Without the mathematical construct, there is no Big Bang theory. It's not possible to decouple the two in any meaningful way.

The language of physics is mathematics. While it's easy to lose sight of this fact for phenomena that happen at ordinary human scales, where we've developed an intuition that corresponds pretty well to the math, the situation is entirely different at scales far outside of our daily experience. Here, intuition utterly fails us and the only thing we can rely on is the math. This is true of both the quantum scale and the cosmological scale, and it is certainly true for the birth of the universe!

It is what it is. The universe's complexity is such that, without the equations, practical explanations are -- at best -- meaningless. More often, they're just simply wrong.

I'm not debating that mathematics is essential to understanding physics. I am debating that math is being overused by the academic industrial complex to add unnecessary complexity to physical problems. Statements like "Without the mathematical construct, there is no Big Bang theory. It's not possible to decouple the two in any meaningful way." sound like an academic religion more than a method of practical reasoning. I get the feeling that many scientists and engineers are more like "theoretical carpenters" who just expound and pontificate rather than come up with a practical explanation for apparently complex phenomenon.

 

I'm not debating that mathematics is essential to understanding physics. I am debating that math is being overused by the academic industrial complex to add unnecessary complexity to physical problems. Statements like "Without the mathematical construct, there is no Big Bang theory. It's not possible to decouple the two in any meaningful way." sound like an academic religion more than a method of practical reasoning. I get the feeling that many scientists and engineers are more like "theoretical carpenters" who just expound and pontificate rather than come up with a practical explanation for apparently complex phenomenon.

Where are these practical explanations supposed to come from? Any practical explanation of an electron that doesn't use the language of quantum mechanics is guaranteed to be both wrong and grossly misleading, because a century of experiments has shown conclusively that electrons are not classical objects. Yet, we have a hundred million years of evolution in a classical environment that's left us with an intuition only for classical notions. So, to avoid the math, pop science books resort to nonsensical phrases such as "wave-particle duality", explaining that sometimes an electron behaves as a wave, and sometimes as a particle. What kind of explanation is that?

When I first read about quantum physics (through pop sci books), I thought it was complete hogwash; no wonder Einstein hated it! Only much later, when I had learned the requisite language, could I understand that classical notions such as position and momentum are mathematical constructs that simply do not apply at the quantum scale. In the quantum world, position and momentum are the eigenvalues of noncommuting unitary operators acting over a Hilbert space of state vectors. While this may initially seem more esoteric than classical position and momentum, both concepts are nothing more than mathematical constructs used to make predictions; we just happen to be more familiar with the classical version. But once you become comfortable with the quantum version, once you can relate the electron to a superposition of quantum states evolving according to the Schrodinger equation, things like the double-slit experiment make perfect sense. More to the point, the ridiculousness and fruitlessness of practical explanations such as "wave-particle duality" becomes clear.

In short, "practical reasoning" stopped being a viable option for physics roughly 200 years ago, when we started exploring impractical things, like electricity, magnetism, energy, and heat. Even before then, Newton had to invent a new and controversial branch of mathematics to show that the cannonball and the Moon obeyed the same laws of motion. As the physics has gotten harder, so has the math. But physicists are not part of some grand conspiracy to make physics unnecessarily complicated. Quite the opposite, the theoretical carpenters spend their days trying to simplify physics. The hope is that the enormous variety of physical processes -- from the dynamics of a spinning top to stellar nucleosynthesis -- can be entirely described by a set of equations small enough to fit on a tee shirt. If that day every comes, pop sci books will provide plenty of practical explanations that will utterly fail to describe what the equations mean.

 

Regarding the definition of wave particle duality, the phenomenon is often misstated as "Sometimes an electron behaves as a wave, and sometimes as a particle". However a better statement would be called the "Wave/particle commonality" and the common features of partilces and waves would be listed as follows:

1. All particles (that consist of matter and have rest mass) move as a wave propagation similar to light.
2. Light carries momentum like a particle.
3. Particle waves and light are both subject to the law of relativity.
4. Particles and light are both subject to the attraction of gravity and the path of their propagation is changed by gravitational field.
5. Particles can be converted to light and light can be converted to particles.

These statements are being refined in my endeavor to write a manual for a high school class on introductory electronics.

