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Proving the Schwartz Inequality and Heisenberg's Uncertainty Principle

In this post we once again derive the Heisenberg uncertainty principle, but this time we make use of the Schwartz inequality and the posit...

Friday, January 27, 2017

Something vs. Nothing vs. Mixed States

Yes, we could ask, "Why is there something rather then nothing?" However, if we take quantum physics into account, we shouldn't limit ourselves to pondering something or nothing--we also need to ponder mixed states. Before we do, let's work out the probability of there being something or nothing using a kind of information theory. We can let '1' represent something, and '0' represent nothing.

Imagine a coin with a '1' on one side and a '0' on the other. We assume the probability of either side being face up is 0.5. However, the coin might favor one side over the other. How do we fix this? We get a trillion coins where one side might be more or less probable than the other. The probabilities could range from 0 to 1, but the average probability will be at or near 0.5 for each side. So it's safe to assume, if we perform the coin toss many many times (and switch coins), on average, we will get each side 50 percent of the time.

To make our analysis easier, let's assume we are using a coin that has the expected (average) value of 0.5 for each side. Half the time we are going to get a universe with something, and half the time we will get nothing. If we define "nothing" as absolutely nothing, we can improve our odds of getting "something" by adding additional coins. Let "n" in the last two equations below be the number of coins:

Using two coins yields the following results:

The [0 0] above is nothing. Those pairs above it have a '1'--so they are something. By only using a pair of coins we have improved the chance of getting something to 0.75. What happens if we use three coins?

The probability of getting something jumps to 0.875. If we use an infinite number of coins, the probability of getting something is one--a sure thing. The probability of getting nothing is zero.

Thus we have a reason why there is something rather than nothing. But what if what we call "something" is really something else: a mixed state? What exactly is a mixed state? You could think of it as being both something and nothing--an undecided state, a state with both ones and zeros. By contrast, a pure state has all ones or all zeros:

If we dare to ask where something or nothing came from, one answer is a mixed state. The mixed state evolves into a pure state. When we flip a coin, we don't know if it's heads or tails when it is in the air. We can say it's both heads and tails--a mixed sate. When it hits the ground and comes to rest, it's in a pure state.

Another example analogous to a mixed state is a ball sitting on a hill. It can roll down one side or the other. It stays in a mixed state until something disturbs it:

In the above diagram, the ball's state can evolve into the pure state of '1' or '0.'

So instead of imagining a universe with just something or nothing, imagine a universe filled with ones and zeros, and, to make things more interesting, let's add an additional state of '-1.'

Instead of coin tosses we have binary number sequences that continuously change over time (t). The above diagram is an example of a two-digit binary sequence. The diagram below models the evolution of these binary numbers:

The diagram above starts with the pure state of [1 1]. It then moves through mixed states to get to [0 0], then moves through more mixed states to get to [-1 -1], and so on. Basically we have something that resembles a sine wave. There is a problem, however: it is too predictable for our probabilistic universe. The evolution of states should be random:

The diagrams above are more realistic. The different states should have varied time duration(t00,t11,etc.). If, say, state [00] jumps to [11] without passing through [01] and [10], We say that these mixed states each have a zero duration. It is also possible for any state to have an exceptionally long duration.

So how do we get the predictable sine wave we started with? We take a trillion random waves and calculate the average time duration for each state. We can then add those average times together to get the total time and average wavelength:

Our predictable binary sine wave is really the expectation value of a bunch of random binary waves:

Now, our binary sine wave isn't very smooth nor continuous. It has a clunky, stair-step quality. We can make it smoother if we add more digits to the states:

In the diagram above, notice the wave with only one digit is the clunkiest. The smooth wave at the bottom has an infinite number of digits or at least a really big number of digits. So let's take the bottom wave and make it our official sine wave:

Sooner or later, our sine wave will interact with another wave (superposition). At one extreme the phase difference between the waves could be 180 degrees, resulting in destructive interference:

At the other extreme the phase difference could be zero degrees, resulting in constructive interference:

If we have a bunch of waves, the average or median phase between each pair will be 90 degrees. Thus, we end up with both a sine and cosine wave.

Having sine and cosine waves lead to wave functions and matter--the stuff we call "something."

However, as we have witnessed, the stuff we call "something" is really a mixture of both something and nothing: zeros and ones. The stuff we call "something" also oscillates between pure and mixed states.

You can find more on this topic if you click here.

Sunday, January 22, 2017

Quantizing Photons and Gravitational Lensing

According to General Relativity theory, curved spacetime causes the mass-less photon to follow a curved geodesic path when it is near a planet, star or black hole. Newton's equation, F = GMm/r^2 seems limited to masses (M and m). The way Einstein figured it, if particles follow a geodesic curve due to warped spacetime, then mass (or the lack thereof) doesn't matter. Gravity attracts photons as well as massive objects. But what exactly is curved spacetime at the quantum scale and how does it move photons around?

