PHYS301 Quantum Mechanics II

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PHYS301 Quantum Mechanics II

• Peter Ratoff
• Lancaster University

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Lecture notes on Web

• Go to Physics Dept Home Page
• Go to Information for Current Students
• Go to Timetables, Teaching Material Course News
• Go to PHYS301 Quantum Mechanics II
• Open the Powerpoint files (15 total)

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Timetable changes

• Week 1 4 lectures (Tue2, Thur, Fri)
• Week 2 3 lectures 1 seminar (Fri)
• Week 3 3 lectures 1 seminar (Fri)
• Week 4 3 lectures 1 seminar (Fri)
• Week 5 2 lectures (Tue)
• Week 6 1 seminar (to be arranged)

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Recommended Text Books

• A.I.M. Rae Quantum Mechanics
• Eisberg/Resnick Quantum Physics

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Course Outline

• Review of basic quantum physics (year 2)
• The 5 postulates of quantum mechanics
• The eigenfunctions of the hydrogen atom
• Quantum measurements Stern-Gerlach expt
• Commutation relations Uncertainty Principle
• Time independent perturbation theory

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Course Outline (contd)

• Degeneracy the Stark effect
• The helium atom the hydrogen molecule
• Identical particles the Exclusion Principle
• Time dependent perturbation theory
• Transition rates and selection rules

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Review of basic quantum physics

• Quantum Mechanics developed to explain the
  failure of classical physics to describe
  properties of matter on atomic scale
• - energy quantization line spectra
• - angular momentum quantization fine
  structure
• - quantum mechanical tunneling
  nuclear ? decay

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Atomic line spectra

• Emission spectra from atoms show characteristic
  lines (unique to each atomic species)
• Spectral lines correspond to transitions between
  discrete atomic energy levels
• Wavelengths follow precise mathematical patterns
• Evidence for energy quantization
• Fine structure (splitting) of spectral lines is
  evidence of angular momentum quantization
• ltltlt Mercury emission spectrum

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Quantum mechanical tunneling

• Wave packet is a spatially localised
  superposition of plane waves - a good description
  of a particle
• if kinetic energy lt potential energy (height of
  barrier)
• then, classically particle cant penetrate
  barrier
• QM allows barrier penetration !
• in nuclei ? particle can tunnel through Coulomb
  potential barrier

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Nuclear ? particle decay

• Nuclear square well coulomb barrier ( 1/r)
• ? particle energy lt coulomb barrier height
  (coulomb energy at nuclear surface)
• classically ? cant escape from nucleus
• QM allows barrier penetration
• nuclear decay lifetime depends on barrier
  thickness

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A Physics joke .
Q What's the difference between a quantum
mechanic and an auto mechanic? A A quantum
mechanic can get his car into the garage without
opening the door!
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Wave particle duality

• Light has wave-like properties (interference,
  diffraction, refraction) and particle-like
  properties (photo-electric effect, Compton
  effect)
• Particles show classical particle behaviour
  (scattering) and also have wave-like properties
  (double slit electron interference, neutron
  diffraction)
• A wave equation can be constructed which
  describes the state of a particle or system of
  particles (the Schroedinger equation)

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The 5 Postulates of QM

• wave function observables
• operators, eigenvalues and eigenfunctions
• position and momentum operators
• states and probabilities
• time dependent Schroedinger equation

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Postulate 1

• For every dynamical system there exists a wave
• function ? that is a single valued function of
  the
• parameters of the system and of time, and from
• which all possible predictions of the physical
• properties of the system can be obtained.

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The wave function

• ? ?(r,t) in general, a complex number
  function
• (Real) Probabilty density P(r,t) ??
  ?(r,t)²
• Probabilty (in volume d³r) P(r,t) d³r
  ?(r,t)² d³r
• Normalisation ? ?(r,t)² d³r 1 over all
  space

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Example ? normalisation

• ?(x) A / ?(1x²)
• ? ? ² dx A² ? dx / (1x²) -? lt x lt
  ?
• A² arctan(x ?) -
  arctan(x -?)
• ? A²
• 1 from the normalisation
  requirement
• gt A (1 /?(?)) . exp(i?) ? is a complex
  phase
• by convention A 1 / ?(?) i.e. ?
  0
• Probability to find particle in range -1 lt x lt 1
  ?
• Prob A² arctan(x 1) - arctan(x -1)
• (1/?) (?/4) - (-?/4)
• 1/2

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Gauge invariance

• Any wavefunction can be multiplied by a phase
  factor exp(i?), where ? is a real number, with no
  physical consequence.
• Wavefunction ? ? exp(i?)
• Probability density ?²
• ?² ?² exp(i?) exp(-i?) ?²
• Probability density unchanged!
• Important in particle physics and the unification
  of fundamental forces

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Example free particle normalisation

• Free particle ? (x,t) A exp i (k x - ? t)
• ? ? ² dx A² ? exp(ikx-i?t-ikxi?t) dx
• A² ? dx (-? lt x lt ? )
  1
• gt A 0 as integral is infinite
• i.e. probability of localising particle to
  position x is zero
• to avoid problem, consider beam of particles with
  average separation L
• unit probability of finding a particle in range 0
  lt x lt L
• ? ? ² dx 1 over range 0 lt x lt L
• gt A² L 1 i.e. A 1/ ?(L)
• ? (x,t) 1/ ?(L) . exp i (k x - ? t)