 

Regarding the definition of wave particle duality, the phenomenon is often misstated as "Sometimes an electron behaves as a wave, and sometimes as a particle". However a better statement would be called the "Wave/particle commonality" and the common features of partilces and waves would be listed as follows:

1. All particles (that consist of matter and have rest mass) move as a wave propagation similar to light.
2. Light carries momentum like a particle.
3. Particle waves and light are both subject to the law of relativity.
4. Particles and light are both subject to the attraction of gravity and the path of their propagation is changed by gravitational field.
5. Particles can be converted to light and light can be converted to particles.

These statements are being refined in my endeavor to write a manual for a high school class on introductory electronics.

I don't know your curriculum or what your teaching goals are, but I like that you're trying to bring a sense of the wonder of modern physics into the classroom. But with all due respect, I believe the statements above won't do anything more than confuse the kids. Are you also going to explain to them rest mass, momentum, special relativity, gravitational fields, and general relativity? A typical high school student has a sense for what a particle is, and what waves are; all the other stuff will be nebulous, at best.

For what it's worth, here's how I would frame the issue:
---
At the scales we're used to dealing with, there's an obvious difference between particles and waves. Particles are localized in space. We can visualize a particle as a tiny marble, and do physics on it just as if it were a regular marble, which has a definite position and velocity. Waves, on the other hand, are of a different nature: they are not localized in space. Picture a surfer riding a wave -- at any point in time, the surfer has a definite position (like a marble), but the wave is spread out. If we want to do physics on the wave, we need to know other parameters about the wave, like wavelength and amplitude (height), which marbles don't have.

At our scales, we call these two types of being classical particles and classical waves. But early in the 20th century, physicists discovered something radically shocking when they started exploring the fundamental nature of light and electricity. At very tiny scales -- the quantum scale -- our familiar ideas of classical particle and classical wave do not apply. Although we call the electron a particle, and draw it as a tiny marble in our chemistry books, electrons do not behave like tiny marbles. If we want to do physics on electrons, we cannot treat them as particles, with a well-defined position and velocity at every point in time. The same is true of photons, the "particles" of light.

So, if electrons and photons are not particles, are they waves? Yes, but not in the classical sense. Classical waves need a medium through which to propagate -- ocean waves need water, sound waves needs air. Furthermore, the height of the wave is a continuous function of time: classical waves develop smoothly. On the other hand, quantum waves need no medium, and their "heights" are not smoothly defined, they come in discrete jumps. If you find this difficult to visualize, you're not alone! The universe behaves so differently from our expectations at the quantum scale that our everyday language and intuition can't help us. If you're curious about this strange and fascinating world, quantum mechanics provides the appropriate concepts and language to help us think about such things.
---

Alternatively, you could just focus on the Heisenberg uncertainty principle, which has some mystique and most kids have probably heard of. Skipping all the math and doing it visually with plots, you could show how the closer we zoom in on a waveform in time, the less we know about its frequency (and vice-versa). This could be related to any conjugate pairs, such as the photon's frequency/energy, or the electron's position/momentum, and it provides a nice opportunity to mention the Fourier transform.

 

I believe in stuff. I think the entire physical universe is made out of stuff. Real Tangible Stuff. I also believe that on an absolute fundamental level..........it is impossible to determine what this stuff is.

That's because the only physical entity there is ....is this stuff. There is nothing else to compare it to. So we have to compare it to itself.

This stuff has some interesting properties. It always has the same amount of stuff. It is just not the only entity.....all the entities are the same amount of this stuff. No more and no less.....AND the amount of stuff remains constant. One can not add any or subtract any of this stuff. This is more constant than c.

Next....this stuff always has two accelerations on it. This is because all this stuff is self repulsive.
This repulsion.....causes an outward acceleration of the stuff....trying to get away from itself. This common direction(origin) acceleration, causes a common perpendicular acceleration that grows with the outward acceleration. At a certain outward velocity.....the perpendicular acceleration will equal the outward acceleration......confining the outward acceleration ....into a constant area spin acceleration.

And this stuff comes in two flavors. Half is right handed and half is left handed. When the perpendicular confining force is applied to the stuff(due to outward acceleration)........half of all stuff will rotate to the right as it turns, and the other half of all stuff will rotate to the left as it turns in confinement. Like the left handed or right handed stripe, on a hula hoop. The hula hoops rotate in the same direction......but the rotational direction of the stripe is the handedness. A rotation within a rotation. Two accelerations from the repulsive powering force. A miniature super nova held and suspended within 2 spins. Or call it particle angular momentum.

