Zero Point Energy and the Charge-Radiation Equation

Gauge Institute Journal Volume 8, No.4, November 2012 H. Vic Dannon 

Zero Point Energy, and the 

Charge-Radiation Equation 

in Bohr’s Atom 

H. Vic Dannon 

vic0@comcast.net 

September, 2012 

Abstract Bohr’s postulate of Quantized Angular Momentum is 

equivalent to the assumption of Zero Point Energy for the 

Hydrogen Atom. 

The confirmation of Bohr’s Atom in spectroscopy, validates the 

existence of Zero Point Energy for the Hydrogen Atom 

For an electron that orbits a proton at the first Bohr orbit r , at 

frequency ν , the following are equivalent 

B 

1) The Angular Momentum is quantized, mv e Br B = � . 

2) The Fine-Structure constant Formula, 

3) The Charge-Radiation Equation, 

4) Ground State Energy is Zero Point Energy 

1 

α 

2 

e 

= 

4πε 

c� 

. 

2 

e 

4πε 

r 

0 

B 

0 

2 

−e 

8πε 

r 

B 

0 

= hν 

. 

B 

1 

2 

B 

=− hν 

, 

B


ν = 6.576928348 × 10 cycles/second 

B 

The Electron’s Magnetic Energy in the Bohr Orbit is 

magnetic 0 B B 2 0 B r 

15 

1 2 2 1 2 2 

μ ν μ v e 

4 (4 π) 

B 

U = r e = 1 

It is negligible compared to its Electric Energy. 

For an electron in the n th Bohr orbit , at frequency , ν 

the following are equivalent 

rn n 

1) The Angular Momentum is quantized, mvr = n�. 

2) The Charge-Radiation Equation, 

3) Orbit Energy is of n photons, 

hνn 

e nn 

2 

e 

4πε 

r 

0 n 

2 

−e 

8πε 

r 

= nhν 

. 

0 n 

n 

=− nhν 

. 

Keywords: Zero Point Energy, Bohr’s Atom, Quantized Angular 

Momentum, Fine Structure Constant, Electron, Photon, Proton, 

Binding Electric Energy, Radiation Energy, Charge-Radiation 

Equation 

2000 Physics and Astronomy Classification Scheme 

03.50.Kk, 03.65.-w. 

2 

1 

2 

� 

1 

n 

2 

hν 

n 

B


Introduction 

In 1901, Planck showed that the radiation-energy density per unit 

volume at frequencies between ν , and ν + δν of an ideal radiator 

(black body) is 

8 2 h 

u(, T) 

3 h 

c e kT 1 

ν 

π ν 

ν = ν . 

− 

The assumption of discrete radiation energy, conflicted with 

Planck’s belief in radiation of continuous waves, and he kept 

searching for a more believable law 

To reconcile his quantum hypothesis with his conception of wave 

radiation, he avoided the conclusion that radiation energy must be 

made of particles, and postulated that radiation is a transition 

between the energy levels of an oscillator. Furthermore, ignoring 

the symmetry between emission and absorption, he maintained 

that the absorption of radiation energy is continuous. 

Under these assumptions, Planck derived in 1912 his second 

radiation law in which zero point energy in the amount of 1 hν is 

2 

added to 

h 

h 

e kT 1 

ν 

ν 

, of his 1901 radiation law. 

− 

In [Dan], we showed that Planck’s derivation of his 1912 radiation 

law only recovers the Zero Point Energy that he unknowingly 

3


assumed in his model for that derivation. 

In particular, we showed that Planck’s ZPE radiation law is 

equivalent to the combined three assumptions of 

1) Zero Point Energy Hypothesis, 

2) the Quantum Law, and 

3) the approximated Boson Statistics distribution law. 

Planck’s 1901 radiation law resolves the Black body radiation 

problem, and confirms the boson statistics distribution law. 

The Quantum law holds independently in the photoelectric effect, 

in Compton’s scattering, and in spectroscopy. 

But Planck’s Zero Point Energy remains a hypothesis. Thus, the 

validity of Planck’s 1912 radiation law, and the existence of 

Planck’s Zero Point Energy are doubtful. 

