Anharmonic Oscillator

[swansont]

1 hour ago, Kartazion said:

If you want to learn quantum mechanics you have to understand the an·harmonic oscillator.

No, not so much.

[Kartazion]

Is learning the anharmonic oscillator mandatory to understand quantum mechanics?
Or is it simply a substitute calculation method compared to others?

[Mordred]

Both the harmonic oscillator and anharmonic oscillator are lessons in QM. Particularly with the HUP and quantum harmonic oscillator. Though you will find oscilators in probability functions.

[Kartazion]

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Is Zero-point energy a coherent state of the oscillator? And there is a link between the Second quantization and the Second quantum theory?

[swansont]

3 hours ago, Kartazion said:

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Is Zero-point energy a coherent state of the oscillator? 

The zero point energy is the energy when the system is unoccupied, or in its lowest state, depending on the system you're describing. It's an energy, not a state.

 

[Mordred]

Second quantization itself is a methodology to handle large multiparticle states. In essence using the Schrodinger or Klien Gordon equation for gets inconvenient when you have 10^(20) particles. So you apply an occupation density through the creation and annihilation operators. Those operators are used in second quantization.

[Kartazion]

Can the second quantization be read along x for one particle as in the diagram below?

[Mordred]

No you need a graph that directly applies the creation and annihilation operators.

Here 

https://en.m.wikipedia.org/wiki/Second_quantization

Edited January 30, 2020 by Mordred

[Kartazion]

We can easily deduce the Fock state and the Fock space in the second quantization.

 

In this approach, the quantum many-body states are represented in the Fock state basis, which are constructed by filling up each single-particle state with a certain number of identical particles. 
...
The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space H.

 

[Kartazion]

On 11/21/2019 at 6:39 PM, Kartazion said:

Good evening

What I wish is to be able to submit a cycle reiteration to the equation. I consider a cycle, to a complete round-trip of the mass m on its axis.

If I multiply the execution of the equation to one million cycles per second, then the mass density, visually speaking, would be distributed at the ends. Indeed, the behavior of the oscillator determines a slowdown, then a certain stop, on each end position of the mass m.

Do you validate this deduction?
 

On 11/22/2019 at 12:20 PM, swansont said:

Probably, but you would have to solve the equation to be sure. Note that you can't arbitrarily change the frequency of the oscillation. That typically comes from the solution to the problem. If you specify it, then some other quantity has to be unknown and free to change. You would not be comparing identical systems. 

Using a simple example, a pendulum's frequency depends on its length, and g. If you dictate that it must have a certain frequency, you have to change the length and/or find a place where g has the proper value to get this result. If it naturally swings at 1 Hz, it's not going to magically swing at 1 MHz

 

 

A new method for constructing squeezed states for the isotropic 2D harmonic oscillator

We introduce a new method for constructing squeezed states for the 2D isotropic harmonic oscillator. Based on the construction of coherent states in [1], we define a new set of ladder operators for the 2D system as a linear combination of the x and y ladder operators and construct the SU(2) coherent states. The new ladder operators are used for generalizing the squeezing operator to 2D and the SU(2) coherent states play the role of the Fock states in the expansion of the 2D squeezed states. We discuss some properties of the 2D squeezed states. 

 

[swansont]

35 minutes ago, Kartazion said:

A new method for constructing squeezed states for the isotropic 2D harmonic oscillator

We introduce a new method for constructing squeezed states for the 2D isotropic harmonic oscillator. Based on the construction of coherent states in [1], we define a new set of ladder operators for the 2D system as a linear combination of the x and y ladder operators and construct the SU(2) coherent states. The new ladder operators are used for generalizing the squeezing operator to 2D and the SU(2) coherent states play the role of the Fock states in the expansion of the 2D squeezed states. We discuss some properties of the 2D squeezed states. 

 

Your point is?

[Kartazion]

21 minutes ago, swansont said:

Your point is?

My point seems right.

Edited February 5, 2020 by Kartazion

[Mordred]

Pray tell which point and how do you believe squeezed states support that point. Your previous post in the quoted section differs considerably to squeezed states. 

 One point you seem to keep missing is the significants of probability wave functions. Ie you need look into greater detail into the mathematics and less on images etc.

 

 

Edited February 5, 2020 by Mordred

[swansont]

10 hours ago, Kartazion said:

My point seems right.

Your example was a quantum system and my example was a classical system. It's also harmonic, not anharmonic.  

Tossing in a link to a paper isn't a rebuttal. You need to discuss details.

[Kartazion]

On 2/5/2020 at 4:48 AM, Mordred said:

Pray tell which point and how do you believe squeezed states support that point. Your previous post in the quoted section differs considerably to squeezed states. 

What I see is the importance of the density at its points *. Namely at the ends of A and B. After that there remains a coherent state, except that for this example it is simply pressed; It is, as I understand it, a real method of application; which we cannot achieve with my classic oscillator, because alternating a particle of mass m to several million cycles per second is not too possible.

On 2/5/2020 at 4:48 AM, Mordred said:

One point you seem to keep missing is the significants of probability wave functions. Ie you need look into greater detail into the mathematics and less on images etc.

* The only probability that I see is that of the presence of a density at a given point, or then a specific part of the wave; namely to isolate an alternation either high or low, and deduce that its probability of being up or down is equal.

The images help me, as much as a graph describes its equation. But in terms of the formulation I count precisely, and hence my presence here, on geniuses or academics who may one day bring the equation to what I wish to present.

17 hours ago, swansont said:

Your example was a quantum system and my example was a classical system. It's also harmonic, not anharmonic. 

