PHYS Quantum Mechanics PHYS Quantum Mechanics Dr Jon Billowes Nuclear Physics Group (Schuster Building, room 4.10) These slides at: Lecture 7

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

Hermitian Operators They have real eigenvalues Eigenfunctions are orthonormal Eigenfunctions form a complete set

Summary of postulates 1.A quantum system has a wavefunction associated with it. 2.When a measurement is made, the result is one of the eigenvalues of the operator associated with the measurement. 3.As a result of the measurement the wavefunction “collapses” into the corresponding eigenfunction. 4.The probability of a particular outcome equals the square of the modulus of the overlap between the wavefunction before and after the measurement.

Example of a “measurement” Photons of unpolarised light polariser50% transmitted 100% polarised Describe each photon as a linear combination of eigenfunctions of dynamic variable being measured: = 50% VERTICAL + 50% HORIZONTAL After measurement photon collapses into the corresponding eigenfunction 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

Example of a “measurement” Photons of unpolarised light Birefringent crystal (eg Icelandic spar) Vertical polarization detector Horizontal polarization detector

Today: 1.5(a) Commutators 1.5(b) Compatibility If then the physical observables they represent are said to be compatible: the operators must have a common set of eigenfunctions: Example (1-D): momentum and kinetic energy operators have common set of eigenfunctions After a measurement of momentum we can exactly predict the outcome of a measurement of kinetic energy. 1.4 Finish off with discussion on continuous eigenvalues