Graduate Course: Quantum Physics

Dinesh Prasad Ketumakha, Dept of Electrical and Electronic Engineering,
and John Venables, Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona


"Anyone who is not shocked by quantum theory has not understood it"
- Neils Bohr

In 1913 Neils Bohr proposed his model of atom which superceded Rutherford's atomic model. Though the planetary model proposed by Rutherford was widely accepted, it fell short on many counts. The nuclear atom proposed by Rutherford was unstable. According to classical theories this atom should collapse. It also failed to explain the discrete spectral lines of elements. Bohr's model of atom could successfully explain the stability of atom by introducing Quantization. It could also explain the Hydrogen spectra. Bohr obtained the value of radius of hydrogen atom and its energy, both of which agree well with experimental results. Was this a coincidence!? Bohr's atomic theory formed the basis for the old Quantum theory. This page concentrates on illustrating the ingredients of the Bohr model.

Bohr's Postulates

Planetary model of the atom

(a) The electron revolves in circular orbits around the nucleus which are restricted by the quantization of angular momentum i.e. they revolve in orbits where the angular momentum of electron is an integral multiple of h/2π, where h is Planck's constant.

  mvr = nh/2π

In these orbits of special radius electron does not radiate energy as expected from Maxwell's laws. These orbits are called  stationary states. This is called as Bohr's quantization rule.

                                                                              source :

(b) The energy of the atom has a definite value in a stationary orbit. The electron can jump from one stationary orbit to another. If it jumps from an orbit of higher energy E2 to an orbit of lower energy E1, it emits a photon. The energy of the photon is E2-E1.The wavelength of the emitted radiation is given by the Einstein - Planck equation.

E2-E1= hν = hc/λ

The electron can also absorb energy from some source and jump from a lower energy level to a higher energy level as shown in the following figure.

source :

The figure above shows the various ways of how an electron can reach ground level after being excited to the third energy level n=3.The total energy that the electron emits as photon is hv30=hv32+hv10=hv32+hv20=hv32+hv32+hv21+hv10.

Energy of a Hydrogen Atom

The above postulates can be used to calculate allowed energies of the atom for different allowed orbits of the electron. The theory developed should be applicable to hydrogen atoms and ions having just one electron. Thus, within the Bohr atom framework, it is valid for He+, Li++, Be3+ etc. Let us consider the case of an ion with the charge of nucleus being Ze and an electron moving with a constant speed v along a circle of radius r with the center at the nucleus. The force acting on the electron is that due to Coulomb attraction and is equal to

F = Ze2/4πε0r2

The acceleration of the electron is towards the center and has a magnitude v2/r. If m is the mass of the electron, from Newton's law we obtain

Ze2/4πε0r2 = mv2/r

Using Bohr's angular momentum quantization rule for the value n, the Principal quantum number, we obtain both the velocity v, and the radius r as:

v = Ze2/2e0hn          r = ε0h2n2/πmZe2 ...(i)

We see that the allowed radii are proportional to n2. For each value of n, we have an allowed orbit. For n=1, we have the first orbit (smallest radius) , for n=2, we have the second orbit and so on.

The kinetic energy of the electron in the nth orbit is

K.E = mv2/2 = mZ2e4/8ε02h2n2 ...(ii)

The potential energy of the atom is 

P.E = -Ze2/4πε0r = -mZ2e4/4ε02h2n2...(iii)

We have taken the potential energy to be zero when the nucleus and the electron are widely separated. The total energy of the atom is

E = K.E+P.E = -mZ2e4/8ε02h2n2  ...(iv)

Equations (i) to (iv) give various parameters of the atom when the electron is in the nth orbit .The atom is also said to be in the nth energy state in this case. In deriving the expression for the total energy E, we have considered the kinetic energy of the electron and the potential energy of the electron-nucleus pair. It is assumed that of the acceleration of the nucleus is negligible on account of its large mass (i.e. that the reduced mass of the system is the same as the electron mass- this can be corrected easily later).

Radii of different Orbits

From equation (i) the radius of the smallest circle allowed to the electron is (n=1)

r1 = ε0h2/πmZe2

For hydrogen atom Z=1 and substituting the values of other constants we get r1=0.0529..nm.This length is called the Bohr radius and is a convenient unit for measuring lengths in atomic physics. It is generally denoted by the symbol a0.The second allowed radius is 4a0 and the third allowed radius is 9a0 and so on. In general, the radius of the nth orbit is

rn = n2a0.

For a hydrogen-like ion with Z protons in the nucleus

rn = n2a0/Z.    ...(v)

Ground and Excited states

From equation (iv) the total energy of the atom in the state n=1 is

E1 = -mZ2e2/8ε02h2

For hydrogen atom Z=1and substituting the values of the constants E1=-13.6 eV. This is the energy when the electron revolves in the smallest allowed orbit r=a0  i.e. the one with radius around 0.053nm. We also see from equation (iv) that energy of an electron is proportional to 1/n2.Thus ,

E n = E1/n2 = -13.6/n2  (eV) ...(vi)

The energy in the state n=2 is E2=E1/4=-3.4 eV. In the state n=3 it is E1/9 = -1.5 eV etc. The lowest energy corresponds to the smallest circle. Note that the energy is negative and hence a larger magnitude means lower energy. The zero of energy corresponds to the state where the electron and the nucleus are widely separated. The state of atom with the lowest energy is called is ground state. The states with higher energies are called excited states. Thus the energy of a hydrogen atom in the ground state is -13.6 eV and in the first excited state = -3.4eV.

Limitations of Bohr's Model

Bohr's atomic model was ultimately not successful. It defied all attempts at improvement over the ten-year period following the original publication, and all 'obvious' improvements lead nowhere; the only 'success' was with the hydrogen atom and similar atoms i.e. those with one electron like He+, Li++ etc. Bohr's atomic model attributes a planetary motion to electrons which means that electrons move around the nucleus in defined circular orbits. This is not the modern view. The electron distribution around the nucleus of an atom is described by a probability distribution, giving rise to 'electron clouds' rather than discrete circular orbits. What survives into the new Quantum Theory is the need for single-valued wave functions, and the fact that, for hydrogen-like atoms, Bohr's model identified various dimensional parameters correctly; the rest is history.

"Every word I utter is to be understood not as an affirmation but as a question."

                                                                                                                   -Neils Bohr

Return to Background Information home page

Related Pages : - Contains applets for spectral lines of elements. - This web page has 3D animations of orbitals. - More about history of Bohr's atom. - Interactive applets to understand Bohr's atom.

References used:

 H.C.Verma - Concepts of Physics-2

Stephen Gasiorowicz - Quantum Physics

Latest version of this document: 24 May 2004