The visualization of the space probability distribution for a moving particle I: in a single ring-shaped Coulomb potential

We first present the exact solutions of the single ring-shaped Coulomb potential and then realize the visualizations of the space probability distribution for a moving particle within the framework of this potential. We illustrate the two-(contour) and three-dimensional (isosurface) visualizations for those specifically given quantum numbers (n, l, m) essentially related to those so-called quasi quantum numbers (n',l',m') through changing the single ring-shaped Coulomb potential parameter b. We find that the space probability distributions (isosurface) of a moving particle for the special case and the usual case are spherical and circular ring-shaped, respectively by considering all variables in spherical coordinates. We also study the features of the relative probability values P of the space probability distributions. As an illustration, by studying the special case of the quantum numbers (n, l, m)=(6, 5, 1) we notice that the space probability distribution for a moving particle will move towards two poles of axis z as the relative probability value P increases. Moreover, we discuss the series expansion of the deformed spherical harmonics through the orthogonal and complete spherical harmonics and find that the principal component decreases gradually and other components will increase as the potential parameter b increases.

' l , ' m ) through changing the single ring-shaped Coulomb potential parameter b. We find that the space probability distributions (isosurface) of a moving particle for the special case l m = and the usual case l m ≠ are spherical and circular ring-shaped, respectively by considering all variables ( , , ) θ ϕ =  r r in spherical coordinates. We also study the features of the relative probability values P of the space probability distributions. As an illustration, by studying the special case of the quantum numbers (n, l, m)=(6, 5, 1) we notice that the space probability distribution for a moving particle will move towards two poles of axis z as the relative probability value P increases. Moreover, we discuss the series expansion of the deformed spherical harmonics through the orthogonal and complete spherical harmonics and find that the principal component decreases gradually and other components will increase as the potential parameter b increases.

Introduction
Since the ring-shaped non-central potentials have potential applications in quantum chemistry and nuclear physics, e.g., they might describe the molecular structure of Benzene and interaction between the deformed nucleuses, it is not surprising that the relevant investigations for them have attracted many attentions [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Based on previous study, we have known that this type of ring-shaped non-central potentials can be solved in spherical coordinates and also the system Hamiltonian with the hidden symmetry makes the bound state energy levels possess an "accidental" degeneracy, which arises from the SU(2) invariance of the Schrödinger Hamiltonian [1]. Generally speaking, the most popular ring-shaped non-central potentials are identified as the Coulomb or harmonic oscillator plus the ring-shaped part ( ) 1/ sin r θ . In this work, we are concerned with only the single ring-shaped Coulomb potential in the limited space.
Many authors have obtained the radial and polar angular differential equations and also got their solutions in recent studies [7,8,13,14,21], but the whole space ( , , ) θ ϕ =  r r probability distributions of the moving particle in the single ring-shaped non-central fields have never been reported because of the complicated computation skills occurred in program. The main contributions mentioned above are either concerned with the radial part in the spherical shell ( , ) r r dr + or with the angular parts in volume angle dΩ [22,23]. This means that these studies are only related to one or two of three variables ( ) , , r θ ϕ . To illustrate comprehensively the space probability distribution of the moving particle confined in the ring-shaped non-central Coulomb potential, the aim of this work is to realize their two-(contour) and three-dimensional (isosurface) visualizations by considering all variables. Such studies have never been done to our best knowledge.
The rest of this work is organized as follows. In Section 2 we first present the solutions of the studied quantum system and give the concrete expressions of the angular wave functions in order to compare with the usual spherical harmonics and also show their distinct properties. In Section 3 we make use of the calculation formula of the space probability distribution to illustrate their visualizations by overcoming the calculation skills in MATLAB program. In Section 4 we discuss the variation of the space probability distribution with the number of radial nodes, the variations with the relative probability value P and those with the negative and positive ring-shaped Coulomb potential parameter b. The expansion coefficients of the deformed spherical harmonics are calculated in Section 5. Some concluding remarks are given in Section 6.

