Anharmonic Oscillator

[Kartazion]

The alternation is then done at the speed of light if you want. Arrived at A or B, the particle marks a definite stop.

[studiot]

1 hour ago, swansont said:

For “instantaneous” I doubt there is a solution.

Indeed for instantaneous implies a step function for which the derivatives used in the equations do not exist.

2 hours ago, Kartazion said:

But you, the experts, do you have a mathematical approach to give me for my equation, namely the alternation of a particle from point A to point B?

If you would go learn some mathematics, as has been suggested, you would know this and not try to misuse equations.

Edited February 21, 2020 by studiot

[Kartazion]

On 2/22/2020 at 12:52 AM, studiot said:

If you would go learn some mathematics, as has been suggested, you would know this and not try to misuse equations.

I must conclude that no expert on this forum has a mathematical approach to suggest to me.

[swansont]

4 hours ago, Kartazion said:

I must conclude that no expert on this forum has a mathematical approach to suggest to me.

One of the obstacles is that you haven’t provided sufficient information about the problem. You have given a vague description of the solution you want.  x = Asinwt is a solution to your conditions

[Mordred]

6 hours ago, Kartazion said:

I must conclude that no expert on this forum has a mathematical approach to suggest to me.

Both harmonic and anharmonic oscillators describe numerous waveforms. They are in essence waveform categories. So as Swansont mentioned is that you need to properly define your particular state you wish to focus on. 

 For example I have no idea if you have the skills to look at the hermite polynomials for the eugenstates and eugenenergies.

 

Edited February 29, 2020 by Mordred

[Kartazion]

1 hour ago, swansont said:

One of the obstacles is that you haven’t provided sufficient information about the problem. You have given a vague description of the solution you want. 

36 minutes ago, Mordred said:

So as Swansont mentioned is that you need to properly define your particular state you wish to focus on.

Simply to alternate a particle from a fixed point A to a fixed point B at high speed, isn't it enough?

37 minutes ago, Mordred said:

 For example I have no idea if you have the skills to look at the hermite polynomials for the eugenstates and eugenenergies.

That's a clue. Thanks

Edited February 29, 2020 by Kartazion

[swansont]

4 minutes ago, Kartazion said:

Simply to alternate a particle from a fixed point A to a fixed point B at high speed, isn't it enough?

But that’s not an accurate summary of the confusing path we’ve been on. The thread is called “anharmonic oscillator” which implies that’s the form of the solution you need.

What’s wrong with Asinwt?

[Mordred]

1 hour ago, Kartazion said:

Simply to alternate a particle from a fixed point A to a fixed point B at high speed, isn't it enough?

 

No that isn't enough. Rather than try to latex all the steps I will link a lesson plan.

https://www.google.com/url?sa=t&source=web&rct=j&url=http://physics.gmu.edu/~dmaria/590%20Web%20Page/public_html/qm_topics/harmonic/&ved=2ahUKEwit56aWkffnAhUiGTQIHdHNDK8QFjAmegQIBRAB&usg=AOvVaw0Xj29xXrtC_pZ9_IruaD_p

If you look at this the solution will depend on the particles principle quantum numbers.

If I were to get the eugenenergies and eugenstates using string theory or QFT the solutions will vary however in all three the principles are the same for orthogonality conditions and hermitean.

Other related articles will employ the ladder operators  (the creation and annihilation operators serve this purpose)

Here is a brute force method using asymptotic analysis.

https://www.google.com/url?sa=t&source=web&rct=j&url=https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/lecture-notes/MIT8_04S13_Lec08.pdf&ved=2ahUKEwit56aWkffnAhUiGTQIHdHNDK8QFjAjegQICBAB&usg=AOvVaw264tFks4Y_ZaWfwThAUR7M

This will add some further details and is a more undergrad level.

https://opentextbc.ca/universityphysicsv3openstax/chapter/the-quantum-particle-in-a-box/

I would identify the commonalities between the three links. Specifically what conditions must be satisfied.

 

 

1 hour ago, swansont said:

But that’s not an accurate summary of the confusing path we’ve been on. The thread is called “anharmonic oscillator” which implies that’s the form of the solution you need.

What’s wrong with Asinwt?

I would think about this question. Specifically identify the difference between a harmonic oscillator vs an anharmonic oscillator there is also an inharmonic oscillator. (Just a side note on the last).

