Quantum Wave Mechanical Model Part One

Bohr's model of the atom was both a success and a failure.  It successfully predicted the frequencies of the lines in the hydrogen spectrum and it adequately explained how atomic spectra worked. 

 

There were several problems that bothered physicists and chemists:

• The model was a total failure when it tried to predict energy levels for atoms with more than one electron. 

 

•  Why should electrons be confined to only specified energy levels?

• Why don't electrons give off light all of the time?
  • As electrons change direction in their circular orbits (i.e., accelerate), they should give off light.

• The Bohr model could explain the spectra of atoms with one electron in the outer shell very well, but was not very good for those with more than one electron in the outer shell.

• Why could only two electrons fit in the first shell and why eight electrons in each shell after that? What was so special about two and eight?

Obviously, the Bohr model was missing something!

 

In 1924, a French physicist named Louis de Broglie suggested that, like light, electrons could act as both particles and waves. De Broglie's hypothesis was soon confirmed in experiments that showed electron beams could be diffracted or bent as they passed through a slit much like light could. So, the waves produced by an electron confined in its orbit about the nucleus sets up a standing wave of specific wavelength, energy and frequency (i.e., Bohr's energy levels) much like a guitar string sets up a standing wave when plucked.

De Broglie's vision of Bohr's atom  

   

The diagram above illustrates an electron standing wave vibrating in an orbit around a nucleus of an atom.  Only integral numbers of wavelengths are allowed. Below is a series of diagrams that illustrate the allowed vibrations of a string fixed on both ends. If a string is fixed on both ends, then the only waves that can occur are those with zero amplitude at those fixed ends; such points of zero amplitude are called nodes. Below we show four of the infinite number of vibrations with a node at each end. These vibrations are called standing waves.

One String Vibrating - 1/2 wavelength - One loop, two nodes quantum number n = 1  - Click on Graphic for animation	One String Vibrating - 1 wavelength - 2 loops, 3 nodes quantum number n = 2 - Click on Graphic for Animation
One String Vibrating - 2 wavelengths - 4 loops, 5 nodes quantum number n = 4 - Click on Graphic for Animation

 

De Broglie carried the idea of standing waves to the Bohr atom. Standing waves in a circular orbit can exist only if the circumference of the orbit is an integral number of the wavelengths (see figure 1).  For a standing wave around the orbit the following must be true

            2Πr = nλ

 

The electrons moving in an orbit have a certain momentum given by the expression

 

P = mv

 

The wavelength of the electron can be expressed as a function of momentum

 

            P = h/λ

 

Substituting for momentum De Broglie got

 

            mv = h/λ

 

Solving for λ de Broglie derived the following relationship

 

            λ = h/mv

 

where h is Planck’s constant, m is the mass of the particle, and v is the velocity.

 

De Broglie proposed this relationship as a general one. With every particle, there is an associated wave.  The wavelength of the particle depends on its mass and how fast it is moving.