Anharmonic Oscillator

[swansont]

11 hours ago, Kartazion said:

For the atomic nucleus I don't know. But the electron can use quantum tunnelling and pass the potential barrier from the false to the true vacuum.
In the QFT the Higgs potential determines whether the Universe is in one of a true vacuum, or a false vacuum.

Doesn't the ground state represent the lowest energy level of an atom?

You just told me it did not.

11 hours ago, Kartazion said:

And so at the bottom of the well at x=0?
So that a large value of n could bring the energy to the highest in the well?

There is no highest level. n can have any value. At large values the levels are close together, approximating a continuum. 

 

11 hours ago, Kartazion said:

Ok.

We have a better chance of finding the particle at the extremities than in the center of the well.

Yes.

[Kartazion]

1 hour ago, swansont said:

You just told me it did not.

Yes but I don't know why yet, because we have the ground state in the false vacuum (our vacuum) and we also have the ground state in the true vacuum even lower in energy (metastability zero-point energy).

1 hour ago, swansont said:

There is no highest level. n can have any value. At large values the levels are close together, approximating a continuum. 

Ok.

12 hours ago, Kartazion said:

We have a better chance of finding the particle at the extremities than in the center of the well.

1 hour ago, swansont said:

Yes.

Here's the part I'm interested in. But in the case of a harmonic oscillation, wouldn't the probability distribution be uniform according to x?
I don't understand why we have a better chance of finding the particle at the extremities. This is why I chose the anharmonic oscillator.
There's a relationship between potential energy and distribution that I need to understand. Or not.

Edited May 2, 2020 by Kartazion

[swansont]

3 hours ago, Kartazion said:

Yes but I don't know why yet, because we have the ground state in the false vacuum (our vacuum) and we also have the ground state in the true vacuum even lower in energy (metastability zero-point energy).

An atom’s ground state is relative to the free state. False vacuum vs tru vacuum would shift both. It would still take 13.6 eV to ionize hydrogen, and the ground state will still be at -13.6 eV

Can you point to some actual physics that says otherwise?

 

3 hours ago, Kartazion said:

Ok.

Here's the part I'm interested in. But in the case of a harmonic oscillation, wouldn't the probability distribution be uniform according to x?
I don't understand why we have a better chance of finding the particle at the extremities. This is why I chose the anharmonic oscillator.
There's a relationship between potential energy and distribution that I need to understand. Or not.

The probability distribution depends on the shape of the potential well. You brought up hydrogen, which has a 1/r potential. That will behave differently than a square well with a constant potential, or some other shape. Each has a different solution to the Schrödinger equation. The 1D simple harmonic oscillator has a potential that varies as x^2

[Kartazion]

1 hour ago, swansont said:

An atom’s ground state is relative to the free state. False vacuum vs tru vacuum would shift both. It would still take 13.6 eV to ionize hydrogen, and the ground state will still be at -13.6 eV

Can you point to some actual physics that says otherwise?

No, I don't know any.

Here's a graph of a decaying continuity under the potential well, but where the particle is never found. 

[swansont]

19 minutes ago, Kartazion said:

No, I don't know any.

Here's a graph of a decaying continuity under the potential well, but where the particle is never found. 

IOW this is something you just made up.

[Kartazion]

1 minute ago, swansont said:

IOW this is something you just made up.

Yes, of course. I don't know what it's worth, but I see the cosmic inflation part in it too.

[Kartazion]

20 hours ago, swansont said:

The probability distribution depends on the shape of the potential well. 

I'm still struggling to understand why. 

The importance that I attach to it, is precisely that the probability distribution is greater at the extremities*, whereas the probability density is greater at the bottom of the well.
What is the factor in the equation that acts on a larger presence of the particle with  the probability distribution, when the potential energy is higher; since the probability distribution depends on the shape of the potential well?

