Add another anchor to the anchored string.

[Spinnor]

Add another "anchor" to the anchored string.

Consider a vibrating string that has an additional sideways restoring force per unit length, the magnitude of the force is proportional to the sideways displacement. This string will have the same dispersion curve of a massive quantum. See:

Iain G. Main, Vibrations and Waves in Physics, 3rd ed., Cambridge University Press, 1993, pages 224-229.

Let us add an additional sideways force per length perpendicular to the first. Let the string move in two dimensions now. Let the displacement of the string in these two dimensions be given by a complex number. The motion of the string is given by a function of time and position along the string. I guessing:


This string will have 2 "orthogonal" sets of solutions which together can be used to come up with a general solution of this system. All solutions to the wave equation of this string, taken as a whole, can be rotated about the length of the string by some angle and this new set can be mapped one to one with the old set, rotating the set of solutions produces the same set of solutions. Any solution to the wave equation of this string can be rotated about the length of the string and we still have the "same" physics. We can have "linearly polarized" waves and "circularly polarized" waves and all polarizations in between.

Now do this again for the three dimensional analog, the 3D bi-anchored string.

Do we get any interesting physics for the 3D case?

Thanks for your thoughts or corrections.

From:

http://en.wikipedia.org/wiki/Quantum...al_formulation

...
In the mathematically rigorous formulation of quantum mechanics, developed by Paul Dirac[7] and John von Neumann[8], the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors") residing in a complex separable Hilbert space (variously called the "state space" or the "associated Hilbert space" of the system) well defined up to a complex number of norm 1 (the phase factor).

 

[Valerie Park]

These state vectors are combined with the operators representing observables to form a mathematical model of the system. This model is then used to make predictions about the possible results of measurements made on the system. Anchoring this model to physical reality requires that the vectors correspond to physical states, and that the operators represent physically meaningful observables. This is called the "eigenstate-eigenvalue" link. The eigenvectors (called "eigenstates") are the vectors corresponding to the possible measurements outcomes, and the associated eigenvalues are the values the observable takes in that state. The measurement of an observable corresponds to the projection of the state vector onto the eigenstates of the operator representing that observable.

 

[sir-pinski]

Adding another "anchor" to the anchored string introduces the concept of a two-dimensional motion, where the string can now move in two perpendicular directions. This adds an additional dimension to the system and allows for the possibility of linearly polarized and circularly polarized waves. In the three-dimensional case, the concept of a "bi-anchored" string further expands the system and may lead to even more interesting physics. This can be seen in the mathematical formulation of quantum mechanics, where the possible states of a system are represented by unit vectors in a complex Hilbert space. By adding another "anchor" or dimension to the system, we open up the possibility for a wider range of states and behaviors, potentially leading to new and intriguing physics. Further exploration and study of these multi-anchored systems could provide valuable insights into the behavior of complex systems in the physical world.