Transcript lect10
PHYS 30101 Quantum Mechanics Lecture 10 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 1.7 Degeneracy 1.7 Degeneracy Orthonormality: it is always possible to construct a set of orthonormal eigenfunctions from a set of non-orthonormal eigenfunctions (“Schmidt orthogonalization” – see Rae) Compatibility: If then a common set of eigenfunctions exist. But we will show an eigenfunction of Q is not necessarily an eigenfunction of R. Nevertheless a set of eigenfunctions can be found that are common to both operators 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 Lz and L2, explicit forms of Lz and L2 in spherical polar coordinates, spherical harmonics Yl,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 Jz and J2. 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 main points so far: There is an (hermitian) operator for every dynamical variable (position, momentum, energy…). Each operator has a complete set of orthonormal eigenfunctions with real eigenvalues: In order to predict the result of a measurement of “q” on a general wavefunction Ψ we need to describe Ψ as a linear combination of the eigenfunctions of the corresponding operator: Ψ = a1φ1 + a2φ2 + a3φ3 ….. + anφn … where The measurement forces Ψ to decide which eigenfunction it is going to “collapse” into (think about unpolarised photons and a polariser). It will collapse into the state φm with probability |am|2 and return the value “qm”. Expectation value (mean value) of q: Example of a “measurement” polariser 50% transmitted Unpolarised light 100% polarised Describe each photon as a linear combination of eigenfunctions of dynamic variable being measured: After measurement photon collapses into the corresponding eigenfunction = 50% VERTICAL + 50% HORIZONTAL After measurement the photon has no memory of its polarization state before the polariser. All subsequent Vertical/Horizontal measurements of transmitted photon will give the definite result: Vertical If then there exists a common set of eigenfunctions: A measurement of “q” will collapse Ψ into one of the eigenfunctions φm and return the result “qm” A subsequent measurement of “r” will now give the definite result “rm” Any further measurement of “q” or “r” for this wavefunction is exactly predictable – the answers will always be “qm” and “rm”. In this case ΔqΔr = 0 Degeneracy If then there exists a common set of eigenfunctions but if there is some degeneracy: and a measurement returns the value “q” the wavefunction collapses into an unknown combination of eigenfunctions that have this eigenvalue: A subsequent measurement of “r” will then force the intermediate wavefunction to collapse into one of the common eigenfunctions, leaving “q” unchanged and returning the eigenvalue “rn” (say). Any further measurement of “q” or “r” for this wavefunction is exactly predictable – the answers will always be “q” and “rn”. In this case ΔqΔr = 0 Example: n=2 level of H-atom L=1 L=0 n=2 (2p level) (2s level) m=0 Measure L2 m=+1 m=0 m=-1 (has eigenvalues L(L+1) ħ2 ) If result is “2ħ2” then we know L=1 but don’t know what “m” will be before the measurement of Lz (its eigenvalues are “m”) Measure Lz, and we force atom to decide its orientation “m” (ie the value of the projection of L on our chosen z-axiz) All subsequent measurements of L2 and Lz will give same results. 3. Orbital Angular Momentum One of the most important aspects of QM with implications in particle, nuclear, atomic, molecular, laser physics. Angular momentum is always quantised because of periodic boundary conditions such as Ψ(r, θ, φ) = Ψ(r,θ, φ+2π). [Background (beyond this course): Paul Dirac (in 1928) “guessed” the relativistic version of the Schrödinger Equation following principle of postulate 3. This is called the Dirac Equation. •“Intrinsic” spin ½ particles emerged from his theory in a natural way We deal with intrinsic spin in this course •Intrinsic magnetic moment correctly predicted •Proton and neutron were not “Dirac” particles but have sub-structure •Antiparticles predicted. Positron discovered in 1933 (Anderson)] Useful formulae TDSE – time dependent Schrödinger Equation Vector operators in spherical polar coordinates Angular momentum operators in spherical polars TISE – time independent S.E.