What is the difference between a classical and quantum oscillator? It must be a particle of mass m against a quanta of light for example. If I chose the harmonic oscillator, it is for its termination in terms of travel along x. Whether the oscillator is harmonic or anharmonic works on the same principle, only its termination.

17 hours ago, swansont said:

Tossing in a link to a paper isn't a rebuttal. You need to discuss details.

On the contrary. I would be happy to see someone find a mathematical approach to the problem posed.

I work on the rest (details).

[swansont]

7 hours ago, Kartazion said:

What is the difference between a classical and quantum oscillator?

Physical motion (i.e. a defined trajectory), for one.

 

 

[Mordred]

The problem with relying too much on graphs though they do serve as a good visual aid is that one can have identical graphs that have nothing to do with one another.

With squeezed states your squeezing the probability amplitude of one operator while increasing the uncertainty in another. 

For example in that paper either the position or momentum operator. Both involve probability functions.

Edited February 6, 2020 by Mordred

[Kartazion]

17 hours ago, Mordred said:

The problem with relying too much on graphs though they do serve as a good visual aid is that one can have identical graphs that have nothing to do with one another.

Graphics are mandatory for me.

I'm not talking about abstract math. But we are talking about an oscillator. It is precise and tangible.
Algorithms and graphs transform well into integrated circuits or into computers and other technological jewels.
Clearly there is always a physical support to the key, and gives a concise representation.
Just as math tries to explain the creation of our universe. Industrially speaking we do not do math to create imagination.
Any formula must be able to represent reality in calories, forces, speed, intensity, ect ...
I'm not saying there is no abstract math in real, but the goal is physical design.

19 hours ago, swansont said:

Physical motion (i.e. a defined trajectory), for one.

Is this not the case for quanta?

[swansont]

5 hours ago, Kartazion said:

 

Is this not the case for quanta?

No, they have a wave nature. Electrons in an atom don’t have trajectories 

[Mordred]

6 hours ago, Kartazion said:

Graphics are mandatory for me.

I'm not talking about abstract math. But we are talking about an oscillator. It is precise and tangible.
Algorithms and graphs transform well into integrated circuits or into computers and other technological jewels.
Clearly there is always a physical support to the key, and gives a concise representation.
Just as math tries to explain the creation of our universe. Industrially speaking we do not do math to create imagination.
Any formula must be able to represent reality in calories, forces, speed, intensity, ect ...
I'm not saying there is no abstract math in real, but the goal is physical design.

 

In physics the goal is to accurately make concise predictions of all measurable physical properties in how they interact or alter. Those mathematics are required to do so.

 In anything dealing with particle physics you require waveforms and wavefunctions as every particle of the SM model is a field excitation. It isn't some abstract choice but an accurate description of all observational evidence.

 The probability functions isn't an arbitrary choice either. The uncertainty principle taught us that the uncertainty is a fact of nature. Though that's not the only reason the probability functions are a necessity.

 It's necessary when you do things like a Fourier transformation to describe their measurable wavefunctions described by their measurable states.

Particles are not solid or corpuscular billiard balls. For example you can have an infinite number of bosons such as the photon exist in the same precise coordinate. They can overlap the same space without any interference or interaction.

Neutrinos can pass through several light years of lead without any interaction or interference.

Edited February 7, 2020 by Mordred

[Kartazion]

2 hours ago, swansont said:

No, they have a wave nature. Electrons in an atom don’t have trajectories 

Yes it's true, nature is just a wave. But there, if I may allow myself, the quantum decoherence comes into play. Because we can see this photon well, and which by its trajectory, see up to our eye. The photon has a trajectory only if we observe it.

[swansont]

19 minutes ago, Kartazion said:

Yes it's true, nature is just a wave. But there, if I may allow myself, the quantum decoherence comes into play. Because we can see this photon well, and which by its trajectory, see up to our eye. The photon has a trajectory only if we observe it.

What does that have to do with a quantum oscillator? What does seeing a photon have to do with quantum decoherence?

[Kartazion]

5 minutes ago, swansont said:

What does that have to do with a quantum oscillator?

You told me about the electron in the atom, I simply answered you with the photon.

[swansont]

3 hours ago, Kartazion said:

You told me about the electron in the atom, I simply answered you with the photon.

Applies to any quantum particle in a bound state. You know, an oscillator.

[Kartazion]

17 minutes ago, swansont said:

Applies to any quantum particle in a bound state. You know, an oscillator.

The oscillator explains a lot. But the oscillator has a trajectory. A trajectory of what? Or at least a periodic charge signal.

6 hours ago, Mordred said:

In physics the goal is to accurately make concise predictions of all measurable physical properties in how they interact or alter. Those mathematics are required to do so.

 In anything dealing with particle physics you require waveforms and wavefunctions as every particle of the SM model is a field excitation. It isn't some abstract choice but an accurate description of all observational evidence.

Yes of course I agree.

6 hours ago, Mordred said:

 The probability functions isn't an arbitrary choice either. The uncertainty principle taught us that the uncertainty is a fact of nature. Though that's not the only reason the probability functions are a necessity.

The oscillator explains it very well with regard to uncertainty.

6 hours ago, Mordred said:

It's necessary when you do things like a Fourier transformation to describe their measurable wavefunctions described by their measurable states.

Excited state Vs Ground state.

6 hours ago, Mordred said:

Particles are not solid or corpuscular billiard balls. For example you can have an infinite number of bosons such as the photon exist in the same precise coordinate. They can overlap the same space without any interference or interaction.

Neutrinos can pass through several light years of lead without any interaction or interference.

If you only have one particle in action, it is sure that there can be no interactions and interference of itself. Matter is made up of vacuum. In the context of the SM, and for the boson, the principle of reiteration makes it possible to superimpose the "particle" several times in the same place, or at least increase its level of density, at a point.