Schroedinger’s Equation ...an historical, heuristic approach

Radical view of light Planck and Einstein re-introduced the particle notion for light a wave “becomes” a particle and still a wave … hmmm

de Broglie... If light (ie “a wave”) can be a particle then maybe a particle (electron?) can “be a wave” - what are the implications of this “leap of reason”?

Compare and contrast: Waves & Particles Waves are extended Waves are continuous Waves conform to wave equations Waves diffract and interfere Waves have amplitude, frequency and velocity Particles are points Particles are discontinuous Particles obey equations of mechanics Particles “bounce” Particles have mass, size(?) and velocity

ParticleWaves, Wavicles or just Weirdness … If de Broglie is correct then we can ascribe a wavelength to a particle:

Schroedinger... Once at the end of a colloquium I heard Debye saying something like: “Schroedinger, you are not working right now on very important problems… why don’t you tell us some time about that thesis of de Broglie’s… in one of the next colloquia, Schroedinger gave a beautifully clear account of how de Broglie associated a wave with a particle, and how he could obtain the quantization rules… When he had finished, Debye casually remarked that he thought this way of talking was rather childish… To deal properly with waves, one had to have a wave equation. Felix Bloch, Address to the American Physical Society, 1976

Schroedinger’s key assumptions concerning a quantum-mechanical wave equation... It must incorporate the relations: Since normal waves “add” linearly (principle of superposition), so too must the solutions to the qm-wave equation. This means the solutions must be linear.

Wavefunctions and wavelengths… The wavelength of the wavefunction depends on energy via: We need a way to determine wavelength at any point We do this with curvature! What are the units for this?

To make a long story short… Units of curvature are Put this together with what we already know and we find

so, without further delay...

Some two-minute problems… What happens to the energy of a quanton-in-a-box as you shrink the box? A) increases B) decreases C) stays the same D) no idea! How would the ground state energy of a quanton in a potential well change as you lower the potential?

Some examples… Compare the particle-in-a-box with the quantum oscillator

Applets and Numerical Examples… Physlet … Link to Moore’s SchroSolver (download of exe)