compatibility, uncertainty principle 1.6 Time-dependence of Ψ

3. Re-cap 1.5(a) Commutators 1.5(b) Compatibility If then the physical observables they represent are said to be compatible: the operators must have a common set of eigenfunctions: Example (1-D): momentum and kinetic energy operators have common set of eigenfunctions After a measurement of momentum we can exactly predict the outcome of a measurement of kinetic energy. The same is not true for a measurement of momentum and total energy (KE+PE).

4. 1.5(c) The Uncertainty Principle If two operators do not commute there is a fundamental limit on the products of the root mean square deviation associated with the measurements of the two dynamical observables: Today 1.6 Time-dependence of Ψ Time-evolution of wavefunction between measurements; Quantum beats

9. Uncertainty Principle - example particles screen P=ħk x z Estimating errors on measuring X and P x

10. Harmonic oscillator wavefunctions Quantum beat between n=0 and n=1 eigenstates