If you take a super thin and elastic conductor......you will observe a right handed coil will contract.....and a left handed coil will expand when you put current thru it.

When the stuff rotates.....the right handed stuff will generate a parallel perpendicular force....this will contract the stuff. The stuff now has more density.....even tho it has no more stuff.

The lefthanded stuff generates an anti parallel perpendicular force.....causing expansion and less density....even tho it has the same amount of stuff.

Half of all stuff is trying to load up with density....the other half is trying to remain empty of it.

Charge is not a property of mass........mass is just the density of charge. Almost all of the mass is right handed angular momentum. Wonder if that makes a difference to anything?

Simple stuff.

 

I don't know your curriculum or what your teaching goals are, but I like that you're trying to bring a sense of the wonder of modern physics into the classroom. But with all due respect, I believe the statements above won't do anything more than confuse the kids. Are you also going to explain to them rest mass, momentum, special relativity, gravitational fields, and general relativity? A typical high school student has a sense for what a particle is, and what waves are; all the other stuff will be nebulous, at best.

For what it's worth, here's how I would frame the issue:
---
At the scales we're used to dealing with, there's an obvious difference between particles and waves. Particles are localized in space. We can visualize a particle as a tiny marble, and do physics on it just as if it were a regular marble, which has a definite position and velocity. Waves, on the other hand, are of a different nature: they are not localized in space. Picture a surfer riding a wave -- at any point in time, the surfer has a definite position (like a marble), but the wave is spread out. If we want to do physics on the wave, we need to know other parameters about the wave, like wavelength and amplitude (height), which marbles don't have.

At our scales, we call these two types of being classical particles and classical waves. But early in the 20th century, physicists discovered something radically shocking when they started exploring the fundamental nature of light and electricity. At very tiny scales -- the quantum scale -- our familiar ideas of classical particle and classical wave do not apply. Although we call the electron a particle, and draw it as a tiny marble in our chemistry books, electrons do not behave like tiny marbles. If we want to do physics on electrons, we cannot treat them as particles, with a well-defined position and velocity at every point in time. The same is true of photons, the "particles" of light.

So, if electrons and photons are not particles, are they waves? Yes, but not in the classical sense. Classical waves need a medium through which to propagate -- ocean waves need water, sound waves needs air. Furthermore, the height of the wave is a continuous function of time: classical waves develop smoothly. On the other hand, quantum waves need no medium, and their "heights" are not smoothly defined, they come in discrete jumps. If you find this difficult to visualize, you're not alone! The universe behaves so differently from our expectations at the quantum scale that our everyday language and intuition can't help us. If you're curious about this strange and fascinating world, quantum mechanics provides the appropriate concepts and language to help us think about such things.
---

Alternatively, you could just focus on the Heisenberg uncertainty principle, which has some mystique and most kids have probably heard of. Skipping all the math and doing it visually with plots, you could show how the closer we zoom in on a waveform in time, the less we know about its frequency (and vice-versa). This could be related to any conjugate pairs, such as the photon's frequency/energy, or the electron's position/momentum, and it provides a nice opportunity to mention the Fourier transform.

I'm trying to find an article that describes what exactly a particle wave is and it's mathematical representation. Seems it would be represented by some form of a sinusoidal function (such as A Sin wt) where A is the maximum value of the wave. My understanding is that a particle wave is represented by a time varying scalar quantity (just the variation of the sheer magnitude of some variable) rather than a transverse or longitudinal displacement of the material.

A simple example of a scalar wave would be to represent how the funds in a bank account vary with time over a one month period. Let's assume the funds can vary in a sinusoidal manner and your account would be represented by A Cos wt where A is the maximum amount of your money (expressed in $$$) deposited when you get a pay check at the beginning and end of the month.

Then t could represent the small time increments (such as seconds) from 0 to 30 days. The lowest point in the wave would be when you're flat broke and you just got the check in your hand, but it still hasn't been deposited.

 

I'm trying to find an article that describes what exactly a particle wave is and it's mathematical representation. Seems it would be represented by some form of a sinusoidal function (such as A Sin wt) where A is the maximum value of the wave. My understanding is that a particle wave is represented by a time varying scalar quantity (just the variation of the sheer magnitude of some variable) rather than a transverse or longitudinal displacement of the material.

