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PHYS 30101 Quantum Mechanics Lecture 3 Dr Jon Billowes Nuclear Physics Group (Schuster Building, room 4.10) [email protected] These slides at: www.man.ac.uk/dalton/phys30101 Plan of action 1. Basics of QM Will be covered in the following order: 2. 1D QM 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 Re-cap: Wavefunctions for a bound particle in a finite well (1-D) Two types of solution: V0 E V=0 Cosine-like tan ka = μ/k Sine-like cot ka = -μ/k exp(-μx) Finite square well: solutions for tan ka = μ/k thus where In graph below, y = ka Compare with wavefunctions for an infinite square well: n=3 n=2 n=1 Useful formulae TDSE – time dependent Schrödinger Equation Vector operators in spherical polar coordinates Angular momentum operators in spherical polars TISE – time independent S.E. 1.3 QM tunnelling through a barrier V=V0 A eikx F eikx B e-ikx V=0 0 a x Consider a flux of particles, momentum ħk, energy E= ħ2k2/2m approaching a barrier, height V0 (V0 > E), width a. We assume that some flux emerges on the far side…