Announcements Exam is Sept from PM in SH Schaeffer by Commonthread


									                          Announcements                                                                              Sigma Aldrich
• Exam 1 is Sept. 21 from 6:30-8:30 PM in 140 SH (Schaeffer Hall).
                                                                                             •    Sigma-Aldrich Corporation will be recruiting Chemists, Biologists and
• Courtney will give a review on Monday, Sept. 20 during regular                                  Biochemists at the University of Iowa for Full-time and Co-operative
  lecture time. (Question and answer)                                                             Education opportunities.
• I will not have office hours on Monday,Sept. 20.
                                                                                             •    Representatives from Sigma will be present at Careers Day '99 on September
• I will have office hours on Tuesday, Sept. 21 from 9-11 AM and from                             23 from 12:30 pm - 6 pm in the Iowa Memorial Union.
  2-3 PM.
• Answers to PS2 are on reserve in library. (electronic and paper).                          •    We will hold an informational meeting to discuss Career Opportunities at
• Graded Problem Set 2 will be available for you to pick up in the                                Sigma on September 23 at 7 pm in the Iowa Memorial Union River Room.
  Chemistry Center (237CB) on Monday.
                                                                                             •    Sigma will be conducting interviews for Full-time and Co-operative
• Regrades on problem sets can be handed in to me or to Courtney.
                                                                                                  Education opportunities on September 24 in the Placement Office, 24
  Please indicate on the front page the problem number to be regraded.                            Phillips Hall.

      Postulates of Quantum Mechanics                                                                                    Postulate II
                                                                                                 • Every physical observable is represented by
     • Postulate I: The state of a system is defined by a
            function, Ψ , the wave function, that contains all the                                 a linear operator,Ω , such that:
                                                                                                                      ∧              ∧           ∧
            information that can be known about the system.                                                          Ω(f + g) = Ω(f) + Ω(g)
            The wavefunction, Ψ, and its first and second                                                            ∧
            derivatives must be:                                                                                     x is linear;         x is not
                  • continuous                                                                   • A quantum mechanical operator corresponding to
                  • finite                                                                         a physical state is constructed by writing down the
                  • single-valued                                                                  classical expression in terms of the variables x, p,
            for all values of x.                                                                   t, and E and converting the expression to an
                                                                                                   operator using the following rules.

   Classical Variable        Quantum        Expression for an          Operation
                                                 Operator                                        Eigenvalue Equations- Example 1
             x                   ˆ
                                 x                   x                 multiply by x                                                         ∧
                                                                                                 Is sin(x) an eigenfunction of Ω and if so what is
             px                 ˆ
                                px                  h d            take derivative with
                                                    i dx         respect to x and multiply       it’s eigenvalue?
                                                                        by hbar/i.
             t                   tˆ                  t                 multiply by t             Ω= 2
                                                                                                 ˆ                 Ψ = sin(x)
                                                                                                    dx                               Yes, sin(x) is an
   Ek=p /2m (kinetic            p2
                                ˆ                  h2 d 2        take second derivative
                                                   2m dx2         with respect to x and          ΩΨ = ωΨ
                                                                                                 ˆ                                   eigenfunction of d2/dx2 with
                                                                                                                                     an eigenvalue of ω = -1.
                                                                  multiply by hbar2/2m.

                               ˆ                                                                 d2
    V(x) = potential           V (x)               V(x)                   V(x)
                                                                                                        sin(x) = ω sin(x)
                                                                                                 dx 2
                           ˆ   ˆ    ˆ
                           H = Ek + V (x)       h2 d 2                                                                                                     ωΨ
                                                                                                 d2                d d         d
                                            −           + V(x)
                                                2m dx 2
                                                                                                      2 sin(x) =       sin(x)  =  cos(x) = −sin(x)
                                                                                                 dx                dx  dx      dx

   Eigenvalue Equations-Example 2                                      Eigenvalue Equations-Example 3
Is x2 an eigenfunction of Ω and if so what is it’s eigenvalue?                       h d
                                                                                  ˆ                       Ψ = Ae ikx
                      d2                                                             i dx
                Ω= 2
                 ˆ              Ψ = x2
                     dx                                                           pΨ = p x Ψ
                  ΩΨ = ωΨ

                                                                                  Postulate IV
                       Postulate III                                        (Average Value Theorem)
   • The measurement of a physical observable
     gives a result that is an eigenvalue of the                         The average or expectation value, <Ω>, of
     operator for that observable according to the                       any observable, Ω, which corresponds to an
     eigenvalue equation below:                                          operator, Ω , is calculated from:

                  ΩΨ = ωΨ
                  ˆ              Ω = operator
                                 Ψ= wavefunction,
                                 ω = eigenvalue
                                                                                        < Ω > = ∫ Ψ*ΩΨdx

Eigenvalues are the allowed values of an observed quantity. This
equation assumes that Ψ is normalized.

What if Ψ is a normalized eigenfunction                                                   Commutators
          of the operator, Ω ?                                                    C = Ωβ − βΩ = [ Ω, β]
                                                                                  ˆ ˆˆ ˆˆ         ˆ ˆ
                                                                                 C = 0 for operators that commute.
        ∞                   ∞                       ∞                                                   ˆ x
                                                                         What is the commutator of [d ,ˆ ]?
Ψ = ∫ Ψ*ΩΨdx = ∫ Ψ*ωΨdx = ω ∫ Ψ *Ψdx = ω
                                                                                       d       d
       −∞                  −∞                       −∞                   [d ,ˆ ]f(x)= xf − x f
                                                                          ˆ x
                                                                                      dx      dx
                                                                                      d          d
                                                                                    = (xf) − x f
  Every observation results in the value, ω.                                          dx        dx
                                                                                        df       df
                                                                                    = x +f −x = f         so [d ,ˆ ] = 1
                                                                                                              ˆ x
                                                                                       dx        dx
                                                                   Chain Rule for Differentiation:
                                                                   d          df   dx    df
                                                                      (xf) = x − f    =x +f
                                                                   dx         dx   dx    dx


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