12.1.8 The Wave Function

• Bohr showed that the angular momentum of an electron orbiting a hydrogen atom was quantised
  • He also showed that the allowed electron energies were also quantised
  • Unfortunately, this original model only worked for hydrogen

• Schrodinger then developed an equation that could be used to calculate the quantised energy levels for other atoms
  • The only information the equation needs is the shape of the particle’s potential energy function

The potential energy curve for hydrogen and the quantised energy levels for n = 1, 2, 3, etc

• On the potential energy curve for hydrogen:
  • The higher the energy levels, the closer together they become in energy

The potential energy curves for (a) an infinite square well and (b) a parabolic well and their associated quantised energy levels

• The shapes of these graphs also produce quantised energy levels but with a different energy distribution
  • For the parabolic well, the energy levels are equally spaced
  • For the infinite square well, the energy levels at larger energies are further apart – this is different to the case for hydrogen

• Finally, as the infinite square well gets wider, the quantised energy levels get closer together

The "swimming pool" square well where the width is very large. For a wide well, the quantised levels get so close together that the allowed levels are effectively “continuous”

• The Schrodinger equation predicts not only the distribution of energy levels for a particular potential energy curve
  • Each energy level also has a wavefunction, usually given the symbol Ψ (Psi)

• Wavefunctions of the infinite square well are shown below
  • They look similar to standing waves on a stretched string

(a) Wave functions for E1 and E2 of an infinite square well. (b) an expanded view of the wavefunctions Ψ, and the probability

• The physical significance of the wavefunction Ψ is that:
  • The square of the wavefunction, Ψ² dr gives the probability of finding the particle in the region of width dr

• The wavefunction is said to be normalised
  • This means the probability of finding a particle somewhere in the well at a particular energy level is equal to one

• The infinite square well shows the wave function going to zero at the edges of the well
  • This means that the probability of finding the particle close to the walls of the well is small

• For the energy level E₁ the probability of finding it at the centre of the well in large
  • However, for E₂ the probability of finding it at the centre of the well is small

 

• A different result occurs for a square well that is not infinitely deep
• For two energy levels in a square well, the wave functions penetrate the barrier regions as decaying exponential curves

 (a) Wave functions for E1 and E2 for a finite square well. (b) the wavefunctions Ψ, and the probability distribution Ψ^2 are seen to penetrate the barrier

• This means that the probability of finding the particle in the classically forbidden barrier region is not zero