Atomic Spectra and Atomic Energy States13.1.8 – 13.1.13

Observing Atomic Spectra

The diagram below shows some of the energy levels of the hydrogen atom.

Calculate the frequency associated with the photon emitted in each of the electron transitions A and B and identify the part of the EM spectrum where they occur.

The Origin of Energy Levels

The electron is bound to the nucleus by the Coulomb force and this force will essentially determine the energy of the electron. If we were to regard the hydrogen atom for instance as a miniature Earth-Moon system, the electron’s energy would fall off with inverse of distance from the nucleus and could take any value. However we can the see the origin of the existence of discrete energy levels within the atom if we consider the wave nature of the electron.

Particle in a Box

• To simplify matters we shall consider the electron to be confined by a one dimensional box of length L.

• In classical wave theory, a wave that is confined is a standing wave.

Quantization of the Electron’s Energy

The Schrödinger Model of the Hydrogen Atom

• In 1926 Erwin Schrodinger proposed a model of the hydrogen atom based on the wave nature of the electron and hence the de Broglie hypothesis. This was actually the birth of Quantum Mechanics. Quantum mechanics and General Relativity are now regarded as the two principal theories of physics.

The mathematics of Schrodinger’s so-called wave mechanics is somewhat complicated so at this level, the best that can be done is to outline his theory. Essentially, he proposed that the electron in the hydrogen atom is described by a wave function, Y.

This wave function is described by an equation known as the Schrodinger wave equation, the solution of which give the values that the wave function can have.

If the equation is set up for the electron in the hydrogen atom, it is found that the equation will only have solutions for which the energy E of the electron is given by E = (n + ½ )hf. Hence the concept of quantization of energy is built into the equation.

Of course we do need to know what the wave function is actually describing. The electron has an undefined position, but the square of the amplitude of the wave function gives the probability of finding the electron at a particular point.

The solution of the equation predicts exactly the linespectra of the hydrogen atom. If the relativistic motion of the electron is taken into account, the solution even predicts the fine structure of some of the spectral lines. (For example, the red line on closer examination, is found to consist of seven lines close together.)

The Schrodinger equation is not an easy equation to solve and to get exact solutions for atoms other than hydrogen or singly ionised helium, is well-nigh impossible. Nonetheless, Schrodinger’s theory changed completely the direction of physics and opened whole new vistas- and posed a whole load of new philosophical problems.

The HeisenbergUncertainty Principle

• In 1927 Werner Heisenberg proposed a principle that went along way to understanding the interpretation of the Schrodinger wave function.

• Suppose the uncertainty in our knowledge of the position of a particle is Δx and the uncertainty in the momentum is Δp, then the Uncertainty Principle states that the product ΔxΔp is at least the order of h, the Planck constant. A more rigorous analysis shows that

Heisenberg and de Broglie

If a particle has a uniquely defined de Broglie wavelength, then its momentum is known precisely but all knowledge of its position is lost.

4h

px p

h

The Principle also applies to energy and time. If ΔE is the uncertainty in a particle’s energy and Δt is the uncertainty in the time for which the particle is observed is Δt, then

This is the reason why spectral lines have finite width. For a spectral line to have a single wavelength, there must be no uncertainty in the difference of energy between the associated energy levels. This would imply that the electron must make the transition between the levels in zero time.

4h

tE

Homework:

• Tsokos– Page 404– Questions 1 to 15