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 3

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: Wavefunctions for a bound particle in a finite well (1-D) V0V0 V=0 E Two types of solution: Cosine-like tan ka = μ/k Sine-like cot ka = -μ/k exp(-μx)

4. Finite square well: solutions for tan ka = μ/k where thus In graph below, y = ka

6. Compare with wavefunctions for an infinite square well: n=3 n=2 n=1

7. Useful formulae TDSE – time dependent Schrödinger Equation TISE – time independent S.E. Vector operators in spherical polar coordinates Angular momentum operators in spherical polars

9. 1.3 QM tunnelling through a barrier A e ikx B e -ikx F e ikx V=0 V=V 0 Consider a flux of particles, momentum ħk, energy E= ħ 2 k 2 /2m approaching a barrier, height V 0 (V 0 > E), width a. 0a x We assume that some flux emerges on the far side…