It's critically important to realize that there is no material or medium through which "quantum waves" propagate. In fact, trying to visualize a physical wave will only add to the confusion because such waves describe the probability amplitudes of finding the object in various quantum states. As Feynman said, there's no use in trying to relate quantum mechanics to stuff we're familiar with; the only path to understanding is to start by just accepting the mathematics as is. Fortunately, the math of quantum mechanics is straightforward and simple. (Quantum field theories, on the other hand... )

So, what exactly do we mean by quantum waves? A reasonably complete answer would amount to an introductory course on QM, but the short of it is that wave functions are solutions to the differential equation (the Schrodinger equation) that describes the dynamics of a quantum object.

Perhaps it would help to see the math in action. Suppose that \({\small \psi}\) represents the quantum state of an electron. The Schrodinger equation for the energy E of that electron is given by

\(\hat{H}\psi = E\psi\)​

where H is the Hamiltonian operator (from classical mechanics). In other words, E is the value of energy we would measure if we measured the electron while it was in state \({\small \psi}\). To keep the math simple, let's suppose that the electron is confined to a line of width L. A solution of the equation is then a stationary sinusoid:

\(\psi(x) = sin(\frac{n \pi x}{L})\)​

with n = 1, 2, 3, .... If we wanted to give \({\small \psi(x)}\) a physical interpretation, we could say that it expresses all the possible standing waves that could fit on a line of length L. Keep in mind, however, that \({\small \psi(x)}\) represents a set of quantum states, not a physical waveform.

The one-dimensional Hamiltonian is given by

\(\hat{H}\psi = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2}\)​

where the 1/2m factor corresponds to classical kinetic energy. Carrying out the partial derivatives, we have

\(\hat{H}\psi = \frac{\hbar^2 n^2 \pi^2}{2 m L^2}sin(\frac{n \pi x}{L})\)​

Thus, from the original Schrodinger equation, we get

\(\frac{\hbar^2 n^2 \pi^2}{2 m L^2}sin(\frac{n \pi x}{L}) = E sin(\frac{n \pi x}{L})\)​

and so

\(E = \frac{\hbar^2 n^2 \pi^2}{2 m L^2}\)​

In other words, the electron's measured energy can never be zero, and it will always by a multiple of n; this is why we say that the electron's energy is quantized. More to our point, we see that the only physical correspondence the wave function \({\small \psi}\) has is to the measured values of energy E. The sine wave used in the calculation was not an actual waveform physically propagating through some medium, rather it mathematically describes the modes of a quantum oscillation in the form of state vectors.

Hope this helps more than it confuses.

 

Hello,

According to S. Hawking, his explanation for the big bang is that it came into existence out of NOTHING. That means there was absolutely nothing there beforehand, yet it 'popped' out of nothing.
The reason he said this was possible was because QM shows that particles can pop in and out of existence without any known reason for them doing so. So it is by pure chance that this happens, and there does not need to be anything there to begin with for the particle to suddenly appear for a reason unknown to us.

Now as i have said before, chance is only chance to someone or some thing that has not yet seen the outcome. This leaves the door open for other types of 'beings' to be able to see the outcome before we do, and even know how and why it happened in the first place.
To illustrate, you roll a single die and hide the die from your opponent so they can not see it but you can. Before you rolled the die it was pure chance to you that it came up as 5, but your opponent still doesnt know it, so it's still pure chance to them. They still have to guess what it is so the chance of them guessing right is still 1/6 although your 'chance' is a perfect 1/1 because you already know the outcome.
Same with computer memory. If you cant tell when a radio station actually transmitted their audio signal then it can be processed over and over again in memory and played back to you an hour later. You would not be able to tell that it was delayed if it had been delayed ever since 1 hour before your birth. You would hear it as just another radio station just like all the other non delayed ones. There could be people though that could tell you every song that was going to come up for play before they played if they were there monitoring the delay system.

 

It's critically important to realize that there is no material or medium through which "quantum waves" propagate. In fact, trying to visualize a physical wave will only add to the confusion because such waves describe the probability amplitudes of finding the object in various quantum states. As Feynman said, there's no use in trying to relate quantum mechanics to stuff we're familiar with; the only path to understanding is to start by just accepting the mathematics as is. Fortunately, the math of quantum mechanics is straightforward and simple. (Quantum field theories, on the other hand... )

So, what exactly do we mean by quantum waves? A reasonably complete answer would amount to an introductory course on QM, but the short of it is that wave functions are solutions to the differential equation (the Schrodinger equation) that describes the dynamics of a quantum object.

