PH 401

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PH 401

• Dr. Cecilia Vogel
• Lecture 21

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Review

• Simple Harmonic Oscillator
• Non-stationary states and w0

Outline

• More about operators and states
• Complete orthonormal sets
• Hermitian Operators
• Commutators and Conservation

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Complete Orthonormal Sets

• recall orthonormal
• eigenstates with different eigenvalues
• are orthogonal,
• and normalized (hopefully)
• Also, if you have all of the eigenstates
• they form a complete set
• which means any state can be written as a linear
  combo of them
• also Sngtltn1-operator

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Time Dependence

• Stationary State wavefunctions
• time dependence
• can be written
• exp(operator) means Taylor expansion of
  exponential in powers of operator

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Time Dependence

• Non-stationary State wavefunctions
• time dependence
• can be written
• Justify
• Suppose
• Then

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Conservation of Energy

• What quantities vary with time, what ones are
  constant
• aka conserved?
• We know ltEgt is conserved
• Proof
• ltEgt
• and all the powers of H commute

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Conservation Laws

• What other quantities are conserved?
• Generally, if
• QHHQ
• i.e. if Q and H commute
• then
• and ltQgt is conserved

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Conservation of Momentum

• Under what circumstances is
• ltpgt conserved?
• ltpgt conserved if p and H commute
• Well
• Generally p does not commute with x or any V(x)
• recall xp?px

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Conservation of Momentum

• Under what circumstances is
• ltpgt conserved?
• p and H commute
• only if V(x) is a constant (everywhere, not PCP)
• ltpgt is conserved if V is constant
• Classically
• p is conserved if F0
• F0 if V is constant

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Matrix Elements

• Recall from homework
• this is not generally true of operators
• Consider
• the first one means
• have the operator act on bgt,
• get a different state cgt,
• then do the overlap ltacgt
• the second one means
• have the operator act on agt,
• get a different state dgt,
• then do the overlap ltbdgt

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Hermitian Operators

• In general, ltacgt? ltbdgt
• so
• For some special operators,
• called Hermitian Operators

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Hermitian Operators

• Since
• for Hermitian Operators,
• expectation values of Hermitian operators are
  real
• eigenvalues of Hermitian operators are real
• All observables correspond to Hermitian operators

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Simultaneous Eigenstates

• For a free particle
• you can have a simultaneous eigenstate of H and p
• For other situations weve seen,
• stationary states (eigenstates of H)
• are not eigenstates of p
• When can you have both?

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Simultaneous Eigenstates

• For a free particle
• H and p commute (since V(x)0
• Generally,
• if O and Q commute,
• you can have a complete set of simultaneous
  eigenstates of both O and Q

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Recall 3-D well

• For the 3-D infinite box
• we had simultaneous eigenstates of
• We called them
• The ps commute, so that works

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Recall x and p

• x and p do not commute
• in fact x,p is imaginary
• so there are NO simultaneous eigenstates of x and
  p
• cannot know x and p at same time
• General,
• if o,q is imaginary
• then O and Q have an uncertainty principle

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For Mon

• continue chapter 11 and continue playing with
  operators
• not in text 3-D