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PHAS2222 Revision Lecture

PHAS2222 Revision Lecture. The plan: First hour: Summary of main points and equations of course Opportunity to request particular topics Second hour: Specially requested topics The 2007 exam paper (mainly Section B)

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PHAS2222 Revision Lecture

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  1. PHAS2222 Revision Lecture • The plan: • First hour: • Summary of main points and equations of course • Opportunity to request particular topics • Second hour: • Specially requested topics • The 2007 exam paper (mainly Section B) • Note: model answers to exams not available on website, but I am happy to give feedback on attempts at past examination papers PHAS2222 Revision Lecture 2008

  2. Angular momentum operators Photo-electric effect, Compton scattering Davisson-Germer experiment, double-slit experiment Particle nature of light in quantum mechanics Wave nature of matter in quantum mechanics Wave-particle duality Postulates: Operators,eigenvalues and eigenfunctions, expansions in complete sets, commutators, expectation values, time evolution Time-dependent Schrödinger equation, Born interpretation 2246 Maths Methods III Separation of variables Time-independent Schrödinger equation Frobenius method Quantum simple harmonic oscillator Legendre equation 2246 Hydrogenic atom 1D problems Angular solution Radial solution

  3. Section 1 – Failure of classical mechanics The photoelectric effect: De Broglie’s relationship for matter waves: Compton scattering: when photon deflected through angle θ, new wavelength is Verifies Diffraction from crystal surfaces and double-slit experiments: Maximum scattering when path difference = nλ

  4. Section 2 – A wave equation for matter waves Time-dependent Schrödinger equation: (for matter waves in free space) Hamiltonian operator (represents energy of particle): (generally) Born interpretation: probability of finding particle in a small length δx at position x and time t is equal to If Hamiltonian is independent of time, can try solution (time-independent SE) Find Uncertainty principle:

  5. Section 3 – Examples of the time-independent SE in 1 dimension V(x) The wavefunction must: III I II 1. Be a continuous and single-valued function of both x and t V0 2. Have a continuous first derivative (unless the potential goes to infinity) 3. Have a finite normalization integral. a V(x) V(x) V(x) III I II V0 V0 -a a x Finite square well Free particle Travelling waves, arbitrary value of energy Matching of solutions: travelling waves (sines or cosines) in well, exponentials in barriers Infinite square well Rectangular barrier Potential step Tunnelling Quantization of energy Transmission and reflection

  6. Section 3 – contd Particle flux (flow of probability): Simple harmonic oscillator: Series solution for H(y) must terminate, so H is a finite power series (polynomial) – called a Hermite polynomial. Termination condition

  7. Section 4 – Postulates of quantum mechanics Postulate 4.1: Existence of wavefunction, related to probability density by Born interpretation. Postulate 4.2: to each observable quantity is associated a linear, Hermitian operator (LHO). The eigenvalues of the operator represent the possible results of carrying out a measurement of the corresponding quantity. Immediately after making a measurement, the wavefunction is identical to an eigenfunction of the operator corresponding to the eigenvalue just obtained as the measurement result. Postulate 4.3: the operators representing the position and momentum of a particle are

  8. Then the probability of obtaining the eigenvalue qn as the measurement result is Section 4 - contd The eigenfunctions of a Hermitian operator belonging to different eigenvalues are orthogonal. If then The eigenfunctions φnof a Hermitian operator form a complete set, meaning that any other function satisfying the same boundary conditions can be expanded as Note that if ψ normalized, φ orthonormal. Postulate 4.4: suppose a measurement of the quantity Q is made, and that the (normalized) wavefunction can be expanded in terms of the (normalized) eigenfunctions φnof the corresponding operator as

  9. Section 4 - contd Commutator: Commuting operators have same eigenfunctions, can have well-defined values simultaneously (‘compatible’) Expectation value: Postulate 4.5: Between measurements (i.e. when it is not disturbed by external influences) the wave-function evolves with time according to the time-dependent Schrődinger equation. Time development in terms of eigenfunctions of Hamiltonian: If and then

  10. Section 5 – Angular momentum Different components do not commute: but In spherical polar coordinates: Lz Their simultaneous eigenfunctions are spherical harmonics: Ly Conserved for problems with spherical symmetry Lx

  11. Section 6 – The hydrogen atom Now look for solutions in the form Angular parts are spherical harmonics Radial part: with Atomic units:

  12. Section 6 - contd Put Series solution for F must terminate: possible only if 0 n is principal quantum number l=0,1,2,…,n-1 -1 l=0 l=1 l=2 l=3

  13. Section 7 - Spin Interaction with magnetic field: Lz Stern-Gerlach experiment: atoms with single outer electron divide into two groups with opposite magnetic moments. |L-S| Coupling of spin to orbital angular momentum: S Ly Quantum numbers describing spin: L L+S Lx

  14. S L Section 7 - contd Total angular momentum (orbital + spin) • Described by two quantum numbers: • j (determining quantity of total angular momentum present); ranges from |l-s| to l+s in integer steps • mj (determining projection of total angular momentum along z), ranges from –j to +j in integer steps

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