Now one of conditions all three links mention can be satisfied by a harmonic oscillator. Without mentioning the specific condition  (as you should study each one ) can those conditions be met with an anharmonic oscillator ?

Edited February 29, 2020 by Mordred

[studiot]

7 hours ago, Kartazion said:

On 2/21/2020 at 11:52 PM, studiot said:

If you would go learn some mathematics, as has been suggested, you would know this and not try to misuse equations.

I must conclude that no expert on this forum has a mathematical approach to suggest to me.

When I was  working in Saudi Arabia a  desert Arab  said to me in all sincerity

 

Our camels don't have speed limits  or traffic controls, so why should our motor vehicles.

 

Would you allow someone who has only ever driven a farm cart the keys to your Porsche?

[Kartazion]

On 2/29/2020 at 5:41 PM, swansont said:

But that’s not an accurate summary of the confusing path we’ve been on. The thread is called “anharmonic oscillator” which implies that’s the form of the solution you need.

What’s wrong with Asinwt?

It is the shape of the sinusoid that is wrong. I would need a more or less square signal to be able to mark the stop at the positions of A and B.
Is it possible to get a square phase portrait with Asinwt?

On 2/29/2020 at 5:42 PM, Mordred said:

Now one of conditions all three links mention can be satisfied by a harmonic oscillator. Without mentioning the specific condition  (as you should study each one ) can those conditions be met with an anharmonic oscillator ?

If I chose the anharmonic oscillator it is for its viscosity characteristic at the end of the travel of the particle according to x.

I have little information on the inharmonic oscillator. It relates elasticity?

[Mordred]

Sigh no you have to recognize that harmonic, inharmonic and anharmonic are descriptives of the characteristics of the oscillator. 

Here is how classical harmonic oscillator is described.

 A simple harmonic oscillator is a sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).

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

Note what stays constant

An anharmonic oscillator

n classical mechanics, anharmonicity is the deviation of a system from being a harmonic oscillator. An oscillator that is not oscillating in harmonic motion is known as an anharmonic oscillator where the system can be approximated to a harmonic oscillator and the anharmonicity can be calculated using perturbation theory. If the anharmonicity is large, then other numerical techniques have to be used

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

Note this oscillator deviates from the harmonic oscillator. Now see the examples they provide for each on the restoring force ?

Now the quantum harmonic oscillator is the Spring example as opposed to the pendulum example.

The first example the restoring force is a linear function. However it is also a symmetric linear function which can be described by the inner product. The inner of two vectors returns a scalar Ie magnitude. 

So the momentum in the spring is described by linear functions.

The latter case the vectors are curvilinear the force follows a spinor rather than a vector. In this case you also require the direction so you would need the cross product of the vector. This is antisymmetric 

The pendulum example requires angular momentum equations which are not linear. 

 You need to understand this to start being able to identify when a ratio of change is symmetric or antisymmetric in wavefunctions. 

Don't worry about inharmonic functions for now let's get this clear first.

Edit this also a vital key to understanding the Maxwell equations as well as GR. As its applicable to all physics models

 

Edited March 4, 2020 by Mordred

[Mordred]

Here is an angular momentum short article buy gives a good example of the above. You will the above applies including the right rule rule. 

https://www.google.com/url?sa=t&source=web&rct=j&url=https://www3.nd.edu/~mhildret/phys10310/lectures/LectureCh11.pdf&ved=2ahUKEwiP6eWF8P_nAhUYGDQIHRZiC6oQFjAAegQIARAB&usg=AOvVaw21GK_qM-EVcYKGr3gnCw60

[swansont]

17 hours ago, Kartazion said:

It is the shape of the sinusoid that is wrong. I would need a more or less square signal to be able to mark the stop at the positions of A and B.
Is it possible to get a square phase portrait with Asinwt?

If I

A square wave can be approximated by the sum of different sine wave harmonics.

[Kartazion]

On 3/4/2020 at 4:05 PM, swansont said:

A square wave can be approximated by the sum of different sine wave harmonics.

I guess it must still be something to put into equation.

But the harmony supposes several pieces of wave to arrive at the result of a square signal.
(In my case, I must be able to locate the particle in a single position, and that without movement for a certain time.)

But I still wish to see how the harmonies are formulated. Would you have a mathematical link on that?

Thanks.

[studiot]

10 hours ago, Kartazion said:

On 3/4/2020 at 3:05 PM, swansont said:

A square wave can be approximated by the sum of different sine wave harmonics.

