Vorlesung 2. Visualisation of wave functions (April 18, 2008)

Transcription

1 Vorlesung 2. Visualisation of wave functions (April 18, 2008) Introduction nobody can tell us how a quantum-mechanical particle looks like at the same time: 1) many quantum mechanical processes can be described by the Schrödinger equation 2) the solutions are called wave functions (because of the oscillatory behavior in space and time) 3) it is not straightforward to understand and interpret graphically the quantum phenomenon Visualisierung II

2 Mathematically: wave function is a complex-valued function of space and time The goal: associate a unique color to each complex number 1. The two-dimensional manifold of complex numbers z=x+iy, x= Re z, y= Im z Complex numbers z C can be represented by pairs (x,y) of real numbers AND visualized as points in the two-dimensional complex plane.

3 1. The two-dimensional manifold of complex numbers Using polar coordinates (r, ϕ) in the complex plane give polar form of a complex number z=r cosϕ + i r sinϕ, r= z, ϕ = arg z One often adds the complex infinity to the complex numbers

4 1b. The stereographic projection you can interpret the complex plane as the xy-plane in the threedimensional space R 3 1) consider a sphere of radius R in the three-dimensional space R 3 2) draw the straight line which contains the point (x,y,0) and the north pole of the sphere (0,0,R) 3) the stereographic projection of z is the intersection of that line with the surface of the sphere gives a unique point on the sphere for each complex number z

5 1b. The stereographic projection θ = π r 2arctan, r = z R a complex number z is mapped 1 )to the northern hemisphere if r>r 2) to the southern hemisphere if r<r 3) the origin z=0 is mapped onto the south pole of the sphere θ=π 4) every point of the sphere (expect the north pole) is the image of some complex number 5) the north pole θ=0 is image of a new element, complex infinity

6 2. The three-dimensional color manifold RGB color system: color manifold is defined as 3D unit cube. The points in the cube have coordinates (R,G,B) intensities of primary colors The corners (Ecken) (1,0,0) - Red, (0,1,0) Green, (0,0,1) Blue are the basic colors (1,1,0) Yellow, (1,0,1) Magenta, (0,1,1) Cyan are complimentary colors (0,0,0) Black, (1,1,1)-White To visualize the complex number by a color mapping from 2D complex plane into 3D color manifold

7 2. The three-dimensional color manifold HSB and HLS color systems: A measure for the distance between any two colors C1=(R1,G1,B1) and C2=(R2,G2,B2) is given by the metric: d(c1,c2)=max{[r1-r2],[g1-g2],[b1-b2]} The distance of a color C=(R,G,B) from the black origin O=(0,0,0) is called the brightness (Helligkeit) b(c)=d(c,0)=max{r,g,b} The saturation(sättigung) s(c) is the distance of C from the gray point on the main diagonal which has the same brightness s(c)=max{r,g,b}-min{r,g,b}

8 2. The three-dimensional color manifold HSB and HLS color systems: The possible value of brightness b varies between 0 and 1: For each value of b, the saturation varies between 0 and the maximal saturation at brightness b s b (max)=b Set of all colors of RGB with the same saturation and brightness is a closed polygonal curve Γ s,b of length 6s which is formed by edges of a cube with length s The hue (Farbton) h(c) of a point C is λ/6s where λ is the length of part Γ s,b between C and the red corner (the corner of Γ s,b with maximal red component) in the positive direction

9 2. The three-dimensional color manifold HSB and HLS color systems: h=0 and h=1 both give the red corner, h is a cyclic variable Then pure colors (red, yellow, green, cyan, blue, magenta) have h=0,1/6,1/3,1/2, 2/3, 5/6 For any color C(RGB) the lightness(leichtigkeit) l(c) is the average of the maximal and minimal component l(c)=(max{r,g,b}+min{r,g,b,})/2 = b(c)-s(c)/2 We have l between 0 and 1, l=0 black, l=1 white (b=1,s=0) The maximal saturation s l max for a given lightness

