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PHYS 30101 Quantum Mechanics Lecture 7 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 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” polariser 50% transmitted Photons of 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 Example of a “measurement” Vertical polarization detector Photons of unpolarised light Horizontal polarization detector Birefringent crystal (eg Icelandic spar) Today: 1.4 Finish off with discussion on continuous eigenvalues 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.