Chapter 15. Bohr's Model of the Hydrogen Atom

Early in the twentieth century, Niels Bohr developed a model for the hydrogen atom. He based his model upon the work of E. Rutherford, J. Nicholson, and J. Balmer among others. It is successful for explaining the basic mechanics of the hydrogen atom and served as the basis for the quantum-mechanical theory, which succeeded it.

Bohr stated that the electron in the hydrogen atom can be bound in a particular circular orbit, n, revolving about the proton, which is the atom's nucleus. The orbit can be relatively close to the nucleus or far away, depending upon the potential electromagnetic energy U_n of the hydrogen atom. This potential energy possesses particular, stepped levels of magnitude and corresponding concentric orbits. All other orbits and energy levels are excluded. From the inner to the outer, we number the orbits as n = 1, 2, 3, 4, ..., i-1, i, i+1, ..., j-1, j, j+1, ....

Just as a free electron makes a light-speed quantum jump over a distance of λ_e in a time span of t_e, so does a bound electron make quantum jumps but, as this chapter shows, at particular, stepped, subluminal speeds.

A. Speed of a Bound Electron

When Bohr created his model, the majority of the magnitudes of the quantum attributes of a free electron had yet to be discovered. He was forced to use the FUPCONs that existed at the time, such as Planck's constant, h, and the permittivity constant, ε₀, which, erroneously, were considered to be true FUPCONs. Therefore, the formulas for the dynamic attributes of an electron that is bound to the hydrogen atom's nucleus were complicated and convoluted.

As an example, we examine the historical formulas for the orbit length, λ_n, and orbit period, t_n, of the bound electron, which, respectively, are:

λ_n = 2 n² h² ε₀ (m_e·q_e²)^-1 (138

and

t_n = 4 n³ h³ ε₀² (m_e·q_e⁴)^-1 (139

These two equations contain Q and M dimensions in addition to all of those dimensions in Planck's and the permittivity constants. Yet, we can reduce each equation to a value possessing, respectively, a single dimension of L and of T. However, instead of performing that tedious process, let us reduce their much simpler quotient, which is the speed of the bound electron, v_n.

The historical formula is:

v_n = λ_n t_n^-1 = [2 n² h² ε₀ (m_e·q_e²)^-1] ×

[4 n³ h³ ε₀² (m_e·q_e⁴)^-1]^-1 = q_e² (2 n h ε₀)^-1 (140

which converts to using the quantum attributes of the electron, as follows:

v_n = q_e² {2 n [h] [ε₀]}^-1 (141

v_n = q_e² {2 n [E_e·t_e] [β q_e² (2 E_e·λ_e)^-1]}^-1 (142

v_n = λ_e (n β t_e)^-1 = c (n β)^-1 (143

Therefore, in the Bohr model of the hydrogen atom, we see that the electron, in its lowest-energy state, revolves around the proton at a speed of one-137^th the speed of light. In higher-energy states, that speed is reduced by dividing the electron's orbit number into it.

The historical formulas for the other dynamic attributes of a bound electron take forms that are similar to those in Equations 138 and 139. Because the process of converting their FUPCONs to electronic attributes is also similar to that for the orbit length and period, we do not concern ourselves with it. Rather, we forge ahead into new concepts.

B. Multiplication Factors

Two multiplication factors affect the bound electron that do not affect free electrons. These factors exist because of the disparity between the magnitudes of the SE inertial, f_ie, and electromagnetic, f_ee, units of force, which introduces the fine-structure constant, β, and the influence of the electron's orbit, which introduces the orbit number, n.

Whenever an equation, which deals with the Bohr model of the hydrogen atom, contains the SE unit of length, λ_e, or time, t_e, or the fine-structure constant, β, we must include the orbit number as a factor. For example: (n λ_e), (n t_e), and (n β). In any orbit, the mass, m_e, of the bound electron remains the same; therefore, we do not multiply it by n.

C. Bohr Units of Length and Time

The historical formulas for the orbit length and period as shown in Equations 138 and 139 convert, respectively, to ones that use electronic attributes, as follows:

λ_n = n² β λ_e = (n β) (n λ_e) (144

t_n = n³ β² t_e = (n β)² (n t_e) (145

We can create and define, respectively, Bohr units of length and time to be the electron-orbit length and period.

D. Attributes of the Bound Electron

By using the SE unit of mass and the Bohr units of length and time, rather than their SE units, the format of the attributes of free and bound electrons are the same, as follows:

Orbit Attribute	Usingλ_e and t_e	Usingλ_n and t_n
Length, `λ_n`	`(n β)(n λ_e)`	`λ_n`
Radius, `r_n`	`(n β)(n λ_e)(2 π)^-1`	`λ_n (2 π)^-1`
Period, `t_n`	`(n β)² (n t_e)`	`t_n`
Rotational frequency, `ω_n`	`(n β)^-2(n t_e)^-1`	`t_n^-1`
Linear speed, `v_n`	`λ_e(n β t_e)^-1`	`λ_n·t_n^-1`
Linear momentum, `p_n`	`m_e·λ_e (n β t_e)^-1`	`m_e·λ_n·t_n^-1`
Angular momentum, `L_n`	`n m_e·λ_e² (2 π t_e)^-1`	`m_e·λ_n² (2 π t_n)^-1`

See Table XIV in Appendix E.

