IB DP Physics: HL复习笔记12.1.8 The Wave Function

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