Yes, if the potential isn't a quadratic function of the position.

@Stam Nicolis: what kind of potential? The one generated by a particle (electrostatic? gravitational?) or the external one? Scalar or vector potential?

The potential energy.

Dear All,

I think yes, and this happens for example via what we called vacuum fluctuations!

The idea is simple when a particle moves from one position to another through disappearance and appearance events, it is according to theoretical calculations must be that the velocity of its jump is much greater than the speed of light (but always the apparent velocity or the mean velocity is less than the speed of light), and therefore the particle much precedes the electromagnetic waves that come from it (in the case we were talking about electromagnetic energy) and thus these waves will reach it later, so it will be affected by it as it comes from the vacuum! but in fact, it comes from itself directly before its jump! this, for example, leads us to the Lamb shift.

For more information please see this paper:

Article A CONCEPT OF UNIVERSAL QUANTUM JUMP

Thanks.

In particular, the illustrious double-slit experiment stands as an exemplary showcase of self-interference. Within this experimental framework, a solitary quantum particle, such as a photon or an electron, traverses through a double-slit apparatus. Astonishingly, during its passage, the particle exhibits a dualistic behavior, behaving both as a discrete particle and an ethereal wave. Consequently, as the particle passes through the slits, it interacts with itself, leading to the emergence of an interference pattern on a distant screen.

No, the double slit experiment is not a good example of self-interaction. You can obtain similar result with Galton board with two inlets. Pure statistics, but not the (self)interaction.

By the way: what could be the observable results of self-interaction? Annihilation or decomposition of such a particle?

The self-interaction of a quantum particle isn't any different from the self-interaction of a classical particle; the difference between the two cases is that the classical particle isn't in equilibrium with any bath, whereas the quantum particle is-it's in equilibrium with the bath of quantum fluctuations.

(A classical particle can be in equilibrium with the bath of thermal fluctuations; in that case the only difference with the quantum particle is in the way the probability density for various quantities is calculated.)

Self-interaction doesn't mean anything more than the property that the potential energy is not a quadratic function of the position.

“Can the quantum particle interact with itself?”

- really cannot; free stable particle is stable – i.e. the same and in the same state – billions of years. More about what are particles see the SS&VT informational physical model, in this case paper https://www.researchgate.net/publication/354418793_The_Informational_Conception_and_the_Base_of_Physics

This couple of examples

“…In particular, the illustrious double-slit experiment stands as an exemplary showcase of self-interference. ….”

“…I think yes, and this happens for example via what we called vacuum fluctuations … it comes from itself directly before its jump! this, for example, leads us to the Lamb shift….”, and

- relate to cases when a particle isn’t free, but interacts with environment – the slits or other particles in an atom.

Though that

“…No, the double slit experiment is not a good example of self-interaction. You can obtain similar result with Galton board with two inlets. Pure statistics, but not the (self)interaction.…..”

- looks as too bold claim, and can well be lesser bold if there would be a reference to published results of such rather simple experiments.

Cheers

Well, well. So the quantum particle on a flat table has to self-interact? Funny ...

The self-interaction of a quantum particle, enclosed in an interacting system, is no different from the self-interaction of a classical particle.

This simple and effective rule is always present in the statistical chain solution produced by the B transition matrix in classical physics and the Q-quantum transition matrix in quantum mechanics.

Here we recall again that B n,n is a true well-defined transition matrix

and that the B-chains are the statistical transfer chain used to find time-dependent solutions for the Poisson, Laplace, and heat diffusion partial differential equations expressed as a simple linear transformation,

U(x,t)=D(N) . b + B^N.U(x,0). . ..(1)

where b is the vector of the real potential energy density applied to the bounded system, U(x,0) is the initial conditions, and D(N) is the transfer matrix expressed by:

D(N)=B + B^2+B^3+ . . .B^N . . . . (2)

For a sufficiently large number of time steps or jumps dt , Eq 2 reduces to,

D=E-I . . . . (3)

For a large enough N.

where the transfer matrix E is given by,

E=1/(I-B) . . . . . (4)

As an application example,

Let's move on to the simplest 3D case of a cube of side L and 8 equidistant nodes (vertices).

The statistical transition of the real 8x8 matrix B is given by,

B=

0 1/6 0 1/6 1/6 0 0 0

1/6 0 1/6 0 0 1/6 0 0

0 1/6 0 1/6 0 0 1/6 0

1/6 0 1/6 0 0 0 0 1/6

1/6 0 0 0 0 1/6 0 1/6

0 1/6 0 0 1/6 0 1/6 0

0 0 1/6 0 0 1/6 0 1/6

0 0 0 1/6 1/6 0 1/6 0

and the steady-state transfer matrix D = E-I is given by Eq 3,4 for N large enough,

1-0.105 0.210 0.210 7.62E-2 7.62E-2 0.210 7.62E-2 7.62E-2 3.81E-2

2- 0.210 0.105 7.62E-2 0.210 0.210 7.62E-2 0.210 3.81E-2 7.62E-2

3- 0.210 7.62E-2 0.105 0.21 0.21 7.62E-2 3.81E-2 0.21 7.62E-2

4-7.62E-2 0.210 0.210 0.105 1.105 3.81E-2 7.62E-2 7.62E-2 0.21

5-0.210 7.62E-2 7.62E-2 3.81E-2 3.81E-2 0.105 0.210 0.210 7.62E-2

6-7.62E-2 0.210 3.81E-2 7.62E-2 7.62E-2 0.210 0.105 7.62E-2 0.210

7- 7.62E-2 3.81E-2 0.210 7.62E-2 7.62E-2 0.210 7.62E-2 0.105 0.210

8- 3.81E-2 7.62E-2 7.62E-2 0.210 0.210 7.62E-2 0.210 0.210 0.105

Note that,

i-All entries of the diagonal D matrix which were originally equal to zero, walk in time, steadily up to the constant value of 1.05.

In the theory of transition matrix solutions, the main diagonal elements represent the residual of the energy density which is in itself the interaction of the elements of the system with themselves.

ii-The sum of all the elements of each row  is precisely equal to 1 as an essential condition for the probabilities in the whole space to be equal to 1.

This condition corresponds to the normalization condition of quantum mechanics.

Now, if we consider the quantum mechanical transition chain, we find a startling similarity.

The numerical statistical solution to SE will be,

Ψ=W. b. . . ..(5)

where b is the vector of the potential applied to the quantum particle and W is the complex quantum transfer matrix expressed by:

W=Q + Q^2+Q^3+ . . .Q^N . . . . (6)

It can be shown that,

Q=Sqrt(B).. . . (7)

where B is the well-known real transition matrix used to find time-dependent solutions for the Poisson, Laplace, and heat diffusion partial differential equations.

Equations 5.6 show that the solutions of Schrödinger's equation depend only on the shape of the potential and the geometry of the boundary conditions as expected.

Let's move on to the same application-specific illustrative cube without loss of generality:

The cube of length L and eight vertices which represent the nodes or quantum states.

and Q n,n is a well-defined complex transition matrix,

defined by, Q=Sqrt(B).

W+I= 1/ (I-Q) . . . . .(8)

Therefore, the complex transfer matrix W + I is given by,

((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)* 6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+ (21-15*i)*6^0.5+176)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35 *i)*2^0.5-...etc

And the stationary numerical solution

Ψ=W. b for the case of the vector of unitary boundary conditions of Constant Real Applied Potential,

b= [1,1,1,1,1,1,1,1] T, becomes,

2^0.5+0 i

2^0.5+0 i

2^0.5+0 i

2^0.5+0 i

2^0.5+0 i

2^0.5+0 i

2^0.5+0 i

2^0.5+0 i

Note that,

i- All eigenvalues ​​of energy are real.

No oscillations in case of constant applied potential.

ii-All eigenvalues ​​of energy are real and they are precisely equal to 2^1/2 showing the special meaning of this irrational figure in quantum mechanics.

to be continued.