E. Bohr Units and SE Units

Upon examination of the characteristics of the Bohr model when using electronic attributes, we notice their simplicity in comparison with the historical presentation, which uses FUPCONs. Many of the FUPCONs disappear. Among these are Planck's, h, and the permittivity, ε₀, constants. Also, electron charge, q_e, and mass, m_e, disappear from the classical definitions of attributes that contain only dimensions of length and time.

We simplify the presentation even further by using the Bohr units of length, λ_n, and time, t_n, instead of their SE units.

F. Bohr-Model Unit of Energy

Let us define the Bohr-model unit of energy, as follows:

E_n = m_e [(n β) (n λ_e)]² [(n β)² (n t_e)]^-2 (146

E_n = m_e·λ_n² (n β t_e)]^-2 = m_e·λ_n²·t_n^-2 (147

We use this factor, E_n, in the definitions of the various energies that are found in the hydrogen atom, as follows.

Energies of the Hydrogen Atom

The magnitudes of the various energies of the hydrogen atom with relation to the Bohr-model unit of energy, E_n, are:

Type of Energy	Using λ_e and t_e	Usingλ_n and t_n
Bohr unit, `E_n`	`m_e·λ_e² (n β t_e)^-2`	`m_e·λ_n²·t_n^-2`
Kinetic, `K_n`	`+ ½ m_e·λ² (n β t_e)^-2`	`+ ½ E_n`
Potential, `U_n`	`- m_e·λ² (n β t_e)^-2`	`- E_n`
Total, `E_T`	`- ½ m_e·λ² (n β t_e)^-2`	`- ½ E_n`

See Table XV in Appendix E.

The magnitudes of all of the energies, E_T, K_n, and U_n, approach zero as n increases because the factor, (n β), exists in the denominator (see Equation 147). In sum, when the distance between the proton and the electron in the hydrogen atom is very large, they are independent particles having no effect upon each other. Their interdependent energies transfer themselves elsewhere.

G. Photon-Emission Spectrum

Whenever an electron that is bound to the nucleus of a hydrogen atom changes from a high orbit, j, to a lower orbit, i, the potential energy of the hydrogen atom decreases. This energy loss transforms into electromagnetic energy in the form of a photon, which possesses the same quantity of energy as that lost by the hydrogen atom. The photon's energy, E_ij, is equal to E_j minus E_i, as follows:

E_ij = E_j - E_i = [- ½ E_e (j β)^-2] - [- ½ E_e (i β)^-2] =

+ ½ E_e (β)^-2 (i^-2 - j^-2) (148

Table XVI in Appendix E pertains to the contents of the rest of this chapter.

Frequency of the Emitted Photon

Historically, a photon's frequency, ν_ij, is equal to its energy, E_ij, divided by Planck's constant, h; therefore:

ν_ij = E_ij (E_e·t_e)^-1 = ½ (β)^-2 (i^-2 - j^-2) t_e^-1 (149

where the factor, E_e·t_e, is Planck's constant.

Wavelength of the Emitted Photon

Photons travel at the speed of light, c; therefore, the wavelength, λ_ij, is equal to c divided by the frequency ν_ij. Because the reciprocal of the difference between the reciprocals of the factors, i² and j² in the frequency equation, ν_ij, is mathematically cumbersome, we use the reciprocal of the wavelength, λ_ij^-1, instead:

λ_ij^-1 = ν_ij c^-1 = ½ β^-2 (i^-2 - j^-2) λ_e^-1 (150

The only differences between the frequency, ν_ij, and the reciprocal of the wavelength, λ_ij^-1, are, respectively, the factors, t_e and λ_e.

The expression:

R_î = ½ (β)^-2 λ_e^-1

is the SE formula for the Rydberg constant. The historical SI formula is:

R_î = m_e·q_e⁴ (8 h³ c ε₀²)^-1 (151

and contains quite an incomprehensible jumble of FUPCONs. When we compare the SE-based formula with the historical SI-based one, the simplicity of the former one is apparent.

Correspondence between `ν_ij` and `ω_n`

The difference between frequencies ν_n-1 and ν_n of adjacent orbits becomes less as the orbit number, n, increases:

(n - 1)^-2 - n^-2 = [n² - (n² - 2n + 1)] [n² (n - 1)²]^-1 =

[2n - 1] [n² (n - 1)²]^-1 =

[2 (n - ½)] [n² (n - 1)²]^-1 (152

Let n approach infinity, î, so that the factors, (n - ½), and, (n - 1), approach (n), then:

[2 (n - ½)] [n² (n - 1)²]^-1 = 2n (n² n²)^-1 = 2 n^-3 (153

where n approaches î.

Therefore, at high values for n:

ν_n-1,n = ω_n = ½ (β)^-2 (2 n^-3) t_e^-1 = (β)^-2 n^-3 t_e^-1 =

[(n β)² (n t_e)]^-1 = t_n^-1 (154

where n approaches î.

We see that the orbital and photonic frequencies approach each other for large values of n.