However, we will show here that Bohr’s postulate of Quantized 

Angular Momentum is equivalent to the assumption of Zero Point 

Energy for the Hydrogen Atom. 

The confirmation of Bohr’s Atom in spectroscopy, validates the 

existence of Zero Point Energy for the Hydrogen Atom. 

0.1 Bohr’s Quantized Angular Momentum 

Consider an electron with mass m , and charge e , in the first 

Bohr orbit of radius r r , at frequency ν = ν , and speed 

1 = B 

1 B 

4 

e


v = v = ω r = 2πν 

rB 

, 

1 

B B B B 

encircling a proton with charge −e 

, 

An electron with charge e , orbiting at frequency ν , and speed 

v = αc = ωr = 2πν 

, a proton with charge e at distance . r 

1 1 1 1 r 1 

1 

− 1 

Bohr postulated that the angular momentum is quantized: 

For the electron’s first, Bohr’s orbit, n = 1, 

and 

mvr 

= � = . 

h 

e 11 2π 

For the electron’s 2 nd orbit, n = 2 , and 

mvr 

2 2 h = � = , 

e 2 2 2π 

For the 3 rd electron orbit, n = 3, 

and 

mvr 

3 3 h = � = , 

e 3 3 2π 

……………………………………….. 

5


Angular Momentum is the vector product 

� � � � � 

mv × r = m( ω × r ) × r , 

and the quantization of that quantity cannot be visualized like the 

quantization of a scalar quantity, such as energy. 

We aim to show that Quantized Angular Momentum is equivalent 

to the assumption of Zero Point Energy for Bohr’s Atom. 

We’ll first show that Quantized Angular Momentum is equivalent 

to the Fine-Structure constant. 

0.2 The Fine-Structure Constant 

Sommerfeld suggested that 

α 

2 

e 

= 

4πε 

c� 

, 

determines the spacing between the fine structure of spectral 

lines. 

α , known as the Fine Structure Constant, is a pure number that 

is approximately 1 

137 

in any unit system. 

The fine-structure constant depends on fundamental constants, 

� e , the electron charge, which was thought to be the smallest 

electric charge, 

� c , light speed in the vacuum, which is thought to be the 

greatest speed, 

6 

0


� h = 2π� 

, Planck Constant of radiation energy, 

ε = , the permittivity of the vacuum. 

1 � 0 2 μ0c 

Consequently, it was suspected to hide some truth of nature. 

Eddington, Pauli, and their contemporaries [Miller] searched for a 

meaning for the appearance of these fundamental constant in one 

formula. 

We show that the Formula for the Fine Structure Constant can be 

written in a form that equates the electron-proton binding electric 

energy to the radiation energy of a photon. 

We call that equivalent form the Charge-Radiation equation. 

0.3 The Charge-Radiation Equation 

The binding electric energy 

e 

2 

4πε0rB of the electron in the first 

Bohr orbit with frequency ν , equals the radiation energy 

rB B 

hνB B 

of a photon of frequency ν . 

The binding electric energy of the Hydrogen Atom in the ground 

state is precisely the radiation energy of a photon with frequency 

that equals the frequency of the electron motion. 

The Charge-Radiation equation is equivalent to the equality of 

the electron’s energy in the ground state to 1 hν of Zero Point 

7 

2 

B


radiation Energy. 

0.4 The Zero Point Energy of Bohr’s Atom 

The Zero Point Energy of Bohr’s Atom is the Ground state energy. 

Its value, 1 hν is the lowest energy of the Bohr Atom. 

2 

B 

The loss of that Zero Point Energy, would mean the destruction of 

the Atom. 

Therefore, at absolute zero temperature, when all thermal 

motions cease, the atom still retains that amount of energy. 

The existence of Zero Point Energy has been always difficult to 

confirm in experiments. But the confirmation of Bohr’s 

quantization of Angular Momentum in spectroscopy, validates 

Zero Point Energy in the amount of 1 hν , for the Bohr Atom. 

8 

2 

B


1. 

