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Recasting a problem in Fourier space Amplitude as a function of time - - PowerPoint PPT Presentation
Recasting a problem in Fourier space Amplitude as a function of time - - PowerPoint PPT Presentation
Recasting a problem in Fourier space Amplitude as a function of time for plucked guitar: Same information is contained in amplitude of frequencies Energy by adding up frequencies/wavelengths Describe energy by where particle is and how fast it is
SLIDE 2
SLIDE 3
Energy by adding up frequencies/wavelengths
Describe energy by where particle is and how fast it is moving
- r by adding up energy in each wavelength given the
distribution of wavelengths
SLIDE 4
2D Fourier transforms of images
SLIDE 5
Expanding wave functions in an HO basis
Single-particle radial wf ψ(r) Expand in harmonic oscillator wfs: ψNmax(r) =
Nmax
- α=0
cαφα(r) Find cαs by diagonalizing HΨ = EΨ
]=
0.5 1.0 1.5 2.0 2.5 r
- 5
5 10 15 V@rD
Eexact'=')1.51'
ψexact(r), ψ0(r), 0.5 ∗ φ0 ψexact(r), ψ0(r), 0.5 ∗ φ0
]=
0.5 1.0 1.5 2.0 2.5 r
- 0.5
0.5 1.0 1.5 2.0 2.5 wf 0.5 1.0 1.5 2.0 2.5 r
- 0.5
0.5 1.0 1.5 2.0 2.5 wf
Nmax = 0, E0 = −1.30 Nmax = 0, E0 = +5.23
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Expanding wave functions in an HO basis
Single-particle radial wf ψ(r) Expand in harmonic oscillator wfs: ψNmax(r) =
Nmax
- α=0
cαφα(r) Find cαs by diagonalizing HΨ = EΨ
]=
0.5 1.0 1.5 2.0 2.5 r
- 5
5 10 15 V@rD
Eexact'=')1.51'
ψexact(r), ψ2(r), 0.5 ∗ φ2 ψexact(r), ψ2(r), 0.5 ∗ φ2
]=
0.5 1.0 1.5 2.0 2.5 r
- 0.5
0.5 1.0 1.5 2.0 2.5 wf 0.5 1.0 1.5 2.0 2.5 r
- 0.5
0.5 1.0 1.5 2.0 2.5 wf
Nmax = 2, E2 = −1.46 Nmax = 2, E2 = −0.87
SLIDE 7
Expanding wave functions in an HO basis
Single-particle radial wf ψ(r) Expand in harmonic oscillator wfs: ψNmax(r) =
Nmax
- α=0
cαφα(r) Find cαs by diagonalizing HΨ = EΨ
]=
0.5 1.0 1.5 2.0 2.5 r
- 5
5 10 15 V@rD
Eexact'=')1.51'
ψexact(r), ψ4(r), 0.2 ∗ φ4 ψexact(r), ψ4(r), 0.2 ∗ φ4
]=
0.5 1.0 1.5 2.0 2.5 r
- 0.5
0.5 1.0 1.5 2.0 2.5 wf 0.5 1.0 1.5 2.0 2.5 r
- 0.5
0.5 1.0 1.5 2.0 2.5 wf
Nmax = 4, E4 = −1.46 Nmax = 4, E4 = −1.04
SLIDE 8
Expanding wave functions in an HO basis
Single-particle radial wf ψ(r) Expand in harmonic oscillator wfs: ψNmax(r) =
Nmax
- α=0
cαφα(r) Find cαs by diagonalizing HΨ = EΨ
]=
0.5 1.0 1.5 2.0 2.5 r
- 5
5 10 15 V@rD
Eexact'=')1.51'
ψexact(r), ψ6(r), 0.2 ∗ φ6 ψexact(r), ψ6(r), 0.2 ∗ φ6
]=
0.5 1.0 1.5 2.0 2.5 r
- 0.5
0.5 1.0 1.5 2.0 2.5 wf 0.5 1.0 1.5 2.0 2.5 r
- 0.5
0.5 1.0 1.5 2.0 2.5 wf
Nmax = 6, E6 = −1.50 Nmax = 6, E6 = −1.40
SLIDE 9
Expanding wave functions in an HO basis
Single-particle radial wf ψ(r) Expand in harmonic oscillator wfs: ψNmax(r) =
Nmax
- α=0
cαφα(r) Find cαs by diagonalizing HΨ = EΨ Extend to many-body system
]=
0.5 1.0 1.5 2.0 2.5 r
- 5
5 10 15 V@rD
Eexact'=')1.51'
ψexact(r), ψ8(r), 0.2 ∗ φ8 ψexact(r), ψ8(r), 0.2 ∗ φ8
]=
0.5 1.0 1.5 2.0 2.5 r
- 0.5
0.5 1.0 1.5 2.0 2.5 wf 0.5 1.0 1.5 2.0 2.5 r
- 0.5