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Review

Review. Free Particle Fourier Synthesis and Analysis group velocity. Momentum Representation free particle time dependence Gaussian wavefunctions. Outline. Representation. The wavefunction contains all the info available about the state of the particle.

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Review

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  1. Review • Free Particle • Fourier Synthesis and Analysis • group velocity • Momentum Representation • free particle time dependence • Gaussian wavefunctions Outline

  2. Representation • The wavefunction contains all the info available about the state of the particle. • momentum info can be found by Fourier analysis. • The momentum amplitude ALSO contains all the info available about the state of the particle. • position info can be found by Fourier synthesis. • Each is just a different representation of the particle’s state.

  3. Momentum Probability • F(p) is found from A(k) by replacing k with k=p/, then dividing the whole function by √ • in order for A(k) and F(p) both to be normalized. • Technically F(p) = F(p,t=0)e-iwt • but for free particle, the momentum probability density is constant

  4. x and p Probability • The momentum probability density is constant (for free particle) • An object in motion remains in uniform motion (constant momentum) • But the position probability density is not!

  5. Probability and time • The position probability density is time-dependent

  6. Gaussian Wavefunctions • Often a Gaussian (aka “normal,” aka “bell-curve”) function is a good approximation for the wavefunction • more mouthwatering mathematics.… • eqn 4.54, 4.55 • The expectation value of x • <x> = x0 • The uncertainty in x • Dx = L

  7. Gaussian Wavefunctions • If the wavefunction is Gaussian, the momentum amplitude will be, too • more mouthwatering mathematics.… • eqn 4.58 and question 4-12 • The expectation value of p • <p> = p0 • The uncertainty in p • Dp = /2L

  8. Uncertainty Principle • At t=0, • DxDp = hbar/2 • Gaussian is “minimum uncertainty wavefunction” • look at graphs • As we will soon see, for t>0 • DxDp > hbar/2

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