Perhaps it would help to see the math in action. Suppose that \({\small \psi}\) represents the quantum state of an electron. The Schrodinger equation for the energy E of that electron is given by

\(\hat{H}\psi = E\psi\)​

where H is the Hamiltonian operator (from classical mechanics). In other words, E is the value of energy we would measure if we measured the electron while it was in state \({\small \psi}\). To keep the math simple, let's suppose that the electron is confined to a line of width L. A solution of the equation is then a stationary sinusoid:

\(\psi(x) = sin(\frac{n \pi x}{L})\)​

with n = 1, 2, 3, .... If we wanted to give \({\small \psi(x)}\) a physical interpretation, we could say that it expresses all the possible standing waves that could fit on a line of length L. Keep in mind, however, that \({\small \psi(x)}\) represents a set of quantum states, not a physical waveform.

The one-dimensional Hamiltonian is given by

\(\hat{H}\psi = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2}\)​

where the 1/2m factor corresponds to classical kinetic energy. Carrying out the partial derivatives, we have

\(\hat{H}\psi = \frac{\hbar^2 n^2 \pi^2}{2 m L^2}sin(\frac{n \pi x}{L})\)​

Thus, from the original Schrodinger equation, we get

\(\frac{\hbar^2 n^2 \pi^2}{2 m L^2}sin(\frac{n \pi x}{L}) = E sin(\frac{n \pi x}{L})\)​

and so

\(E = \frac{\hbar^2 n^2 \pi^2}{2 m L^2}\)​

In other words, the electron's measured energy can never be zero, and it will always by a multiple of n; this is why we say that the electron's energy is quantized. More to our point, we see that the only physical correspondence the wave function \({\small \psi}\) has is to the measured values of energy E. The sine wave used in the calculation was not an actual waveform physically propagating through some medium, rather it mathematically describes the modes of a quantum oscillation in the form of state vectors.

Hope this helps more than it confuses.

I have to relax and drink an ice cold beer after reading this because it sounds like the physicist's equivalent of the IRS Tax Code (which is over 18,000 pages of bureaucratic nonsense). As a practical matter however, I'm wondering how Louis Debroglie (who first proposed the idea that matter propagated as a wave) would explain his theory.

 

I have to relax and drink an ice cold beer after reading this because it sounds like the physicist's equivalent of the IRS Tax Code (which is over 18,000 pages of bureaucratic nonsense). As a practical matter however, I'm wondering how Louis Debroglie (who first proposed the idea that matter propagated as a wave) would explain his theory.

I may not have done a good job of showing it, but the math of quantum mechanics really is very straightforward, basically just complex numbers and some linear algebra.

In any case, every physicist agrees with the math of quantum mechanics. Where disagreement happens is in the interpretation of what the math is telling us. Fundamentally, QM shows that quantum objects can be, and generally are, in a superposition (sum) of quantum states. This sounds innocent enough until we realize that, for example, position is a quantum state. In other words, a literal interpretation suggests that quantum objects can be in more than one place at the same time. Most people don't like that idea -- after all, when we actually measure the position of a quantum object, it is always in a single place. So several ways of interpreting the math have been developed to explain this so-called measurement problem.

From a bird's eye perspective, the act of doing quantum mechanics looks like this:

1. Define a wave function that represents a system's quantum state at some initial time.
2. The system's state evolves in time according to the Schrodinger equation.
3. When we measure the system, we find that the probability of finding it in some particular (classical) state is equal to the square of the state's amplitude.

Mathematically,

\(\psi(x, t_0) \rightarrow i\hbar \frac{\partial}{\partial t} \psi \rightarrow P(x, t) = |\psi(x, t)|^2\)​

We can call this the quantum algorithm. The problem is the third step, where we measure the system and get a classical result. In step 2, the system is chugging along interacting with the universe in a superposition of quantum states, yet the instant we measure it the wave function collapses to a definite state. Not only is this unlike any other physics we've ever seen, we can't explain why or how it happens. The old trope about observation disturbing the system isn't enough -- after all, our measurement systems themselves are also quantum systems.

Why should one particular type of quantum system (the probe of a measurement device) have such a profound and unusual effect, when innumerable other types of quantum systems don't? To try to suss this out, lots of very clever experiments have been performed, where we try to be sneaky with the measurements, and they always come to the same result: the act of observation collapses the wave function. This is where the interpretations come in.