Exact solutions to single ring-shaped Coulomb potential
The single ring-shaped Coulomb potential is given by and the Schrödinger equation is written as Take the wave function of the following form where 0, 1, 2, m = ± ±  . Substitute (3) into (2) and get the respective radial and angular differential equations as Take a new variable transform cos x θ = and equation (4a) becomes ( ) is also the solution of Eq. (5) and its definition is given by [24,25] ( On the other hand, in terms of the spherical harmonics ( , ) We can define deformed spherical harmonics ( , ) with the following property * ( Obviously, m′ is equal to m when 0 b = , and Eqs. (8), (10), (11) will reduce to the results in central field. It is worth pointing out that all states ( , ) . This arises from the fact that the ring-shaped potential reduces the symmetry of the system and the degeneracy.
In Table 1 we list the analytical expressions of some given deformed spherical in the special cases of potential parameter 0 b = and 0 b ≠ .
Its solutions are nothing but the Coulomb case, i.e. [28] Z r a n n l r r Z n l Zr Z r u r e F n l l a n n a n a n

Isosurface and contour visualizations of space probability distributions
It is well known that the space probability distributions for a moving particle at the position ( , , ) θ ϕ =  r r can be calculated by Obviously, this formula is independent of the azimuth angle ϕ and thus symmetric with respect to the axis z. In order to display the space probability distribution we will transform (15) from original spherical coordinates to the popular Cartesian coordinates through the coordinate transformations ( , x y , z ) and calculate density block, say den (N, N, N), which is composed of all values n l m w ′ ′ ′ for all N N N × × positions. In this work, we take N=81. For given quasi quantum numbers ( , , ) n l m ′ ′ ′ , we realize their isosurface (three dimensional) and contour (two dimensional) visualizations of the space probability distributions for different states ( 5) n ≤ by using MATLAB program (see Tables 2 and 3).

Variation of space probability distribution with respect to the numbers of radial nodes
In Table 2 we display the space probability distributions for three different cases b = 0 corresponding to the Coulomb potential, b = 0.5 and b = 10 corresponding to ring-shaped potentials. The unit in axis is taken as the Bohr radial a 0 . To clearly visualize the internal structure of the graphics, we generate a section plane without considering those numerical values in the regions 0, 0, 0 It is found that the graphics becomes compressed, i.e., the space probability distributions elongate along with the axis x and y and also the hole formed in the ring-shaped potentials expand towards outside as the ring-shaped potential parameter b increases. This can be well understood by the relations given in Eq. (7). We know that the value of the quasi quantum number ' m increases relatively for a given quantum number m. When l m = , the isosurface of the density distribution is spherical, but for the case l m ≠ , its isosurface is circularly ring-shaped.
In Table 3, the space probability distribution is projected to plane yoz and is shown to be symmetric with respect to the axis y and z . Here we only plot the graphics in the first quadrant through magnifying proportionally the space probability 2 | ( ) | ′ ′ ′ Ψ  n l m r and making it the maximum value to be 100, while the interval is taken as 10. There exists a corresponding balance among the density distributions in axis directions x, y and z since the sum of density distributions is equal to unit according to the normalization condition.

Variation of the space probability distributions with respect to different relative probability values P
In order to display the isosurface of the space probability distributions for different chosen relative probability values (0,100)% P ∈ , we take the quantum numbers (n, l, m)=(6, 5, 1) as a typical example in Table 4. We find that for the smaller P the particle is distributed to almost all position spaces but for the larger P, we notice that the particle moves towards to two poles of axis z.

Variations of the space probability distribution with respect to different potential parameter b
For fixed quantum numbers m and n θ , it is known from Eq. (7) that the quasi quantum number m′ becomes larger with the increasing b and thus leads to the increment of the quasi quantum number l′ . In Table 5  In Table 6, we give the comparison between the special cases for 0 b < and 0 b > . Obviously, we find that the space probability distributions for the negative

Expansion coefficients of the deformed spherical harmonics
Since the spherical harmonics (9) where the expansion coefficients can be calculated by In Fig. 1  For =1 m , each value of lm a is also equal to zero for the case of odd value l m − .
Thus, in Fig.1 we only present the case of even value l m − for =1 m . Notice that 1 3 ( ) P x is always biggest, but the principal component 1 3 ( ) P x decreases gradually, and other components will increase as the parameter b increases.

Concluding remarks
In this work we first presented the exact solutions to the single ring-shaped Coulomb potential and then realized the visualization of the space probability distributions for a moving particle within the framework of this potential. We have illustrated the two-(contour) and three-dimensional (isosurface) visualizations for some given quasi quantum numbers ( ', ', ') n l m by taking different ring-shaped potential parameter b.
We have found that the space probability distributions of the moving particle in the cases of the l m = and l m ≠ are spherical and circularly ring-shaped, respectively.