amplitude as a function of time for plucked guitarsingle-particle radial wf (r) expand in harmonic...
TRANSCRIPT
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 movingor by adding up energy in each wavelength given thedistribution of wavelengths
2D Fourier transforms of images
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
Out[532]=
0.5 1.0 1.5 2.0 2.5r
-5
5
10
15V@rD
Eexact'=')1.51'
ψexact(r), ψ0(r), 0.5 ∗ φ0 ψexact(r), ψ0(r), 0.5 ∗ φ0
Out[476]=
0.5 1.0 1.5 2.0 2.5r
-0.5
0.5
1.0
1.5
2.0
2.5
wf
0.5 1.0 1.5 2.0 2.5r
-0.5
0.5
1.0
1.5
2.0
2.5
wf
Nmax = 0, E0 = −1.30 Nmax = 0, E0 = +5.23
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
Out[532]=
0.5 1.0 1.5 2.0 2.5r
-5
5
10
15V@rD
Eexact'=')1.51'
ψexact(r), ψ2(r), 0.5 ∗ φ2 ψexact(r), ψ2(r), 0.5 ∗ φ2
Out[480]=
0.5 1.0 1.5 2.0 2.5r
-0.5
0.5
1.0
1.5
2.0
2.5
wf
0.5 1.0 1.5 2.0 2.5r
-0.5
0.5
1.0
1.5
2.0
2.5
wf
Nmax = 2, E2 = −1.46 Nmax = 2, E2 = −0.87
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
Out[532]=
0.5 1.0 1.5 2.0 2.5r
-5
5
10
15V@rD
Eexact'=')1.51'
ψexact(r), ψ4(r), 0.2 ∗ φ4 ψexact(r), ψ4(r), 0.2 ∗ φ4
Out[485]=
0.5 1.0 1.5 2.0 2.5r
-0.5
0.5
1.0
1.5
2.0
2.5
wf
0.5 1.0 1.5 2.0 2.5r
-0.5
0.5
1.0
1.5
2.0
2.5
wf
Nmax = 4, E4 = −1.46 Nmax = 4, E4 = −1.04
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
Out[532]=
0.5 1.0 1.5 2.0 2.5r
-5
5
10
15V@rD
Eexact'=')1.51'
ψexact(r), ψ6(r), 0.2 ∗ φ6 ψexact(r), ψ6(r), 0.2 ∗ φ6
Out[489]=
0.5 1.0 1.5 2.0 2.5r
-0.5
0.5
1.0
1.5
2.0
2.5
wf
0.5 1.0 1.5 2.0 2.5r
-0.5
0.5
1.0
1.5
2.0
2.5
wf
Nmax = 6, E6 = −1.50 Nmax = 6, E6 = −1.40
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
Out[532]=
0.5 1.0 1.5 2.0 2.5r
-5
5
10
15V@rD
Eexact'=')1.51'
ψexact(r), ψ8(r), 0.2 ∗ φ8 ψexact(r), ψ8(r), 0.2 ∗ φ8
Out[493]=
0.5 1.0 1.5 2.0 2.5r
-0.5
0.5
1.0
1.5
2.0
2.5
wf
0.5 1.0 1.5 2.0 2.5r
-0.5
0.5
1.0
1.5
2.0
2.5
wf
Nmax = 8, E8 = −1.50 Nmax = 8, E8 = −1.43