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Advanced Studies in Theoretical Physics Vol. 13, 2019, no. 8, 433 - 437 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2019.91246

The Extent of Acceptance of the Non-separable Solution in Cylindrical Coordinates through the Hydrogen

Mohammad F. Alshudifat

Dept. of Physics, Al al-Bayt University, Mafraq 25113, Jordan

This article is distributed under the Creative Commons by-nc-nd Attribution License. Copyright c 2019 Hikari Ltd.

Abstract

The has been used to measure the extent of accep- tance of the non-separable solution in cylindrical coordinates. This work is not representing a new solution for the hydrogen atom, but to see how the non-separable solution in cylindrical coordinates gives acceptable results when compared with the known separable solution in spherical coordinates. The current work shows that the energy states (labeled by n = 1, 2, 3, ...) were found to be consistent with the Rutherford-. The azimuthal number ` was found to have a single value for each energy state, while the magnetic m was found to have two allowed values ` and ` − 1 for each energy state, this shrank the degeneracy of the excited states (n`m) to only two different states (n``) and (n` ` − 1), this result veers from the degeneracy cal- culated using spherical coordinates. A comparison between this work and the known solution in spherical coordinates show some acceptable results especially for the energy of atomic states, which is promising to use cylindrical coordinate in quantum problems when the spherical symmetry is deformed.

Keywords: Cylindrical coordinates, non-separable solution, hydrogen atom, energy states, azimuthal quantum number, angular momentum 434 Mohammad F. Alshudifat

1 Introduction

Schrodinger equation is used to extract the properties of the physical states of atomic and subatomic systems. Problems with spherically symmetric po- tentials are usually treated using spherical coordinates. This work used the non-separable Shrodinger equation in the cylindrical coordinates to reexamine the properties of the hydrogen’s atomic states and to see how the accept- able are these properties to the known properties in the spherical coordinates. Non-separable Schrodinger equation of the hydrogen atom has been solved in cylindrical coordinates by L. Gold [1], but the solution was corresponding to the S states only. This work generalizes the non-separable solution of the hy- drogen atom in cylindrical coordinates, where the energy, and degeneracy of each and eigenvalue of the angular momentum L2 have been calculated and discussed in the results.

2 The hydrogen atom in cylindrical coordi- nates

Start from Schrodinger equation of hydrogen atom in spherical coordinates

 h¯2 e2 1 ∇2 + ψ(r) = k2ψ(r), (1) 2m 4πo r where r−2m E k = . h¯2 In cylindrical coordinates, r = ρ + z. Assume ρ → kρ, z → kz, and s −2m  e2 2 ξ = , E 4πoh¯

Eq. (1) in cylindrical coordinates becomes ! ∂2 1 ∂ ∂2  1 ∂2  ξ + + − 1 − + ψ = 0. (2) ∂ρ2 ρ ∂ρ ∂z2 ρ2 ∂φ2 pρ2 + z2

Equation (2) is separable in φ and its solution is Φ(φ) = exp(im φ), where m can be 0, ±1, ±2, .... Equation (2) becomes " #! ∂2 1 ∂ m2 ∂2 ξ + − + − 1 − ψ = 0. (3) ∂ρ2 ρ ∂ρ ρ2 ∂z2 pρ2 + z2 The extent of acceptance of the non-separable solution ... 435

Assume the solution for Eq. (3) is  p  ψ(ρ, z) → F (ρ, z) exp − ρ2 + z2 , so the solution vanishes as r = pρ2 + z2 → ∞. Substitute in Eq. (3)

∂F ∂F m2 ∂2F ∂2F 1 ∂F p (ξ − 2) F − 2 ρ − 2 z = F − − − ρ2 + z2. (4) ∂ρ ∂z ρ2 ∂ρ2 ∂z2 ρ ∂ρ

Since the right hand side of Eq. (4) is multiplied by pρ2 + z2, an acceptable solution that both sides of Eq. (4) are equal zero, we end up with separable partial differential equations. Assume F (ρ, z) = R(ρ)Z(z), the right hand side gives d2Z + α2Z = 0, (5) dz2 and d2R 1 dR m2  + − − α2 R = 0, (6) dρ2 ρ dρ ρ2 where α is a constant. While the left hand side gives 2ρ dR 2z dZ (ξ − 2) − − = 0. (7) R dρ Z dz

The z-part of Eq. (7) forces α to be zero. Accordingly, the solution of Z(z) and R(ρ) are 1 X s Z(z) = csz , s=0 and m −m R(ρ) = c2ρ + c3ρ ,

The constant c3 must be zero so R(ρ) will not blow up as ρ → 0. It turns out that the of the hydrogen atom in cylindrical coordinates is √ m s − ρ2+z2 imφ ψnsm(ρ, z, φ) = Ansm ρ z e e . (8)

When the Hamiltonian operator (Eq. (2)) acts on the wave function (Eq. (8)) it gives √  2 2 ((ξ − 2(m + s + 1)))zse − ρ +z = 0. (9) pρ2 + z2 The solution of this equation indicate that

ξ − 2(m + s + 1) = ξ − 2n = 0, (10) 436 Mohammad F. Alshudifat where n = m + s + 1. (11) The energy equation of the hydrogen atom states is derived from Eq. (10)

"  2 2# me e 1 En = − 2 2 , 2¯h 4πo n and it is consistent with the Rutherford-Bohr model.

3 Angular momentum

The orbital angular momentum operator L in cylindrical coordinates is

 −z ∂  ∂ ∂  ∂  L = −ih¯ rˆ + φˆ z − r + kˆ , r ∂φ ∂r ∂z ∂φ where the scalar L2 operator can be calculated to be " #  ∂ ∂ 2 ∂ z2 ∂ z 2  ∂2 L2 = −h¯2 z − r − z + + + 1 , (12) ∂r ∂z ∂z r ∂r r ∂φ2 or " #  ∂ ∂ 2 ∂ z2 ∂ z 2  L2 = −h¯2 z − r − z + − + 1 m2 . (13) ∂r ∂z ∂z r ∂r r

When L2 operator acts on the hydrogen wave function (Eq. (8)), the resul- tant eigenvalue is

(m + s) ((m + s) + 1)h ¯2 = `(` + 1)¯h2, (14) where ` = m + s is the azimuthal quantum number. Accordingly, the wave equation Eq. (8) becomes

pρ2 + z2  ρ m  z `−m − ψ (ρ, z, φ) = A e na eimφ, (15) n`m n`m na na where the previously defined k constant was written in term of the Bohr ra- dius a as 1 k = . na The extent of acceptance of the non-separable solution ... 437

4 Conclusion and Discussions

So far, the hydrogen atom states have been reexamined in cylindrical coordi- nates. The energy of the states was found to be consistent with the Rutherford- Bohr model. Table 1 shows the first three energy states with the related quantum numbers and number of degeneracies. According to these quantum numbers, the azimuthal quantum number ` became a single value for each state and equal to n − 1, which assign a unique value of angular momentum for each energy state, while the m degeneracy re- duced to two values ` and ` − 1 each energy state, this shrank the degeneracy of the excited states to twofold degeneracy, while the is not de- generate. Although some of these results are veer from the known atomic state properties of the hydrogen atom, the main properties like state’s energy and the eigenvalue of L2 are staying the same, this conclusion makes the non- separable solution in cylindrical coordinates acceptable for the sake of general states properties especially when the spherical symmetry of the potentials are deformed like the interaction potential of deformed nuclei.

Table 1: The ` and m quantum numbers versus the first three energy states of the hydrogen atom using cylindrical coordinates, these results are veer from the correct results calculated using spherical coordinates

Energy state n ` m ψn,`,m(r, z, φ) Num. Degeneracy Ground state 1 0 0 ψ1,0,0 1 First excited state 2 1 0 ψ2,1,0 1 ψ2,1,1 2 Second excited stat 3 2 1 ψ3,2,1 2 ψ3,2,2 2

References

[1] Louis Gold, Wave mechanics for hydrogen atom in cylindrical coordinates: Non-separable eigensolutions, Journal of the Franklin Institute, 268 (2) (1959) 118-121. https://doi.org/10.1016/0016-0032(59)90455-7

Received: December 15, 2019; Published: December 30, 2019