In the scientific literature dedicated to discoveries in quantum mechanics, there are some ex-tensions of known problems, such as in [1]-[13], in which we can see the identification of some previously disregarded aspects that escaped analysis . Likewise, in the literature of mathematical physics, there are new findings that show that several mathematical approaches can be adapted to evaluate several complementary aspects in quantum mechanics [14]-[23].

Here we are interested in the so-called “electric Stern-Gerlach” effect, an analog of the Stern-Gerlach effect . We begin by reviewing references [24]-[26], which address the problem of a beam of ammonia molecules (N H3), initially traveling along the positive X direction (Fig. 1), and then splitting into two secondary beams in the Y = 0 plane, within a region where a non-homogeneous electric field acts, whose spatial configuration presents a gradient along the Z coordinate direction.

Regarding references [24]-[26], it is worth noting that these do not contain analytical expres-sions for the corresponding eigenfunctions; furthermore, the gradient of the electric field does not appear explicitly in their equations, only the square of the electric field, which is considered intense. In [27], on the other hand, considering the ammonia molecule as a system with only two energy levels (of the vibrational type) and the action of a perturbative potential with a specific shape, it was determined that the expressions of the corresponding energy levels depend on the applied electric field, which is why the Stark effect was recognized. It should be mentioned, in the case of [24], that these absences are consistent with a previous comment by the author . It can be inferred from figure 1 above, and from other similar figures, that the expression “electric Stern-Gerlach” effect, uses the spatial separation of a beam of ammonia molecules into two secondary beams as the only argument to support the analogy with the Stern-Gerlach effect. It is shown here, however, that this is not the case. Based on a result shown in [3],[13], that the gradient of a Stern-Gerlach-type magnetic field explicitly contributes to the eigenenergies of the corresponding Hamiltonian, it is found, in a certain physical context and under certain conditions, that the non-zero gradient of the electrostatic field also contributes to the respec-tive eigenenergies (and not only the gradient, as we will see later), which extends the analogy between both effects.

In this paper, we follow almost completely the general physical framework defined in [24]- [26] for the ammonia molecule, as well as for the corresponding beam. In particular, we model the ammonia molecule as an effective quantum particle , without internal structure, with mass equal to the molecule’s mass and with the same permanent electric dipole moment. Furthermore, and unlike these references, the electric field gradient appears here explicitly in the equations; in particular, we assume that this gradient is sufficiently weak, so that ∆z, in Fig. 1, corresponding to the vertical aperture of the initial beam into two beams, is sufficiently small, with which the linearization of the term proportional to z (z within the interval ∆z) is a good approximation. Based on this, we construct the explicit eigenfunctions for the “electric Stern-Gerlach” effect, for the decoupled equations for the state vector components. In addition, we explicitly show that the eigenenergies of the model incorporate the contribution of the electrostatic field’s gradient and the density of the electric charges that generate this field; results not found in [24]-[26].

Finally, regarding the beams unfolded, contained in the plane Y = 0, the following situations are possible: (i) All molecules in a beam are in the same steady state; that is, the molecules in the first beam have a different state from the molecules in the second beam; (ii) In the same beam there are two sets of molecules, each in a steady state. The first case corresponds (see section 3) to the value A = 0 and the second to the value A ̸= 0; in particular, non-zero and small values of A correspond to situations in which there is a majority presence of molecules in the same steady state.

mathematics of partial differential equations".

A point particle, so that in the energy levels there will be no contributions that could come from vibrations, deformations, etc. that can manifest themselves in a non-point structure.

From classical electromagnetism we know that polar molecules, such as ammonia, have a per-manent electric dipole moment. This type of molecule undergoes deformation proportional to the acting local field (assuming a linear regime), in the presence of an external electric field, and tends to orient itself parallel to the direction defined by the external field at the location.