Quantized Angular Momentum 

and the Fine Structure Constant 

1.1 

Proof: 

mv r 

B B 

e 

= � ⇔ α = 

4πε 

c� 

Ignoring the magnetic Field of the proton on the electron’s charge, 

the centripetal repulsion is balanced by the electric attraction 

2 

v 2 

B 1 e 

e 

4 2 

B πε0 

B 

m = . 

r r 

1 

mv r v = e 

�� 4πε 

e B B 

� 

B 

v 

B 

0 

2 

0 

1 e 

= 

4πε 

� , 

v 2 

B 1 e 

= 

�c 

4πε 

� c 

. 

α 

9 

0 

2 

, 

2 

0


2. 

The Fine-Structure Constant and 

the Charge-Radiation Equation 

2.1 

Proof: 

Hence, 

Therefore, 

2 2 

e e 

α = ⇔ = hν 

4πε c�4πε r 

0 0 

α 

B 

2 

0 

B 

e 

= 

4πε 

c� 

, 

2 

e 

� = 

4πε 

αc 

= 

e 

0 

2 

4πε0vB 2 

e 

= . 

4πε0ωBrB 2 

e 

� ωB 

= . 

4πε 

r 

10 

0 

B


3. 

The Charge-Radiation Equation 

and Zero Point Energy 

3.1 

2 2 

1 2 2 − 

1 

= hνB ⇔ U1 st Orbit = mω 2 BrB + = − 

2 

0r �� 

B 0rB 

U �� 

e e 

4πε 4πε 

Proof: 

Ignoring the magnetic forces, 

rotation 

U electric 

�� 

2 

−e 

8πε 

r 

2 

v 2 

B 1 e 

e 

4 2 

B πε0 

B 

0 B 

m = , 

r r 

1m 2 e 

2 

v 

ω 

= 1 

2 

�B 2 2 

BrB 0 

hν 

2 

1 e 

, 

4πε 

r 

�� B 

The electron’s energy at the ground orbit is the sum of the 

electron’s kinetic energy of rotation, 

1 2 2 1 2 

rotation ω 

2 B B 2 B 

U = m r = mv , 

and the electron’s binding electric energy 

11 

B 

hν 

B


0 

U 

electric 

2 

−e 

= , 

4πε0rB 2 

1 2 2 −e mω 2 BrB+ �� 4πε 2 0rB 1 e 

2 

−e 

= 

8πε0rB 

1 = − hν 

2 B 

4πε 

r 

B 

In the Bohr orbit, the electron’s energy is of photon with energy 

− hν . 

1 

2 

That is, the Ground state Energy of the Bohr Atom is Zero Point 

Energy of 1 hν . 

2 

B 

3.2 Bohr’s Atom Zero Point Energy 

3.3 

1 1 e 

EB =− hν 

13.6eV 

2 B =− =− 

8πε 

r 

13.6eV 

15 

ν B = 2 = 6.576928348 × 10 cycles/second 

h 

Computation: h = 4.1356692 × 10 eV , [Woan]. 

B 

−15 

12 

0 

2 

B 

.


4. 

The Electron’s magnetic Energy 

4.1 Electron’s Total Quantized Energy 

Bohr’s quantization of the Angular Momentum ignores the 

electron’s Orbital Magnetic Energy, and the electron’s Spin 

Magnetic Energy. 

Including orbital magnetic forces, in the force balance, the 

electron’s total quantized energy in the Bohr orbit is 

2 

e 

− + Umagnetic = hνB. 

8πε 

r 

4.2 The Electron’s Magnetic Energy 

0 

B 

1 2 2 

magnetic = μ 

4 0 Be νB 

U r 

1 

= 

(4 π) 

2 

μ 

1 

2 2 

0vBe 

rB 

Proof: The current due to the electron’s charge e that turns 

cycles/second is 

I = eνB 

The Magnetic Energy of this current is [Benson, p.486] 

1 

2 

2 

LI . 

13 

νB


By [Fischer, p.97] 

2 πrB 

1 

0 2 0 B 

2πrB 

L = μ = μ r . 