The oldest is the Copenhagen interpretation, also known as the "Shut up and calculate" interpretation. In this line of thought, the wave function is just a probability density function and not a physical thing, in fact nothing about the system is physical until we measure it. Before measurement, quantum systems are indefinite and so we can't say anything about them at all. Hence, ignore the ontological aspects of quantum systems and just get on with the measurements and calculations.

A radically different interpretation was given by Hugh Everett, which came to be called the many-worlds interpreation (MWI). To solve the measurement problem, Everett said that instead of interpreting the superposition of quantum states as probabilistic, we should take the Schrodinger equation at face value: every superposition is physically real. When we measure a quantum system, there is no wave function collapse; our measurement device gets entangled with the system just like any other, i.e., the device (and indeed us) becomes part of the superposition of states. There is only one wave function with one initial condition -- the initial condition at the start of the universe -- and it has been evolving ever since. That we appear to be experiencing a single reality, in which every experiment has a definite (classical) outcome, is only an illusion: every possible result does happen, but these realities form separate, independent branches of the wave function.

Bohmian mechanics takes a different approach. Early in QM history, in the late 1920s, de Broglie came up with a pilot-wave theory, in which quantum particles are actually physical particles (like little marbles). These particles are "guided" by the wave function, which is a physical wave whose dynamics match that of the probability wave function in the Copenhagen interpretation. It was an ad-hoc theory that produced more questions than answers and was quickly abandoned, even by de Broglie himself. Much later, in the 1950s, David Bohm reformulated the pilot-wave theory into Bohmian mechanics. Like MWI, it solves the measurement problem by denying wave function collapse. But instead of positing an infinitely-branching wave function, Bohmian mechanics posits hidden variables. All the measurable properties, such as position and momentum, are invisibly encoded in the pilot waves that guide each particle. Note, however, that the de Broglie-Bohmian interpretation does not advocate wave-particle duality. Particles are always particles, and waves are always waves. Furthermore, the Bohmian pilot waves are not physical three-dimensional spatial waves; in order to encode all the properties of a particle, pilot waves are highly-dimensional abstract waves in configuration space.

There are several other interpretations out there, but MWI and Bohmian mechanics are the two biggies. Of course, all interpretations have their problems, none of which prevent anyone from doing quantum physics. Since the experimental outcomes of all interpretations agree, they are more philosophical than physical. In any case, it is hopefully clear that the waves in quantum mechanics -- whether we interpret them to be probability amplitudes, or state-space evolutions, or expressions of configuration space -- are mathematical objects, not physical.

 

I may not have done a good job of showing it, but the math of quantum mechanics really is very straightforward, basically just complex numbers and some linear algebra.

In any case, every physicist agrees with the math of quantum mechanics. Where disagreement happens is in the interpretation of what the math is telling us. Fundamentally, QM shows that quantum objects can be, and generally are, in a superposition (sum) of quantum states. This sounds innocent enough until we realize that, for example, position is a quantum state. In other words, a literal interpretation suggests that quantum objects can be in more than one place at the same time. Most people don't like that idea -- after all, when we actually measure the position of a quantum object, it is always in a single place. So several ways of interpreting the math have been developed to explain this so-called measurement problem.

From a bird's eye perspective, the act of doing quantum mechanics looks like this:

1. Define a wave function that represents a system's quantum state at some initial time.
2. The system's state evolves in time according to the Schrodinger equation.
3. When we measure the system, we find that the probability of finding it in some particular (classical) state is equal to the square of the state's amplitude.

Mathematically,

\(\psi(x, t_0) \rightarrow i\hbar \frac{\partial}{\partial t} \psi \rightarrow P(x, t) = |\psi(x, t)|^2\)​

We can call this the quantum algorithm. The problem is the third step, where we measure the system and get a classical result. In step 2, the system is chugging along interacting with the universe in a superposition of quantum states, yet the instant we measure it the wave function collapses to a definite state. Not only is this unlike any other physics we've ever seen, we can't explain why or how it happens. The old trope about observation disturbing the system isn't enough -- after all, our measurement systems themselves are also quantum systems.

Why should one particular type of quantum system (the probe of a measurement device) have such a profound and unusual effect, when innumerable other types of quantum systems don't? To try to suss this out, lots of very clever experiments have been performed, where we try to be sneaky with the measurements, and they always come to the same result: the act of observation collapses the wave function. This is where the interpretations come in.