10 2. The three-dimensional color manifold HSB and HLS color systems: The maximal saturation s l max for a given lightness s l max = 2l if l < or = ½ s l max = 2(1-l) if l > or = ½ in the HSB color system is characterized by the triple (h,s,b) of hue, saturation, and brightness Color manifold is a cone (Kegel) with vertex at the origin The values (2πh,s,b) are cylindrical coordinates: 1) b corresponds to z 2) s specifies the radial distance from the axis of the cone 3) ϕ=2πh gives the angle

11 2. The three-dimensional color manifold HSB and HLS color systems: The coordinates (h,l,s) determine hue, lightness, and saturation of a color in HLS colorsystem Interpretation: double cone, the position of the color is given by 1) l corresponds height(höhe) 2) s specifies the radial distance from the axis of the cone 3) ϕ=2πh gives the angle

12 3. A color code for complex numbers Finally the mapping from complex plane into the manifold of colors in a unique way 2D complex plane is mapped into the surface of the threedimensional color manifold Each point in the complex plane will receive the color of its stereographic image on the surface of the sphere

13 3.1 Color map of the sphere 1) Every point (θ,ϕ) of the sphere (except for the poles) will be colored with a hue given by ϕ=2π 2) the lightness of the color depends linearly on θ l (θ) = 1- θ/π, 0 θ π 3) one chooses the maximal saturation s(θ) = s max (l (θ)) This defines the homeomorphism (one-to-one mapping) A) the north pole (0 =θ, z = ) is white B) the south pole (π =θ, z =0) is black C) the equator (π/2 =θ, z =R) has lightness ½ (all colors with saturation 1)

14 3.1 Color map of the complex plane Color map of the sphere defines a coloring of the complex plane 1) each complex number is colored with a hue determined by its phase h=ϕ/2π positive real values are red negative real values are in cyan (green-blue) z and z has complimentary hue z=0 (black) and z=infinity (white) additive elementary colors: red, green, blue (0, 2π/3, 4π/3) complimentary colors: yellow, cyan, magenta (π/3, π, 5π/3) imaginary unit i has ϕ= π/2 and h=1/4 (between yellow and green)

15 4 Visualization of the complex valued function 4.1 Complex-valued function in one dimension The simplest quantum system single spin-less particle in one space dimension 1) at each point x a complex number ψ(x) is given. Consider stationary plane wave in one dimension ψ k = exp, ( ikx) x R

16 4 Visualization of the complex valued function 4.1 Complex-valued function in one dimension Several methods of visualization 1) method 1: real and imaginary parts separate plots of the real part and the imaginary part Re Im ψ ψ k k = = cos sin ( kx) ( kx) does not have a lot of physical meaning. It is important to know the absolute value of the wave function

17 4 Visualization of the complex valued function 4.1 Complex-valued function in one dimension Several methods of visualization 1) method 2: plot the graph 1D wave function can be always visualized using three-dimensional plot: For each point x we may plot Re ψ as a y-coordinate and Im ψ as a x-coordinate --- space curve (graph of the function ψ) A) the orthogonal distance of the curve from x-axis will be the absolute value ψ B) plots are sometime difficult to interpret, difficult for higher dimensions

18 4 Visualization of the complex valued function 4.1 Complex-valued function in one dimension Several methods of visualization 1) method 3: use the color code for the phase One plots the absolute value and fills the area between x axis and the graph with a color indicating the complex phase at the point x (one can use simplified color map, absolute value is displayed as height) all colors at maximum saturation and brightness: Hue is given by h=arg(ψ(x))/2π

19 4 Visualization of the complex valued function 4.2 Higher-dimensional wave functions Function of two variables 1) method 4: real and imaginary part: ψ(x,y) is a complex-valued function of two variables then the real valued functions Re ψ(x,y) and Im ψ(x,y) can be visualized as threedimensional surface plots of the real and the imaginary part 2) method 5: plot of vector fields: A complex number z can be interpreted as two-dimensional vector with components (Re z, Im z). Hence the function can be regarded a vector field Disadvantage: method is not able to show a very fine details

20 4 Visualization of the complex valued function 4.2 Higher-dimensional wave functions Function of two variables 1) method 6: use a color map for defining the phase and the height (or absolute value) can be determined by lightness Can be easily generalized for three dimensions use isosurface to represent the absolute value of the function The surface can be colored according to the phase