On the other hand, in a semi-classical treatment, the components µ1, µ2 and µ3 of ⃗µ, the induced electric dipole moment in the ammonia molecule (in the presence of an external electric

The only one to be considered here, since this induced electric moment is sufficient to reveal the gradient dependence on the eigenenergies of the Hamiltonian in (2.3).

field E ⃗, are defined by a polarizability matrix:

      µ1 σ11 σ12 σ13 E1       ⃗µ =  µ2  =  σ21 σ22 σ23   E2  . (2.1) µ3 σ31 σ32 σ33 E3

In our case, from (1.6), we have that E1 = 0, E2 = −αy and E3 = (α + (ρ/ϵ0))z.

According to expression (10), the components of ⃗µ are:

µ1 = σ12E2 + σ13E3 ⇒ µ1 = − σ12 αy + σ13 α + (ρ/ϵ0) z,

µ2 = σ22E2 + σ23E3 ⇒ µ2 = − σ22 αy + σ23 α + (ρ/ϵ0) z,

µ3 = σ32E2 + σ33E3 ⇒ µ3 = − σ32 αy + σ33 α + (ρ/ϵ0) z. (2.2)

Furthermore, the quantum Hamiltonian for an “effective particle ”, with induced electric dipole moment ⃗µ, in the electrostatic field given in (1.6), is written as:

( ) ! !

1 1 0 ⃗µ·E − ⃗ A ˆ ˆ 2 ˆ 2 ˆ 2 H = Px + Py + Pz + , (2.3) 2m 0 1 −A −⃗µ·E ⃗

where A is the probability amplitude that a particle that is initially in the stationary state φ1 passes to the stationary state φ2.

For the rightmost term in (2.3), corresponding to the interaction between the electric dipole moment and the externally applied electric field, we write,

⃗µ·E − ⃗ A H = , (2.4) −A −⃗µ·E ⃗

where, ⃗µ·E ⃗ = − σ αy + σ α + (ρ/ϵ ) z (αy) + (2.5)

+ − σ32 αy + σ33 α + (ρ/ϵ0) z (α + )z.

In the case drawn in Fig.1, and as considered in [14, 15, 16], there is no gradient along the Y direction, and the separation of the initial beam into two beams only occurs in the plane Y = 0 (from which it follows, according to (1.6), that E2 = 0); in this case, expression (2.4) is simplified to the following,

σ33 α + (ρ/ϵ0) z −A ′′   H =  2  . (2.6) −A −σ33 α + (ρ/ϵ0) z

For what follows, and this is a key procedure in this development, we proceed to linearize the term: α + (ρ/ϵ0) z , which corresponds (except for the coefficient −σ33) to one of the elements on the diagonal of the matrix H , in (2.6), within a “suitable” interval. Then we do:

2 2 ˜ α + (ρ/ϵ0) z → (α + (ρ/ϵ0))z → (ω) → aω + b = α + (ρ/ϵ0) az + b .

where we have performed the linearization in the third step, from left to right, in the schematic sequence above, keeping the factor α + (ρ/ϵ0) explicit.

Corresponding to an ammonia molecule, as discussed previously.

In this situation, where we use the linear approximation above, we have a simpler expression for the electrical interaction matrix (i.e., the interaction between the induced electric dipole mo-ment and the external electrostatic field):

σ33 α + (az + b) −A

′′ ϵ0

H =   . (2.7)

linear ρ

−A −σ33 α + (az + b)

Thus, for the approach and assumptions considered, the stationary equation, ! ( ) ! !

φ1 1 2 2 ′′ φ1 φ1 H ˆ = (P ˆ + P ˆ )I + H = E , (2.8)

x z linear

φ2 2m φ2 φ2

where I is the identity matrix, can be written explicitly as a system of two coupled equations for the components φ1 and φ2 of the (non-relativistic) spinor,

1 2 2 ρ (P ˆ + P ˆ ) + σ α + ( ) (az + b) φ − Aφ = E φ (2.9)

x z 33 1 2 1 1

2m ϵ0

1 2 2 ρ (P ˆ + P ˆ ) − σ α + ( ) (az + b) φ − Aφ = E φ (2.10)

x z 33 2 1 2 2

2m ϵ0

or, after grouping like terms, we have:

1 2 2 ρ (P ˆ + P ˆ ) + a σ α + ( ) z φ − Aφ = E ˜ φ (2.11)

x z 33 1 2 1 1

2m ϵ0

1 2 2 ρ (P ˆ + P ˆ ) − a σ α + ( ) z φ − Aφ = E ˜ φ . (2.12)

x z 33 2 1 2 2

2m ϵ0

where the eigenenergies are redefined (which now include a dependence on the gradient), as follows, E ˜ ≡ E − σ α + ( ) b, (2.13)

E ˜ ≡ E + σ α + ( ) b. (2.14)

Equations (2.11) and (2.12) can also be written compactly as,

O ˆ φ − Aφ = E ˜ φ (2.15)

1 1 2 1 1

O ˆ φ − Aφ = E ˜ φ (2.16)

2 2 1 2 2

where O ˆ is the corresponding differential operator. In the next section we will construct the solutions to equations (2.11) and (2.12) assuming the particular case when there is no coupling (A = 0) between them.

the state vector

Before we begin to solve equations (2.11) and (2.12), for A = 0, it should be noted that our purpose is to determine whether or not the gradient of the electrostatic field contributes to the eigenenergies of the physical system considered.

The independent equations that we have to consider are:

1 2 2 ρ (P ˆ + P ˆ ) + a σ α + z φ = E ˜ φ , (3.1)

x z 33 1 1 1

2m ϵ0

In [10] what was initially considered a system of two coupled Schrödinger equations was considered, showing that they admitted exact decoupling, but for two functions distinct from the initial ones.

1 2 2 ρ (P ˆ + P ˆ ) − a σ α + z φ = E ˜ φ . (3.2)

x z 33 2 2 2

2m ϵ0

We begin with equation (3.1). Assuming that a solution has the form corresponding to the separation of its variables, we write:

φ1(x, z) = X(x) Z(z), (3.3)

Next, since the incident beam moves along the X direction and the unfolded beams move away from each other symmetrically with respect to the same direction, we will impose, given

(i/ℏ)pxx

the presence of the leftmost term in (3.1), that: X(x) = e , where px is an eigenvalue of P ˆ . In this case, we obtain the following equation for the function Z:

( ! ) ′′ 1 2m px Z (z) − 2m a σ α + (ρ/ϵ ) z − E ˜ − Z(z) = 0 (3.4)

2 33 0 2 1

ℏ ℏ 2m

where, for brevity, we write:

1 2m px λ ≡ 2m a σ α + (ρ/ϵ ) , λ ≡ E ˜ − . (3.5)

2 33 0 1 2 1

ℏ ℏ 2m

Note that the expression in (3.4) is an equation not so different from the Airy equation. We can

1/3 −2/3

arrive at the same Airy equation by changing the variable: ζ = λ z − λ1λ ; that is, we go from (3.4) to the equation:

Z (ζ) − ζZ(ζ) = 0, (3.6)

the well-known Airy equation; that is, Z ≡ Ai. Then we have, from (3.3):

(i/ℏ)pxx 1/3

φ1(x, z) = e Ai λ z + ξ0 . (3.7)

where ξ0, representing the coordinate of the point of an absolute maximum of the Airy function (Ai), is incorporated in (3.7) to move this maximum to the origin of coordinates (the same procedure will be done, but symmetrically, for the function φ2), so that the absolute maximum of φ1 and φ2 coincide since their separation is only due to their interaction with the electrostatic field. Note, according to what has already been seen, that one would expect the function Ai in

1/3 −2/3

(3.7) to have as an argument the expression (λ z−λ1λ ), but this is not the case; the correct expression is (λ z + ξ0).

On the other hand, it is known that the Airy function presents, on one of its sides, decreasing oscillations that do not vanish at infinity and, on its opposite side, a continuously decreasing behavior that begins at its absolute maximum. Also, from (3.7), we have that the function: |φ1| = |Ai| is not integrable. Therefore, to represent a physically acceptable wave function, we should eliminate the part of the function Ai that produces this inconvenience. Using the same resource used in [13], that is: we appropriately multiply the function in (3.7) by the Heaviside function (Θ).

(i/ℏ)p x 1/3 λ0 φ1(x, z) = e Ai λ z + ξ0 Θ z − + ξ0 , (3.8)

which cuts the Airy function at its first zero, λ0.

With regard to determining the eigenenergy E ˜ , or E , it is sufficient to consider (3.7). So, after calculating the spatial derivatives of the function in (3.7), we can write:

ℏ ∂ φ1 ∂ φ1 ρ − + + al σ33 α + z φ1 = 2m ∂x2 ∂z2 ϵ

2 2 2/3

px 1/3 (i/ℏ)p x ℏ λ ′′ 1/3 (i/ℏ)p x = Ai λ z + ξ0 e − Ai λ z + ξ0 e + 2m 2m ρ 1/3 (i/ℏ)p x + al σ33 α + z Ai λ z + ξ0 e . (3.9) In (3.9), we will add and subtract the quantity:

n 2 o1/3 2 2 ρ 2 1/3 (i/ℏ)p x ξ0 a σ33 α + ℏ /2m Ai λ z + ξ0 e , (3.10)

Then we have: !

ℏ ∂ φ1 ∂ φ1 ρ − + + a σ33 α + z φ1 = 2m ∂x2 ∂z2 ϵ

2 . 1/3 p ρ 2

x 2 2 2 1/3 (i/ℏ)pxx

= − ξ0 a σ33 α + ℏ 2m Ai λ + ξ0 e + 2m ϵ0

2 2/3

ℏ λ ′′ 1/3 (i/ℏ)p x − Ai λ z + ξ0 e +

. 1/3 ρ ρ 2

2 2 2 1/3 (i/ℏ)pxx

+ al σ33 α + z + ξ0 a σ33 α + ℏ 2m Ai λ z + ξ0 e . (3.11) ϵ0 ϵ0

or, also: !

ℏ ∂ φ1 ∂ φ1 ρ − + + a σ33 α + z φ1 = 2m ∂x2 ∂z2 ϵ

2 . 1/3 p ρ 2

x 2 2 2 1/3 (i/ℏ)pxx

= − ξ0 a σ33 α + ℏ 2m Ai λ z + ξ0 e + 2m ϵ0

2 2/3

ℏ λ ′′ 1/3 1/3 − Ai λ z + ξ0 − κ Ai λ z + ξ0 (3.12)

where,

( ρ ) 2 2 ρ 2 !

2m a σ α + a σ α + ℏ 1/3 33 ϵ 2m 33 ϵ κ = z + ξ0 . (3.13) 2 2/3 2 2/3 2m ℏ λ ℏ λ

Considering (3.5), we find that κ is rewritten as:

κ = λ z + ξ0. (3.14)

Thus, equation (3.12) can be written as follows:

ℏ ∂ φ1 ∂ φ1 ρ − + + a σ33 α + z φ1 = 2m ∂x2 ∂z2 ϵ

2 . 1/3 p ρ 2

x 2 2 2 1/3 (i/ℏ)pxx

= − ξ0 a σ33 α + ℏ 2m Ai λ z + ξ0 e + 2m ϵ0

2 2/3

ℏ λ ′′ 1/3 1/3 1/3 − Ai λ z + ξ0 − λ z + ξ0 Ai λ z + ξ0 . (3.15)

But, the second term on the right-hand side in equation (3.15) equals zero, since Ai λ z + ξ0 satisfies the Airy equation. Therefore, we have the simplified expression:

ℏ ∂ φ1 ∂ φ1 ρ − + + a σ33 α + z φ1 = 2m ∂x2 ∂z2 ϵ

2 . 1/3 p ρ 2

x 2 2 2

= − ξ0 a σ33 α + ℏ 2m φ1. (3.16) 2m ϵ0 Comparing (3.16) with (3.1), we have that the eigenenergy associated with the function φ1 is given by:

( 2 2 2 )1/3 2 a σ α + (ρ/ϵ0) ℏ p 33 E1 = − ξ0 . (3.17) 2m 2m

Replacing the expression E1 ≡ E1 − σ33 α + ( ) b, found in (2.13), in expression (3.17), we obtain the following formal expression for the eigenenergy E1:

