Superposition

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Superposition

• Quantum Mechanics
• PHYS481

2
Problems for Section 5.1

• 5.2 State in a 1-d box
• 5.5 Energy probabilities in a 1-d box
• 5.9 In the middle of a box
• 5.10 Not in the right half of a box

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Problems for Section 5.2

• 5.12 Commutators for 3 operators
• 5.13 Commutator for Hermitian operators
• 5.15 Linear independence of functions
• 5.18 Commutator for common eigenstates

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Problems for 5.3

• 5.20 Phase factors

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Problems for 5.4

• 5.22 Momentum energy for trig states
• 5.24 Operator product commutators
• 5.25 Miscellaneous uncertainty relations
• 5.28 Deriving energy-time uncertainty

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Problems for Section 5.5

• 5.35 Taylor expansion
• 5.40 Averages and sqrt variances
• 5.41 Uncertainty in 1-d box

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Commutators

• Commutator of A B A,B AB BA
• To find A,B g(x) commutator ? g(x)
• For A B to commute, their commutator 0
• Also A B are said to be compatible
• Always commute with constants
• Always commute with their squares
• Always commute with functions of themselves
• TRICK expansion ep exp (p) ?(1/n!) pn
  (see eq 5.58)
• NB all functions are trigonometric summations
  (Fourier)
• Nonzero commutator examples
• x,p i?
• x,p2 2i ?p
• x2,p 2i ?x
• E,t i?

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Common eigenfunctions

• THEOREM If A B commute, A and B have
  nontrivial common eigenfunctions.
• Be able to prove this
• APPLICATION Momentum and energy have common
  eigenfunctions

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Linear independence

• Set of functions are linearly independent IF
  linear combination is 0 only if all coeff are 0
• In Hilbert space, if two vectors are linearly
  independent, they dont point in same direction
• One must rotate to align them
• How many eigenvectors are there corresponding to
  any given eigenvalue? How many linearly
  independent eigenvectors?

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Degeneracy

• An eigenvalue is degenerate if it corresponds to
  more than one eigenvector (provided they are
  independent of one another)
• Doubly degenerate if there are two independent
  eigenvectors
• N-fold degenerate if there are N independent
  eigenvectors
• An operator is degenerate if it has degenerate
  eigenvalues

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Sharing eigenvectors

• If A B commute, an eigenfunction of one may or
  may not be an eigenfunction of the other,
  depending on the degeneracy of A B.
• degenerate operators have more eigenstates than
  nondegenerate ones
• Eigenvectors may be degenerate wrt one operator
  and not in the other (Refer to Venn diagrams on
  p.138)
• THEOREM One can form N linear combinations
  which are N linearly independent eigenstates of
  both A and B
• Eg. free particle energy eigenvalue (doubly
  degenerate)
• Eigenvector of H is also eigenvector of p2
• Eigenvector of H can be cos, sin, or exp
• p,H 0 but sin and cos are not eigenvectors of
  p

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Uncertainty

• Uncertainty relations have origins in
  compatibility properties
• ?A ?B ? ½ lt A,B gt
• Eg. ?x ?p ? ½ lt i? gt ½ i?
• Recall ?A ?A ltA2gt - ltAgt2 ? 0
• Alternate way of calculating uncertainty/errors
• If A and B dont commute, then measuring A
  doesnt necessarily limit range of B