I guess it must still be something to put into equation.

But the harmony supposes several pieces of wave to arrive at the result of a square signal.
(In my case, I must be able to locate the particle in a single position, and that without movement for a certain time.)

But I still wish to see how the harmonies are formulated. Would you have a mathematical link on that?

Thanks.

 

Harmonic approximations will not help you.

You should study the Impulse function and particularly the Heaviside step function.

 

Google France should give you lots of references in French.

[swansont]

11 hours ago, Kartazion said:

I guess it must still be something to put into equation.

But the harmony supposes several pieces of wave to arrive at the result of a square signal.
(In my case, I must be able to locate the particle in a single position, and that without movement for a certain time.)

But I still wish to see how the harmonies are formulated. Would you have a mathematical link on that?

Thanks.

Using sine waves to form other waves is part of Fourier analysis.

[studiot]

47 minutes ago, swansont said:

Using sine waves to form other waves is part of Fourier analysis.

 

You said it better the first time.

 

On 3/4/2020 at 3:05 PM, swansont said:

A square wave can be approximated by the sum of different sine wave harmonics.

 

 

Fourier analysis is a linear process.

A linear process can only ever approximate a non linear one and the OP specifically want a non linear one.

In particular square, pulse and rectangular waveforms are non linear.

As Fourier approximation becomes a more and more accurate one, the Gibbs phenomenon becomes more and more pronounced, in the limit with an infinity at every transition.

Edited March 6, 2020 by studiot

[Kartazion]

The very useful Dirac-Delta Impulse functional has a simple Fourier Transform and derivation   f(t) = δ(t-a)  (shifted impulse).

The question is whether it is possible to oscillate the impulse function or the Heaviside step function?

Edited March 6, 2020 by Kartazion

[Kartazion]

Are there any sequential equations? So I should invent the sequential equations ... a bit like the sequential logic in digital circuit theory.

Just associate the function f with the sequential logic.

[studiot]

9 hours ago, Kartazion said:

Are there any sequential equations? So I should invent the sequential equations ... a bit like the sequential logic in digital circuit theory.

Just associate the function f with the sequential logic.

There are quite a few ways of implementing non linear oscillators in electronic circuit theory, rather fewer in mechanical systems.

The difference between a linear and non linear oscillator is that a linear oscillator is based on a characterisitc equation that has only one solution at any one point, say the value of current at a particular applied voltage for linear oscillators.
But non linear ones have some values of voltage which have (at least) two possible values of current.
This leads to circuits or mechanical devices able to switch rapidly between the two possible values.

 

But none are perfect.

[Mordred]

Another classical example of a momentary non linear oscillator is the quarterly amplitude decay in PID temperature controllers when they are stabilizing to set point. Unless the controller is fine tuned for critically damped setpoint.

[Kartazion]

The principle of my oscillator is not that of an RLC circuit. 

Here, if you spoile, is an example of a pseudo CERN-style oscillator, but linear, not looped, very arcuate, but which represents a particle sensitive to magnetic susceptibility, which oscillates step by step in a tube.

You will see a very basic diagram, but its principle is that there. 

Its clock is clocked at regular intervals, is a simple harmonic oscillator.
The bottom Johnson counters integrated circuit advances the particle from left to right, and the top one from right to left.

Spoiler

Ridicule does not kill.

 

[Kartazion]

4 hours ago, Kartazion said:

... very arcuate, ...

I meant archaic.

[studiot]

7 hours ago, Kartazion said:

The principle of my oscillator is not that of an RLC circuit. 

I did not say it was.

I suggested you look at some real world non linear oscillators and the mathematical reason (which I gave you) why they are non linear.

Particularly the sort of oscillators I was thinking of are not RLC oscillators anyway.

[Kartazion]

On 3/10/2020 at 11:17 AM, studiot said:

There are quite a few ways of implementing non linear oscillators in electronic circuit theory, rather fewer in mechanical systems.

2 hours ago, studiot said:

I did not say it was.

I suggested you look at some real world non linear oscillators and the mathematical reason (which I gave you) why they are non linear.

Particularly the sort of oscillators I was thinking of are not RLC oscillators anyway.

Ok. But it was for information. What electronic circuits are you talking about?

Impulse function sometimes uses RC without L, and the Heaviside step function uses a simple switch.