Thus, the magnetic energy due to the electron charge is 

11 2 1 2 2 

μ 

22 0rB( eνB) = μ 

4 0rBe 

νB 

= 

1 

4 

2 2 

μ0erB ν 

�B 

1 

ω 

2 

4π 

2 

B 

1 2 2 1 

= μ 

2 0vBe 

. 

(4 π) 

r 

4.3 The Electron’s Magnetic Energy in the Bohr’s Orbit is 

Proof: 

negligible compared to its Electric Energy 

1 1 

U r v 

2 2 μ0v 

2 Be 

2 

magnetic (4 π) 

B 1 B 1 2 1 

−6 

= = = α ≈ ≈ 8.5 × 10 

2 2 2 2 (137) 

electric 1 

2π 2π 

π 

U e c 

8πε 

0 

r 

B 

14 

B


5. 

The n th Orbit Charge-Radiation 

In the n th orbit, 

the Angular momentum is mvr = n�, 

The orbit energy is 

5.1 The Electron’s Energy 

Proof: 

5.2 

Proof: 

5.3 

E 

n 

e nn 

1 

=− 

8πε 

2 2 

1 2 2 −e −e 

1 2 2 

m 

2 eωnrnm 2 eωnrn 

4πε0rn 8πε0rn 

0 n 

α 

+ = = − = − � 1 c 

2 r 

r v 

+ = = = − = − 

1 2 2 

meω 2 nn r 

�� 

2 e 

2 

−e 4πε0rn 2 

−e 8πε0rn 2 

−e 

B 

8πε0rB 

r 

�� n 

1 B � 

2 rn 

4πε 

r 

− �ωB 

v 

2 2 ( e nn) n 

= = � 

n 

vB 

α 

= = c 

n n 

m ω r r αcα vnc mvr n� n 

= 

e n n 

2 

n = B 

r n r 

15 

1 

2 

0 

e 

r 

2 

n 

n 

1 

2 

�αc 

r 

n


Proof: 

n� n� 

� 

r n 

2 2 

n = = = = n r 

1 

B 

mv e n m mv 

e v 

n B 

e B 

5.4 ν = 1 ν 3 

Proof: 

n n B 

v 

ν = ω = = = ω = ν B 

1 

1 1 vn 

1 n B 1 1 1 

n 2π n 2π 2 2 3 2 B 

r π π 

3 

n nrB n n 

5.5 The Orbit Energy 

Proof: 

E 

n 

1 1 1 

En = E ( ) 

2 B = − hν 

2 2 B 

n n 

2 2 

1 e 1 e rB 

=− =− 

8πε0 rn 8πε0 

rB 

r 

�� n 

1 

2 

hν 

5.6 The Charge-Radiation Equation 

The binding electric energy 

B 

1 

n 

2 

e 

2 

4πε 0rn 

of the electron-proton 

Atom, where the electron is in the n th Bohr orbit r with 

frequency , equals the radiation energy nhν of n photons 

of frequency ν . 

νn n 

n 

16 

n


Proof: 

1 

4πε 

0 

1 

4πε 

0 

e 

r 

e �v 

r 

r r nr 

2 

n 

= nhν 

2 

n 

= B 

n 

= � ω 

�B 3 

n ω 

B 

2 

B 

= nhνn 

n 

17 

n


References 

[Benson], Benson Walter, Harris John, Stocker Horst, Lutz Holger, 

“Handbook of Physics”, Springer, 2002. 

[Born] Max Born, “The Mechanics of the Atom”, Ungar Publishing, 1924 

[Dan] H. Vic Dannon, “Zero Point Energy: Planck Radiation Law” Gauge 

Institute Journal, Vol.1 No 3, August 2005, 

[Fischer] Fischer-Cripps, A., C., “The Physics Companion”, IoP, 2003. 

[Miller] Arthur I. Miller, “Deciphering the Cosmic Number” Norton, 2009 

[Polyanin] Andrei D. Polyanin, Alexi I. Chernoutsan, “A Concise Handbook of 

Mathematics, Physics, and Engineering Sciences” CRC, 2011. 

[Woan] Graham Woan, “The Cambridge Handbook of Physics Formulas”, 

Cambridge 2000. 

18

Zero Point Energy and the Charge-Radiation Equation

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