Quantum Mechanics

The 2 images on the right hand side of these animations may be a way to visually represent quantum mechanics. The wave function is the superposition of all possible states a system governed by a complex-valued wave function can be in. The story is complete, even if uncertain due to the complex numbers which are not a totally ordered set. The upper right image is literally a superposition of every frame that appears in the animation that is front and center. Below all that is possible is a random sampling of those what is possible. All that can be is contrasted with what happens to be, a core issue in the foundations of quantum mechanics.

But why must this be so? We can write complex valued equations which are accurate representations of measurements we make of Nature, yet those equations blissfully ignore if one collection of events could cause another set of event to occur. The equation in fact describes clusters of events which are independent of each other due to a spacelike separation. Such an equation is valid math. Yet to make it apply to Nature, we need to take the norm of the expression. When we take this step, the resulting expression is not a description of one event that appears after another as happens in classical physics. Instead the norm is the probability that an event will be seen at a location in spacetime.

The Wave Function of a Wave Equation

summary: 

The wave function of a wave looks simple, but simple can be deceptive.

description: 

The wave function of a wave equation in spacetime does not move in a bumpy wave through space. It moves along military straight lines (unless you choose different coordinates, which is perfectly valid). The movement starts with pair creation, an agreed apon parting of ways. Movement is slowest at the maximal separation. There is a rush to collide and destroy. Because the animation going in looks like the one going out, there is time reflection. Because there are always two points dancing toward or away from each other, there is a reflection in space. Much of the mystery in interference experiments of quantum mechanics centers around this symmetric spacetime function.

command: 
q_graph -out amp -dir int10 -box 1.6 -command 't_function -t_function cos -x_function sin -y_function zero -z_function zero -n_steps 199 -pi 4 -n_t_cycles 300 -n_t_step 0 1 1 0 0' -color yellow
math
equation: 

\phi = (cos(\omega t), sin(\omega t), 0, 0)

tags
Physics Tag: 
simple harmonic oscillator
Math Tag: 
sine
cosine
Programming Tag: 
command line quaternions
t_function

The Wave Function of a Wave Equation Shifted

summary: 

A phase shift is added to a wave function, but you cannot tell due to the periodic boundary condition.

description: 

A change in the phase was included. Each event above another event has a different starting time due to the phase shift, but given enough time, all the same locations in spacetime are experienced, so the pattern looks the same.

command: 
q_graph -out amp_shifted -dir int14 -box 1.6 -command 't_function -t_func cos -x_func sin -y_func zero -z_func zero -n_steps 1000 -pi 10 -n_t_cycles 1000 -n_t_step 0.0314 1 1 0 0 | q_add 0 0 -1.5 0 | q_add_n_m 0 0 0.003 0 1000 1000' -color yellow
math
equation: 

\phi = (cos(\omega t + \delta), sin(\omega t + \delta), k \delta, 0) \, \textrm{with} \, \delta: 0 \to 10 \pi

tags
Physics Tag: 
quantum mechanics
wave function
wave equation
Programming Tag: 
command line quaternions
t_function
q_add
q_add_n_m

The Wave Function of a Wave Equation Shifted and Marked

summary: 

Each and every photon is identical to every other, but by coloring in those where t = 0, we can cheat and see the phase.

description: 

Part of the great mystery of quantum mechanics is that all particles are identical. There is no adding a tag or painting one red while the rest are yellow. As a programmer, we can cheat, do something not allowed in Nature, and mark all those where t = 0 in red. The shift is the same as before, but now we can spot its trail.

What is so tricky in quantum mechanics is not the vectors - those we can always point at with our fingers. Instead it is the scalars that provide the challenge, the unpointables. Each scalar is connected to three vectors to make 3 complex numbers, but the scalar can be shared by other events. The scalar become the thread within a pattern of events, and between separate patterns of events. It is wonderfully ironic that the simplest core component can be so confusing by playing many roles.

command: 
q_graph -out amp_shifted_marked -dir int14 -box 1.6 -command 't_function -t_func cos -x_func sin -y_func zero -z_func zero -n_steps 0 -pi 10 -n_t_cycles 1000 -n_t_step 0.0314 1 1 0 0 | q_add 0 0 -1.5 -.1 | q_add_n_m 0 0 0.003 0 1 1000' -color red -command 't_function -t_func cos -x_func sin -y_func zero -z_func zero -n_steps 1000 -pi 10 -n_t_cycles 1000 -n_t_step 0.0314 1 1 0 0 | q_add 0 0 -1.5 0 | q_add_n_m 0 0 0.003 0 1000 1000' -color yellow
math
equation: 