The oldest is the Copenhagen interpretation, also known as the "Shut up and calculate" interpretation. In this line of thought, the wave function is just a probability density function and not a physical thing, in fact nothing about the system is physical until we measure it. Before measurement, quantum systems are indefinite and so we can't say anything about them at all. Hence, ignore the ontological aspects of quantum systems and just get on with the measurements and calculations.

A radically different interpretation was given by Hugh Everett, which came to be called the many-worlds interpreation (MWI). To solve the measurement problem, Everett said that instead of interpreting the superposition of quantum states as probabilistic, we should take the Schrodinger equation at face value: every superposition is physically real. When we measure a quantum system, there is no wave function collapse; our measurement device gets entangled with the system just like any other, i.e., the device (and indeed us) becomes part of the superposition of states. There is only one wave function with one initial condition -- the initial condition at the start of the universe -- and it has been evolving ever since. That we appear to be experiencing a single reality, in which every experiment has a definite (classical) outcome, is only an illusion: every possible result does happen, but these realities form separate, independent branches of the wave function.

Bohmian mechanics takes a different approach. Early in QM history, in the late 1920s, de Broglie came up with a pilot-wave theory, in which quantum particles are actually physical particles (like little marbles). These particles are "guided" by the wave function, which is a physical wave whose dynamics match that of the probability wave function in the Copenhagen interpretation. It was an ad-hoc theory that produced more questions than answers and was quickly abandoned, even by de Broglie himself. Much later, in the 1950s, David Bohm reformulated the pilot-wave theory into Bohmian mechanics. Like MWI, it solves the measurement problem by denying wave function collapse. But instead of positing an infinitely-branching wave function, Bohmian mechanics posits hidden variables. All the measurable properties, such as position and momentum, are invisibly encoded in the pilot waves that guide each particle. Note, however, that the de Broglie-Bohmian interpretation does not advocate wave-particle duality. Particles are always particles, and waves are always waves. Furthermore, the Bohmian pilot waves are not physical three-dimensional spatial waves; in order to encode all the properties of a particle, pilot waves are highly-dimensional abstract waves in configuration space.

There are several other interpretations out there, but MWI and Bohmian mechanics are the two biggies. Of course, all interpretations have their problems, none of which prevent anyone from doing quantum physics. Since the experimental outcomes of all interpretations agree, they are more philosophical than physical. In any case, it is hopefully clear that the waves in quantum mechanics -- whether we interpret them to be probability amplitudes, or state-space evolutions, or expressions of configuration space -- are mathematical objects, not physical.

Hello there,

Some interesting stuff there, thanks.

An interesting geometrical interpretation, which you can probably explain better than i can, is a "pea pod" that is traveling though space. The pea pod is sort of an ellipsoid but may have pointed ends so it's only approximately ellipsoid and that's just because i cant think of a better shape right now.
To generate the actual 3d pea pod shape, "something" rotates around it's outer skin, in an orbit perpendicular to it's axis of travel but always moving forward. This creates a helical spiral which is similar to a fine pitch screw thread. Now lay this pea pod (or thread) down on it's side and look at an angle perpendicular to the axis of travel and the vertical, whatever that is that is moving and forming that helix forms a projection on the front 2d plane of view and that is a sine wave with amplitude and frequency.

As i said you may be able ot explain this better than i can because i havent had to deal with this kind of thing in years and years. Back some 20 years ago or maybe a little less i posted a complete example of an interaction on another site, but cant even remember now what kind of interaction it was
That's one reason i like sites like this so that we can 'refresh' a little here and there.

 

I have to relax and drink an ice cold beer after reading this because it sounds like the physicist's equivalent of the IRS Tax Code (which is over 18,000 pages of bureaucratic nonsense). As a practical matter however, I'm wondering how Louis Debroglie (who first proposed the idea that matter propagated as a wave) would explain his theory.

Hello there,

I dont think you really mean it is nonsense, but i thought i would interject that the quantum view of the world sure does explain things that i dont think have any other explanation. Things that actually exist yet depend on the so called quantum states to operate AT ALL. Case in point, the single atom transistor which is a single atom that is capable of performing functions that usually require multiple regular Si transistors.

 

A properly orientated atom or molecule could function as a multiple input gate.

 

To generate the actual 3d pea pod shape, "something" rotates around it's outer skin, in an orbit perpendicular to it's axis of travel but always moving forward. This creates a helical spiral which is similar to a fine pitch screw thread. Now lay this pea pod (or thread) down on it's side and look at an angle perpendicular to the axis of travel and the vertical, whatever that is that is moving and forming that helix forms a projection on the front 2d plane of view and that is a sine wave with amplitude and frequency.

