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Understanding Quantum Mechanics: Tunneling Phenomena and Postulates

This lecture by Dr. Jon Billowes focuses on key concepts of Quantum Mechanics (QM), exploring the principles of tunneling phenomena through potential barriers and the time-independent Schrödinger equation (TISE). Topics include infinite and finite wells, barriers, postulates of QM, the significance of wave functions (Ψ), Hermitian operators, and the uncertainty principle. The lecture dives into practical applications like nuclear decay and thermonuclear fusion, expanding on how quantum mechanics explains these processes in the universe.

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Understanding Quantum Mechanics: Tunneling Phenomena and Postulates

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  1. PHYS 30101 Quantum Mechanics Lecture 4 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

  2. Plan of action • Basics of QM • 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 from lecture 3 1.3 QM tunnelling through a barrier V=V0 A eikx F eikx B e-ikx V=0 x 0 a 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…

  4. Reflection and transmission at a potential barrier: Quantum mechanical tunnelling http://www.sgi.com/fun/java/john/wave-sim.html

  5. Simple theory of α decay The α particle is preformed in the nucleus and bouncing around within the walls formed by the Coulomb barrier. Classically, it is impossible for the particle to escape but in reality it can tunnel through the energy-forbidden region to escape with final kinetic energy equal to the Q value. The chance of tunnelling through depends strongly on the width and height of the barrier, so the higher the Q value is, the greater the chance of escape. Q value The α-particle makes about 1020 “assaults” on the barrier every second. It can take years before it escapes.

  6. Half-lives of alpha-emitters Age of Universe 1 microsecond Energy of α particle

  7. Scanning Tunnelling Microscope No applied E-field V With applied field Potential energy of electron near the surface of a metal

  8. Image of a surface obtained with by scanning tunnelling microscopy

  9. Thermonuclear fusion in stars proton proton p + p d + e + + νe The reverse of α-decay. It happens at surprisingly low temperatures – the average thermal energy of protons is well below the (Coulomb) barrier and fusion takes place by barrier penetration – a slow process, so nuclear fuel lasts for astronomically long times. Sir Arthur Eddington (BSc in Physics, 1st Class, Owens College, Manchester 1902) To doubters that stars were hot enough for fusion he would say “Not hot enough? Go and find a hotter place!”

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