( 2 2 2 )1/3 2 a σ α + (ρ/ϵ0) ℏ p ρ 33 E1 = + σ33 α + ( ) b − ξ0 . (3.18) 2m ϵ0 2m

Following a similar procedure, we obtain for φ2:

(i/ℏ)p x 1/3 λ0 φ2(x, z) = e Ai − λ z + ξ0 Θ − z − + ξ0 , (3.19)

With eigenenergy:

( 2 2 2 )1/3 2 a σ α + (ρ/ϵ0) ℏ p ρ 33 E2 = − σ33 α + ( ) b − ξ0 , (3.20) 2m ϵ0 2m

where it is clear that the eigenenergies depend on the magnitude of the gradient α and the electric charge density ρ. These results are valid only within the context of linearization that we have assumed.

In the context that we have considered, we note, on the one hand, that in (3.18) and (3.20) there are two types of contributions of the electrostatic field gradient to the eigenenergies; on the one hand, the term:

± σ33 α + ( ) b,

produces the energetic separation between the molecules, in relation to the energies of the trans-lational movement, while the term,

( 2 2 2 )1/3 a σ α + (ρ/ϵ0) ℏ − ξ0 ,

produces the same displacement of the energetic levels unfolded by the linear term in: α+(ρ/ϵ0).

In the case with a very weak gradient, we have that the spatial separation of the initial beam of molecules into two beams, in the plane Y = 0, in the region where the electric field acts, would be small.

On the other hand, we have from (3.18) and (3.20), that for the values of the geometric parameters a = 0 and b = 0, only the kinetic energy term remains in the eigenenergies, which means that these values would characterize, in an operational manner, and after the linearization of the quadratic term, the absence of an electric field acting on the beam of molecules; and then we would have to:

px E1 = E2 = .

Also note an interesting fact. In the magnetic case, the density of magnetic charges can only have a null value , so that taking the value α = 0 is enough to cancel the gradient effect

Since magnetic monopoles do not exist as elementary particles; however, magnetic monopoles resulting from a collective effect have been experimentally observed in the structure called “spin ice” [28].

on the eigenenergies in the Stern-Gerlach effect [9]. On the other hand, in the “electric Stern-Gerlach” case, the density of electric charges is non-zero; thus, α = 0, alone, could not cancel the additional effect of displacement in the energy levels given in (3.18) and (3.20), unless, together with a null value attributed to the density of electric charges ρ = 0 (which would then not generate an electrostatic field). In other words, in the case of the “electric Stern-Gerlach” effect, the gradient effect on eigenenergies becomes a combined effect of the gradient and the density of electric charges, a type of result that the authors claim to be unaware of in the general literature on quantum mechanics.

This article considers a beam of ammonia molecules within a region where an inhomogeneous electric field acts. These molecules were modeled as quantum particles that can access only two stationary states; without internal structure, however, with a permanent electric dipole moment. Assuming a weak gradient regime and via a linearization process of a quadratic term, we obtain explicit analytical expressions, (3.8) and (3.19), for the eigenfunctions of the reduced Hamil-tonian (2.7), as well as for the corresponding eigenenergies, (3.18) and (3.20), which present explicit dependence on the gradient of the electrostatic field and the density of electric charges that generate this field, results that do not appear in the references [14, 15, 16].

The results found for the eigenenergies, in particular, show that the analogy between the spatial separation observed in the Stern-Gerlach effect and the spatial separation of a beam of ammonia molecules, into two secondary beams due to their interaction with this electrostatic field, is broader than considered, since an analogy can also be established between the corre-sponding eigenenergies, in the sense of the manifestation of the gradient effect in both cases.

The spatial separation of the incident beam into two secondary beams, in the “electric Stern-Gerlach” effect, results from the molecules presenting a permanent electric dipole moment. Here, we were particularly interested in the energetic part of this effect, where it was sufficient to consider the induced electric dipole moment.