Good point, lots of phenomena that we wouldn't normally describe as wave-like can be characterized in terms of sinusoids. The obvious connection is Euler's identity between a complex exponential and a sum of sinusoids:

\(e^{ix} = cos(x) + i sin(x)\)

This relation is ridiculously useful and ubiquitous. Since cosine and sine are orthogonal functions, we can use them as basis functions in function space to decompose arbitrary functions into a linear combination of sines and cosines (Fourier transform). So, though a function such as f(x) = x^2 isn't wave-like by any means, we can still describe it using "waves".

Or, we can take the set of complex numbers with magnitude 1 (the unit circle on the complex plane) and form the multiplicative circle group. This group is isomorphic to U(1), the unitary group of rotation transformations -- any pure rotation can be described as a transformation by an element of U(1). In quantum mechanics, we call state transitions "wave functions", but they're really rotations in state space -- waves just happen to be a mathematically convenient description.

The physicist Peter Woit said it best: "In a very real sense, the reason for the 'quantum' in 'quantum mechanics' is precisely because of the role of U(1) groups acting on the state space. Such an action implies observables that characterize states by an integer eigenvalue of an operator Q, and it is this 'quantization' that motivates the name of the subject."

In quantum field theories, U(1) is the gauge group of quantum electrodynamics. In QED, electrons correspond to an irreducible representation of U(1); an individual electron is then a unit vector in that representation (modulo phase).

It goes much further, of course. We see complex exponentials everywhere in EE, in the solutions to differential equations, in the Laplace transform...

 

On a somewhat related subject of quantum mechanics, I'm wondering how an electron "orbital" is formed in the first place. If an electron is removed (ionized) from a Hydrogen atom and released near the perimeter of what would be the outermost orbital, why doesn't the electron just "fall" straight into the nucleus and collide like a meteor?

I understand that the wave length of the electron's orbital must be 2 TT r, but in order to form an orbital, it seems the electron must have some tangential momentum in the first place.

 

On a somewhat related subject of quantum mechanics, I'm wondering how an electron "orbital" is formed in the first place. If an electron is removed (ionized) from a Hydrogen atom and released near the perimeter of what would be the outermost orbital, why doesn't the electron just "fall" straight into the nucleus and collide like a meteor?

I understand that the wave length of the electron's orbital must be 2 TT r, but in order to form an orbital, it seems the electron must have some tangential momentum in the first place.

I think it's been explained several times that electrons don't orbit like classic particles in the Bohr model.
https://chem.libretexts.org/Textboo...7:_Why_Don't_Electrons_Fall_into_the_Nucleus?

http://physics.bu.edu/~duffy/py106/PeriodicTable.html

Electron probability density clouds
A very important difference between the Bohr model and the full quantum mechanical treatment of the atom is that Bohr proposed that the electrons were found in very well-defined circular orbits around the nucleus, while the quantum mechanical picture of the atom has the electron essentially spread out into a cloud. We call this a probability density cloud, because the density of the cloud tells us what the probability is of finding the electron at a particular distance from the nucleus.

In quantum mechanics, something called a wave function is associated with each electron state in an atom. The probability of finding an electron at a particular distance from the nucleus is related to the square of the wave function, so these electron probability density clouds are basically three-dimensional pictures of the square of the wave function.

 

if You rewound part of your life, how could it unfold any different than it has, unless we assume that we transport our current mindset along with us. Undoubtably, in my mind, everything there is, oscillates at it’s resonant frequency.

 

So what's the difference between an electron traveling as a wave in a linear momentum VS an electron traveling as wave in an atomic orbital which has angular momentum?

Also, if an electron were traveling through free space, what would it look like VS being in an atom?

 

Imagine you do go back with mindset. And you try to make an alternate decision. BUT....if you do that.....it tries to unwind all the people it effects........and they are un-windable. So...no matter what you do...or how hard you try.......you are forced to make the same decision.

A Boring Nightmare.

 

I think first you need to rethink what you think you know about wave-particle duality. The concept of "wave particle duality" was dumped some 80+ years ago. There is no wave particle duality because quantum objects are not waves and they are not particles. Wave-particle duality is just a crutch to help you move up the hill to modern quantum theory.

https://arxiv.org/pdf/quant-ph/0101012.pdf