\phi = (cos(\omega t + \delta), sin(\omega t + \delta), k ~ \delta, 0) \, \textrm{with} \, \delta: 0 \to 10 \pi
\textrm{red} = (cos(\delta), sin(\delta), k ~ \delta, 0) \, \textrm{with} \, \delta: 0 \to 10 \pi

tags
Physics Tag: 
quantum mechanics
wave function
wave equation
phase
Math Tag: 
sine
cosine
Programming Tag: 
command line quaternions
t_function
q_add
q_add_n_m

The Wave Function Squared of a Wave Equation with Phase Shifts

summary: 

The product of one wave equation with another whose phase has been shifted results in an interference pattern.

description: 

The product of one wave function with another that has been shifted is animated. The order of this product is done both ways, resulting in the red and blue animations. In the 3 complex planes, we see 3 patterns. The circle in the t-y plane indicates that a wave equation with harmonic boundary conditions is on display. The straight line in the t-z plane shows nothing is happening except the march of time. Really the same happens in the t-x plane, but the change in phase has been included.

command: 
q_graph -out interference_ty_red_blue -dir int14 -box 1.6 -command '221 "t_function -t_function cos -x_function zero -y_function sin -z_function zero -n_step 1799 -pi 124 -n_t_cycles 1000 -n_t_step 0 1 0 1 0 | q_conj" "t_function -t_function cos -x_function zero -y_function sin -z_function zero -n_step 1799 -pi 124 -n_t_cycles 1000 -n_t_step 0.031 1 0 1 0" | q_x | q_add 0 -1.5 0 0 | q_add_n_m 0 0.003 0 0 1800 1000' -color red -command '221 "t_function -t_function cos -x_function zero -y_function sin -z_function zero -n_step 1799 -pi 124 -n_t_cycles 1000 -n_t_step 0.031 1 0 1 0 | q_conj" "t_function -t_function cos -x_function zero -y_function sin -z_function zero -n_step 1799 -pi 124 -n_t_cycles 1000 -n_t_step 0 1 0 1 0" | q_x | q_add 0 -1.5 0 0 | q_add_n_m 0 0.003 0 0 1800 1000' -color blue
math
equation: 

\phi'^* \phi = (cos(\omega t + \delta), 0, -sin(\omega t + \delta), 0)(cos(\omega t), 0, sin(\omega t), 0) + (0, k \delta, 0, 0) \, \textrm{with} \, \delta: 0 \to 10 \pi in red
\phi^* \phi'= (cos(\omega t), 0, -sin(\omega t), 0)(cos(\omega t + \delta), 0,  sin(\omega t + \delta), 0) + (0, k \delta, 0, 0) \, \textrm{with} \, \delta: 0 \to 10 \pi in blue

tags
Physics Tag: 
wave function
wave equation
interference
Math Tag: 
sine
cosine
Programming Tag: 
command line quaternions
t_function
q_add
q_add_n_m

Gamma Matrices

summary: 

The gamma matrices are a tool to systematically look through all possible paths through spacetime given 4 numbers.

description: 

The path in yellow is multiplied on both sides by all 16 combinations of the basis vectors on the left and the right. An Italian physicist named DeLeo figured out how to map the gamma matrices (also referred to as the Dirac matrices) to the triple product ("Quaternions and Special Relativity", J. Math. Phys., 37:6, 2955-2968, 1996). A team in Mexico, José López-Bonilla, L. Rosales-Roldán, and A. Zúñiga-Segundo detailed the process - and made me aware of the connection via email. The gamma matrix machinery can be hard to appreciate, there being all kinds of combinations of matrices and spinors that play roles. With quaternions the story is much more straightforward: multiply on the left and the right by (1, i, j, k).

Let's look at 1 triple product, i (t, x, y, z) k = (z, -y, -x, t). The algebra is simple, but the results are odd. This function swaps the value of time into the z position, and visa versa. The values of x and y trade places and signs. While you and I might like to consider values for time and space to be solid, in relativistic quantum field theory, Nature has a need to take what ever numbers are in the house and systematically shuffle them so that the sum of all possible paths can be calculated. This animation shows four points coming in form four directions, all paths possible with these 4 numbers. The 16 paths can be seen. Now the work done by the 16 Dirac matrices does not seem so utterly abstract.

command: 
q_graph -out gamma -dir gamma -box 2.5 -loop 0 -command 'q_add_n 2.9 3.1 3.2 2.8 -0.006 -0.0059 -0.0061 -0.0062 1000' -color yellow -command 'q_add_n 2.9 3.1 3.2 2.8 -0.006 -0.0059 -0.0061 -0.0062 1000| gamma -almost' -color blue
youtube embed: 
Outside video: 
math
equation: 

( 1 |  i |  j | k) (t,x,y,z) ( 1 |  i |  j | k)

tags
Physics Tag: 
quantum field theory
gamma matrices
Programming Tag: 
command line quaternions
q_add_n
gamma