ConcepTest #72 Look at the potential well sketched to the right. A particle has energy E which is less than the energy of the barrier U 0 located at 5 < x < 6. Consider the following regions: Region A: to the left of the barrier, x < 5 Region B: to the right of the barrier, x > 6 Compare the wavelengths in the two regions: E U0U0

ConcepTest #73 A system is in the (superposition) state If you make a measurement of the system, what is the probability you would find the system in the state ? (Alternate: What is ?) Follow-up examples: If you make a measurement of the system, what is the probability of finding the system in the state ?

Example: States and Probabilities A system is in the (superposition) state Determine the probability of finding the particle to be in the state Alternate wording: Calculate

ConcepTest #74 Consider a particle of mass m trapped in an infinite square well (“box”) of length L. The state, where, represents a “stationary” or “pure” state of the particle, with energy. The particle is in the superposition state. a) What is the energy of the superposition state? Hold up as many cards as needed. b) You measure the energy of this particle. What value will you measure? Hold up as many cards as needed. c) You find the energy of the particle to be E 1. You immediately measure the energy of this particle again. What value will you measure? Hold up as many cards as needed.

Example: Expectation Value is Average!

Quantum States & Quantum Numbers: mass m, length L ; 0 < x < L Pure state:, quantum number Superposition state mass m, length L 1, L 2 ; 0 < x < L 1 ; 0 < y < L 2 Pure state: Quantum numbers: mass m, length L 1, L 2, L 3 ; 0 < x < L 1 ; 0 < y < L 2 ; 0 < z < L 3 Pure state: Quantum numbers: Particle in a 1-D boxParticle in a 2-D box Particle in a 3-D box Each spatial dimension “gets” one quantum number

More than just spatial dimensions: Spin Angular Momentum QUANTUM SPIN Classical Angular Momentum Review Earth around axis (day/night); Ball spins on own axis, etc. Orbital Angular Momentum Circular orbits, reference point center of circle Earth around sun (year)  Particles have INTRINSIC SPIN spin quantum number Integer spin quantum number  BOSONS examples: photons (s = 1) alpha particle( 4 He nuclei) Half - Integer spin quantum number  FERMIONS examples: electrons, protons, neutrons, (s = ½)