*And above all, is this form of distribution valid for a classical harmonic oscillator?
The distribution must be uniform following the shape of the well, because:

On 5/1/2020 at 1:47 PM, Mordred said:

If all your sinusoidal waves have equal amplitude you will have equal probability of locating a particle at each amplitude

This is the case for a classical harmonic oscillator.

Clearly, is there a probability distribution equation for a a classical oscillator?

[swansont]

1 hour ago, Kartazion said:

I'm still struggling to understand why. 

Do you understand why, for a classical system?

A flat-bottomed well has a particle moving at constant speed. In QM that’s equal probability (i.e. amplitude) for any anti-node 

For a parabolic well, the probability is higher near the turn-around, because the particle moves slower. In QM you have a wave function higher amplitude near the turnaround.

The QM solutions are quantized, but the overall trend is the same.

1 hour ago, Kartazion said:

The importance that I attach to it, is precisely that the probability distribution is greater at the extremities*, whereas the probability density is greater at the bottom of the well.

What is the distinction between distribution and density?

1 hour ago, Kartazion said:

What is the factor in the equation that acts on a larger presence of the particle with  the probability distribution, when the potential energy is higher; since the probability distribution depends on the shape of the potential well?

*And above all, is this form of distribution valid for a classical harmonic oscillator?
The distribution must be uniform following the shape of the well, because:

The factor is the value of the potential; it’s a term in Schrödinger’s equation. It should not be a surprise that the wave function will differ if V(x) is different.

 

1 hour ago, Kartazion said:

This is the case for a classical harmonic oscillator.

Clearly, is there a probability distribution equation for a a classical oscillator?

The time spent in some region dx would be dependent on speed, and be proportional to the probability. Divide that time by the period of oscillation.

[Kartazion]

49 minutes ago, swansont said:

For a parabolic well, the probability is higher near the turn-around, because the particle moves slower. 

The time spent in some region dx would be dependent on speed, and be proportional to the probability. Divide that time by the period of oscillation.

That's precisely why I chose an anharmonic oscillator. To accentuate the slowness of the particle near the turn-around.

50 minutes ago, swansont said:

What is the distinction between distribution and density?

The number of peaks over an interval.

Modified image source: https://www.academia.edu/8950511/Molecular_Quantum_Mechanics_4th_ed_ATKINS-FRIEDMAN

[Phi for All]

On 5/2/2020 at 10:43 AM, swansont said:

IOW this is something you just made up.

On 5/2/2020 at 10:50 AM, Kartazion said:

Yes, of course. I don't know what it's worth, but I see the cosmic inflation part in it too.

!

Moderator Note

Invention is NOT about making things up. Please stick to science you can support with evidence and critical reasoning. 

 

 

[Kartazion]

3 hours ago, Phi for All said:

!

Moderator Note

Please stick to science you can support with evidence and critical reasoning. 

 

I made a mistake. And I do not know if my reasoning is correct, especially in the case of a classical oscillator.

Here is the explanation of my nonsense:

I was based on the quantum perturbation of the ground state.
Indeed by the energy of the disturbance and the oscillation of the flow at the bottom of the well, does not allow the particle to fall under the well.
- In quantum: the attenuation of the perturbation allows the sliding by tunnel effect of the passage of the particle under the well.
- In classic: if there is an attenuation of the kinetics at the bottom of the well, then the particle falls under the well.

[swansont]

1 hour ago, Kartazion said:

I made a mistake. And I do not know if my reasoning is correct, especially in the case of a classical oscillator.

Here is the explanation of my nonsense:

I was based on the quantum perturbation of the ground state.
Indeed by the energy of the disturbance and the oscillation of the flow at the bottom of the well, does not allow the particle to fall under the well.
- In quantum: the attenuation of the perturbation allows the sliding by tunnel effect of the passage of the particle under the well.

What perturbation?

Under the well?

1 hour ago, Kartazion said:

- In classic: if there is an attenuation of the kinetics at the bottom of the well, then the particle falls under the well.

