Quantum mechanics I Fall 2012 Physics 451 Quantum mechanics I Fall 2012 Oct 19, 2012 Karine Chesnel

Next homework assignments: Phys 451 Next homework assignments: HW # 14 due Friday Oct 19 by 7pm Pb 3.7, 3.9, 3.10, 3.11, A26 HW #15 due Tuesday Oct 23 Pb 3.13, 3.14, 3.17, 3.18, 3.22, 3.23 Practice test 2 M Oct 22 Work with your group! Test 2 : Tu Oct 23 – Fri Oct 26

Generalized statistical interpretation Phys 451 Generalized statistical interpretation Particle in a given state We measure an observable (Hermitian operator) Operator’s eigenstates: eigenvector eigenvalue Eigenvectors are complete: Discrete spectrum Continuous spectrum

Generalized statistical interpretation Phys 451 Generalized statistical interpretation Particle in a given state Fourier’s trick to find Cn Normalization: Expectation value

If you measure an observable Q on a particle in a certain state , Phys 451 Quiz 18 If you measure an observable Q on a particle in a certain state , what result will you get? the expectation value one of the eigenvalues of Q the average of all eigenvalues A combination of eigenvalues with their respective probabilities

Generalized statistical interpretation Phys 451 Generalized statistical interpretation Operator ‘position’: Probability of finding the particle at x=y:

Generalized statistical interpretation Phys 451 Generalized statistical interpretation Operator ‘momentum’: Probability of measuring momentum p: Pb 3.11: probability of measuring p in a given range

Different notations to express the wave function: Phys 451 The Dirac notation Different notations to express the wave function: Projection on the position eigenstates Projection on the momentum eigenstates Projection on the energy eigenstates

The uncertainty principle Quantum mechanics The uncertainty principle Finding a relationship between standard deviations for a pair of observables Uncertainty applies only for incompatible observables

The uncertainty principle Quantum mechanics The uncertainty principle Position - momentum

The uncertainty principle Quantum mechanics The uncertainty principle Position - Energy Pb 3.14

The uncertainty principle Quantum mechanics The uncertainty principle Energy - time Special meaning of Dt

An excited particle emits a certain radiation of energy E Quantum mechanics Quiz 19 An excited particle emits a certain radiation of energy E with a band width DE. What can we say about the characteristic lifetime of excited state? Lifetime is a least Lifetime is a most

Heisenberg equation of motion Quantum mechanics Heisenberg equation of motion Pb 3.17 constant Ehrenfest’s theorem

Heisenberg equation of motion Quantum mechanics Heisenberg equation of motion Definition for Dt: when To evaluate Dt: choose an appropriate operator calculate and Pb 3.18 Application: use Q = x, in the case of the infinite square well

The Dirac notation Inner product: “bra” “ket” Operator: Pb 3.22 Quantum mechanics The Dirac notation Inner product: “bra” “ket” Operator: Pb 3.22 Projection operator: for orthonormal basis Pb 3.23 or