PH 401

1
PH 401

• Dr. Cecilia Vogel
• Lecture 4

2
Review

• Requirements on wavefunctions
• Free Particle

Outline

• Free Particle
• Fourier Synthesis and Analysis
• group velocity
• Momentum amplitude

3
Free Particle Simple Solution

• Problem 1
• Probability density
• This particle is as likely to be 20 light-years
  away as it is to be here.
• This wavefunction is not normalizable.
• Note often used anyway as an approximation for
  beams that have large extent.

4
Free Particle Simple Solution

• Problem 2
• phase velocity
• why is p2mv instead of pmv??

5
Wavepackets

• Solve the localization/normalization problem by
  creating a wavepacket.
• wavepacket localized wavefunction
• made by combining the simple wavefunctions with
  different ks (and ws)
• This mathematical process is called Fourier
  synthesis.
• Any well-behaved function can be synthesized in
  this way

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Fourier Amplitude

• A(k) is the amplitude of each k in the combo
• depends on k,
• independent of x,
• constant in time for free particle.
• For some functions A(k)
• Y(x,t) is localized
• and normalizable.

7
Fourier Synthesis

• Given A(k)
• know Y(x,t)
• at all time and position
• Knowing Y(x,t)
• all info that you can know is known.

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Fourier Analysis

• Given Y(x,0)
• you can mathematically analyze the function to
  determine A(k).
• After determining A(k),
• you can then determine Y(x,t)
• at all x and t.
• From initial state, can predict state at later
  times.

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Aside Notes

• Here we took simple TDSE solutions for free
  particle and combined them.
• Always true that if you find a set of solutions
  to TDSE, any linear combo will be a solution,
  too.
• Each partial wave moves at different speed
• Part with wavenumber k moves at speed

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Group Velocity

• If each partial wave moves at different speed
• what is the speed of the wavepacket?
• The speed of the peak of the wavepacket is the
  group velocity

but

• So group velocity
• and p mv !!!
• group vel is classical particle vel

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Interpret Fourier Amplitude

• If A(k) is the amplitude of each k in the combo
• then this function tells us the amplitude for
  each momentum, F(p).
• since p?k
• If we know the amplitude of each p,
• wwe know the probability density for that momentum

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Momentum Probability

• To be continued.

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For Wednesday

• chapter 5.1-5.4, 5.6