There is no "under the well"

In a classical system, if there is no energy, the particle is at rest at the bottom of the well. The only way to go to a lower energy is to change the well.

"Perturbation" typically implies you are adding energy in some way. "Vacuum" is not a term I associate with a classical oscillator. It sounds like you are still making things up.

 

[Kartazion]

3 hours ago, swansont said:

What perturbation?

Quote

Since the lowest allowed harmonic oscillator energy, E0, is ℏω2 and not 0, the atoms in a molecule must be moving even in the lowest vibrational energy state. This phenomenon is called the zero-point energy or the zero-point motion, and it stands in direct contrast to the classical picture of a vibrating molecule.

 

3 hours ago, swansont said:

Under the well?

Only the extension of the energy under the well. From the false to the true vacuum. 
Conventionally this pattern is in a cascade, followed by a potential barrier.

 

3 hours ago, swansont said:

There is no "under the well"

In a classical system, if there is no energy, the particle is at rest at the bottom of the well. The only way to go to a lower energy is to change the well.

My idea is that there is a particle size slit. Either in the center, or two extremities not far from the center.
Depending on the kinetic force, the particle may or may not pass through the slit to finally change the well.

I'm not sure it works very well. Because the particle has to hit the chamfer at each pass.

I have now learned the lesson.

[swansont]

13 hours ago, Kartazion said:

I have now learned the lesson.

No, I think you have combined bits and pieces of different theories and created an abomination. There’s no evidence you’ve learned anything.

[Kartazion]

6 minutes ago, swansont said:

There’s no evidence you’ve learned anything.

“The only thing I know, is that I know nothing.”

Socrates.

 

[studiot]

12 minutes ago, Kartazion said:

“The only thing I know, is that I know nothing.”

Socrates.

 

Quote

Alexander Pope

A little learning is a dangerous thing

This famous quote is often said as 'a little knowledge'................

 

 

[Kartazion]

14 minutes ago, studiot said:

This famous quote is often said as 'a little knowledge'................

"Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution."

Albert Einstein

[StringJunky]

2 hours ago, Kartazion said:

"Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution."

Albert Einstein

Einstein's statement is not binary; it is not either/or.

[Kartazion]

6 hours ago, StringJunky said:

Einstein's statement is not binary; it is not either/or.

And I thought there was one thing more important than the other.

[swansont]

Einstein wasn’t advocating having no knowledge whatsoever.

[Kartazion]

2 minutes ago, swansont said:

Einstein wasn’t advocating having no knowledge whatsoever.

I have all my life to learn.

[swansont]

1 minute ago, Kartazion said:

I have all my life to learn.

Please don’t waste my time while you do so.

[Kartazion]

11 minutes ago, swansont said:

Please don’t waste my time while you do so.

Then ignored me, and don't answer.

I was talking to StringJunky.

9 hours ago, studiot said:

A little learning is a dangerous thing 

I didn't know going to college or university was dangerous.

[Kartazion]

...

Here is the part I was advised to study in order to be able to develop a probable mathematical solution to the given problem (which is, and with the help of an anharmonic oscillator, to highlight the duplication of a single particle, in two distinct points).

Those are the Partial Differential Equations.

I'll be back in a little while to give you the first forms of scripture.

Thanks.

[Kartazion]

Before arriving at the PDE, the probability distribution that I want as a function of point x and time t₀ would have the following form:

[math] P(x,t_0) = \delta(x-f(t)) \delta(t_0-t) [/math]

Where [math]\delta(z)[/math] is the Dirac delta function. With this distribution, we can know the probability of finding the particle between the points x₁ and x₂, and in the time interval (t₁, t₂) in an analogous manner:

[math] P(x_1 < x < x_2, t_1 < t < t_2) = \int_{x_1}^{x_2}\int_{t_1}^{t_2}P(x,t)dxdt [/math]
 

Edited June 27, 2020 by Kartazion