PH 401

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PH 401

• Dr. Cecilia Vogel
• Lecture 9

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Review

• notation operators, states
• commutators
• eigenstates and eigenvalues

Outline

• Review four postulates
• Review braket notation
• eigenstates and eigenvalues
• exam 1

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4 Postulates

• Wavefunction contains all accessibel information
  about particles state
• position probability density absolute square of
  wavefunction
• every observable has a corresponding operator
• The Time Dependent Schroedinger Equation

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Review Braket Notation

• state ket, agt
• operator acts on a state
• operators dont necessarily commute
• Some properties
• ltabgt number (found by integral)
• ltaagt1
• expectation value of observable

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Eigenstates

• If the state of the particle is such that
• some observable has a definite value,
• zero uncertainty,
• then the state is an eigenstate of that
  observable
• the definite value is called the eigenvalue.
• If an ensemble of particles is in an eigenstate
  of an observable,
• such as energy,
• then every measurement of that observable will
  yield the eigenvalue.

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Eigenstates

• If the state q1gt is an eigenstate of observable
  Q with eigenvalue q1,
• then when the operator Q-hat acts on that state,
• the result is
• Examples
• eikx and p
• e-iwt and E

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Eigenstates Expectation Values

• If the state q1gt is an eigenstate of observable
  Q with eigenvalue q1,
• then the expectation value of operator Q-hat
• If every measurement of that observable will
  yield the eigenvalue,
• then the average will be the eigenvalue.

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Eigenstates Uncertainty

• If the state q1gt is an eigenstate of observable
  Q with eigenvalue q1,
• then the uncertainty of operator Q-hat
• If every measurement of that observable will
  yield the eigenvalue,
• then there will be no deviation.

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Simultaneous Eigenstates

• If the state o1q1gt is an eigenstate of
  observable Q and O,
• then when either operator acts on that state,
• the result is a constant (the eigenvalue) times
  the state
• Examples Can we have a simultaneous eigenstate
  of x and p?
• Consider (xp-px)Y
• (x1p1-p1x1) Y0, but
• -ihbar(x(d/dx)-(d/dx)x)Y is not 0

NO!
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Simultaneous Eigenstates

• How about simultaneous eigenstates of p and E,
• Eigenstates of p
• Y(x, t)f(t)eipx/hbar
• Y(x, t) g(x)e-iEt/hbar
• Eigenstate of both
• Y(x, t)Aeipx/hbar e-iEt/hbar
• can only have definite p and E, if E depends on
  p, but not E
• Free particle Ep2/2m

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For Wed

• chapter 6.1-7