The diagram below depicts a photon we will call "a." It's entering the gravitational field of a planet surrounded by circular field lines. The red circles have less energy than the blue circles. You could say the field lines are blue-shifted towards the planet's surface and red-shifted away from it.

Photon "a" will pick up energy, increase its wave frequency if it moves toward the surface, and will lose energy if it goes up and away. Since we are covering quantum physics, let's assume photon "a" can go in any direction it wants within the gravitational field:

Photon a's position (y) is very uncertain. Uncertainty Principle to the rescue! Equation 1) below shows that if energy (E) increases, the position of "a" becomes more certain. Thus, a's position will most likely be where there is the most energy, and more energy is found if the photon heads down.

Equations 2) and 3) show the relationship between energy (E), the wave number (k) and Einstein's field equations. At the quantum scale, curved spacetime is wave numbers that become larger when a photon moves down and smaller when a photon moves up. Equation 4) enables us to determine the probability that photon "a" will be at position "y." As we hinted at above, the probability is large where energy (E) is large. That seems reasonable when you consider it takes energy to make a photon.

Even though photon a's position is uncertain, let's imagine it is somewhere just outside the planet's gravitational field. We plot its probability distribution below. It makes a nice bell curve. We could even think of the photon as a distribution of energy.

The expectation value is at the bell curve's center. This is where there is the most energy; this is where we will most likely find "a." Let's add some energy to the right side of the bell curve:

The energy distribution is now changed. This creates a new expectation value for "a.".

Photon "a" has shifted to the right, or, more precisely, photon "a" can more likely be found to the right of its previous most-likely position. If we turn the graph on its side, we get a picture of "a" falling towards the energy field.

But it's not just the single, classical-looking photon "a"--the entire energy and probability distribution of "a" is falling towards the energy field.

A new bell curve is formed, but it is only temporary because "a" has moved down into a greater energy field, so the bell curve must shift downward again.

And again. See a pattern yet?

If photon "a" had enough momentum it could have whizzed right passed the planet along a geodesic curve:

We can use a's bell curve to model the geodesic at the quantum scale:

This was part six of the gravity series. To read the other parts, click here, here, here, here, and here.

Thursday, January 19, 2017

The Quantum Differences Between Gravity and Electromagnetism

Welcome to part five of the gravity series. To read the other parts, click here and here. Are gravity and electromagnetism (EM) the same? There are some who think they are the same force. There may have been a time in the early universe when all the forces were one force, and, as time passed, the one force evolved into the ones we are familiar with. If gravity and EM are the same, they have some distinct differences we will examine at the quantum and cosmic levels.

In previous posts we discovered that gravity is a field with ever shortening spacetime wavelengths, or, increasing energy as a falling body moves closer to where there is greater mass or energy density. In addition, particle-waves' wave numbers increase. These wave numbers have the same units as curved spacetime. Gravitational acceleration is due to the difference in energy or wave number between an upper surface layer of spacetime and a lower one.

Imagine a particle falling in a gravitational field. We can model this using Schrodinger's equation with a twist: we take the difference between the Hamiltonian in the upper layer and the Hamiltonian in the lower layer:

We want to find the change in wave number (k):

We perform a couple of more steps to get the particle's change in kinetic energy:

If we take the wave number (k) and multiply it by Planck's constant (h-bar) and divide it by the particle's mass, we get the particle's change in velocity (v).

At equation 12) notice that the first term has momentum (p). Here are two kinds of momentum: mv (mass X velocity) and fh/c (frequency X Planck's constant/light speed). Velocity (v) could be a function of one or both of these momenta--or we could have two types of velocity: equations 13 and 14 below are derived from the first term of equation 12).

Equation 13) isn't fully simplified. We want to emphasize that the particle's mass cancels itself. The velocity is the same whether the mass is big or small. We can label this velocity the change in velocity due to gravity (Vg). Take note that momentum is not conserved: a big falling mass has more momentum than a smaller falling mass.

Equation 14) tells a different tale. A change in mass does change the velocity (Ve). Momentum is conserved. This equation fits the EM force.

Equations 13) and 14) reveal that when mass (m) is very large, gravity's influence stays the same; whereas, EM's impact diminishes. When mass (m) is small, EM becomes the dominant force--gravity becomes less significant.

Another key difference between gravity and EM is EM is a function of charge; whereas, gravity is a function of mass or energy density. Gravity attracts but EM obeys Column's law (like charges repel, opposite charges attract).

The crude diagrams below demonstrate EM interactions. When the field arrows are pointing towards each other, particles A and B push apart. When the arrows point away from each other, A and B separate. Particles A and B attract each other when the field lines (arrows) flow in the same direction, as if B is flowing towards A.

The following diagrams show how A and B share gravitational field lines. This sharing causes the energy field between A and B to become more intense than the fields at the far right and left of A and B. A and B want to move away from each other and move closer together. The shared energy between them makes the latter more probable. To see how this works in more detail, click here. As the distance between A and B decreases, the shared energy between them becomes stronger and the gravitational acceleration increases (the inverse square law).

