PHYS Quantum Mechanics “the dreams stuff is made of” PHYS Quantum Mechanics “the dreams stuff is made of” Dr Jon Billowes Nuclear Physics Group (Schuster Building, room 4.10) These slides at:

PC3101 Quantum Mechanics Recommended texts: A.I.M Rae, Quantum Mechanics (4 th edition, IOP)* F. Mandl, Quantum Mechanics P.C.W. Davies, Quantum Mechanics A.P. French & E.F. Taylor, An Intro. To Quantum Mechanics Beyond level of course: S. Gasiorowicz, Quantum Mechanics (but recommended for PC3602) *supplementary material has moved to

Syllabus 1.Basics of quantum mechanics (QM) Postulate, operators, eigenvalues & eigenfunctions, orthogonality & completeness, time-dependent Schrödinger equation, probabilistic interpretation, compatibility of observables, the uncertainty principle. 2.1-D QM Bound states, potential barriers, tunnelling phenomena. 3.Orbital angular momentum Commutation relations, eigenvalues of L z and L 2, explicit forms of L z and L 2 in spherical polar coordinates, spherical harmonics Y l,m. 4.Spin Noncommutativity of spin operators, ladder operators, Dirac notation, Pauli spin matrices, the Stern-Gerlach experiment. 5.Addition of angular momentum Total angular momentum operators, eigenvalues and eigenfunctions of J z and J 2. 6.The hydrogen atom revisited Spin-orbit coupling, fine structure, Zeeman effect. 7.Perturbation theory First-order perturbation theory for energy levels. 8.Conceptual problems The EPR paradox, Bell’s inequalities.

The Schrödinger Equation was guessed by induction: Seemed plausible test works OK, accept until falsified Classical plane wave (sound or light) obeys the wave equation The solution requires But does not work for matter waves where we want Requires 2 nd derivative of x but only 1 st derivative of t Idea! Let’s try (TDSE: Time-dependent Schrödinger equation) No prediction of quantum mechanics has ever been experimentally falsified

Wave-particle duality applies to all objects: particle screen Particle detected at single point on the screen – the probability wave instantaneously collapses to zero everywhere else. If undisturbed, the particle propagates as a (probability) wave. Development of the wave with time is exactly described by the TDSE Electron interference: Quantum interference experiments: Single photons electrons neutrons atoms Buckminster fullerene (C 60 )

Conceptual problems with quantum mechanics Quantum mechanics works - but there are many worries on interpretation that tend to become matters of opinion; debate enters realm of philosophy. Conceptual basis of QM is fundamental to our understanding of the nature of the physical universe – so we should try and learn more by experiment, not debate (EPR paradox and Bell’s inequalities). Alastair Rae: (in 4 th edition of his text book) “my own understanding continues to grow…” Richard Feynman: “I think I can safely say that nobody understands quantum mechanics.” Niels Bohr: “Anyone who is not shocked by quantum mechanics has not understood it.” Feynman (again): “…shut up and calculate.”

After revision of basics, first new topic will be Quantum Mechanical Tunnelling Consider a roller-coaster… Classically the car can only go as far As C before rolling back – but quantum- fluctuations in energy could allow the car through the energy-forbidden region and appear at E. This quantum process controls the rates of alpha- decay and spontaneous fission in nuclear physics. Application: Scanning tunnelling microscope Iron atoms on copper

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

Plan of action 1.Basics of QM 2.1D QM Will be covered in the following order: 1.1 Some light revision and reminders 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