One thing EM and gravity have in common are mass-less bosons. Since they are mass-less, these bosons have unlimited range and they travel at light speed. This raises a troublesome paradox, for we know EM is much much stronger than gravity. How can atoms and molecules ever get together via gravity when their outer-shell electrons have a repulsive force far greater than gravity's attractive force?

Consider two hydrogen atoms that are close together. We fully expect the electrons to repel each other, same goes for the protons. But could the proton in one atom be attracted to the electron in the other? If so, that attraction could, to some extent, cancel the repulsive force. We can use the following equations to see if gravity is stronger or weaker than the net EM repulsive force.

When we crunch the numbers we find that as distance (d) between the atoms increases, the EM force (Fe) drops more quickly than gravity (Fg). In the diagram below, where the atoms are close together, the distance between the electrons is very short compared to the distances between opposite charges, so the repulsive force is strong. This is good news! It means the atoms will remain distinct and separate. It means the ground will be solid beneath your feet.

By contrast, when the atoms are far apart, the different distances between opposites and same-charge particles are less dramatic. The charges cancel each other and gravity dominates.

Monday, January 16, 2017

Quantizing Gravity: Why Different Masses Fall at the Same Rate

This is part four of the gravity series. Click here to find parts 1 through 3. In this part we show how quantum gravity explains why particles that have different masses (or wave frequencies) fall down (not up) at the same acceleration rate when they're in a gravitational field.

Why do different masses fall at the same rate? It's as if gravity knows the mass of each falling object and adjusts its force to overcome each object's inertia so acceleration (g) is nearly constant. Believe it or not, we can derive the answer to this puzzling phenomenon from Heisenberg's Uncertainty Principle:

Variable p is momentum, and normally x is position, but for our purposes x shall represent a particle wavelength. Of course h-bar is good old Planck's constant. In equation 4) we end up with k--the wave number. In addition to providing the number of waves in a particle wave, the wave number has the same units as curved spacetime, so do the other terms in equation 4). Below one or more of the terms are set equal to Einstein's tensor (G) and his field equations.

If you are wondering how curved spacetime moves matter, the far left side of equation 4 provides a clue. It contains momentum (p). Momentum is all about movement, so any particle that interacts with something with momentum is bound to be set in motion. But why should the motion be necessarily down? We'll cover that later; let's finish what we started:

If velocity (v) is held constant, equations 10) and 11) show that when the mass (m) of a particle increases, the wavelength (x) decreases, and vice versa. This anti-correlation between mass and wavelength may explain why different-massed objects fall at the same rate. Equation 12) shows how a photon's velocity (c) is maintained: the higher the frequency (f), the shorter the wavelength (x), and vice versa. So if you take any two particles (A and B), they will have the same velocity from point to point because the A-mass times the A-wavelength equals the B-mass times the B-wavelength.

They also have the same acceleration (g). This is due to decreasing spacetime and particle wavelengths (increasing wave numbers (k)) as particle A (see diagram 1 below) moves further into the field where there is more energy density.

Equations 13) and 14) above and 15) through 20) below show how the wave number (k) is derived and how it relates to Schrodinger's and Einstein's equations:

Now let's deal with the problem of motion. Which way does particle A go when it encounters gravity. Why should it go down and not up or sideways? Since we are covering quantum mechanics, it stands to reason that particle A's motion is probabilistic--it can go in an infinite number of directions.

The probability of A going in any one direction is 1/infinity--virtually zero. Predicting A's direction seems hopeless, but there is a way to simplify the problem: we divide the possible directions into four sectors--each with a .25 probability density.

We can also add up all the vectors in each sector and find the mean or median; i.e., the expectation value:

Now we add a gravitational field. Diagram 5 has a blue section (blue shift) and a red section (red shift) to model an increase in energy from top to bottom. We expect particle A to move further and faster in the field's bottom portion than the top portion.

It looks like we've simplified the problem to the point where we can model what particle A is most likely to do when it moves through a gravitational field.

In diagram 6 above, particle A starts at the center and goes either up or down at each node. If it heads up, its distance and velocity shrinks due to decreasing field energy. If it goes down its distance and velocity grow due to increasing field energy. The vertical lines in the middle represent the average ups and downs. Note that the average down exceeds the average up, so particle A has a downward bias. Particle A has also added to its net downward velocity--it has accelerated downward or decelerated upwards depending on the direction of its initial velocity. Each time particle A reaches a new field level, on average, it repeats diagram 6 and adds more to its down velocity. Thus, we have gravity.

Update: The diagrams below show how different masses fall at the same rate in a gravitational field:

As you can see, each particle within each atom interacts with gravity separately (i.e., has its own squiggly field line), so gravity does not distinguish between different masses. The more massive particles have more field lines per time (t) which is equivalent to higher frequency and shorter wavelengths--so big mass times short wavelength falls at the same rate as small mass times long wavelength.

Update: The following diagram summarizes how mass and mass-less particles behave in a gravitational field: