The free wavicle: motivation for the Schrödinger Equation

1. The free wavicle: motivation for the Schrödinger Equation • Einstein showed that hitherto wavelike phenonomenon had distinctly particle-like aspects: the photoeffect • photon energy is E = hf = ħw(h = Planck’s constant; f = frequency; w = angular frequency = 2pf) • let f(x,t) be a wave’s amplitude at position x at time t • the Classical Wave Equation reads (v = phase speed) • first pass at a solution is any function in the form f(± x – vt); a pattern that moves at speed v to R (+) or to L (–) with fixed shape • can build linear combinations that satisfy CWE with different phase speeds, too, so the pattern may in fact evolve as it moves • consider the simple harmonic solution [wavelength l; period T

2. Going to a complex harmonic wave • sines are as good as cosines, so if we take a complex sum as follows, it also works (assume A is real): • the real and imaginary parts are 90° out of phase • in the complex plane, for some x, f orbits at w CW on a circle of radius A much simpler than ‘waving up and down’ • this will be the frequency of the quantum oscillations Einstein on waves  de Broglie on particles

3. Non-relativistic wavicle physics • free Non-Relativistic Massive Particle has • non-free NRMP has • we connect this energy to the photon energy • let the wavicle amplitude function be written Y(x,t) and for a free wavicle we take the earlier complex form • compare to classical wave equation (order, reality..)

4. What is this thing, the wavefunction Y(x,t)? • it contains information about physics: position, momentum, kinetic energy, total energy, etc. using operators • we assume that TDSE also works for a non-free particle if energy is conserved (V = V(x) so its operator is trivial) • Born (Max) interpretation of complex wavefunctionY(x,t) • -- Y*Y = r(x,t) probability density at time t; Y = probability amplitude • -- Y*Ydx = probability that, at time t, particle is between x and x + dx

5. Some peculiarities of the free wavicle f(x,t) • it exists finitely everywhere so does not represent a ‘bound state’ • we will revisit these ideas again but we need a lot more insight into the subtle distinctions between bound states (which are discrete) and free states (which form a continuum) • for now, the normalization integral IS infinite but it will turn out that a free wavicle with any other wavenumber is orthogonal – so the infinity is really a dirac delta function in ‘k-space’

6. Mathematical attributes of Y • it must be ‘square-integrable’ over all space, so it has to die off sufficiently quickly as x± ∞, to guarantee normalizability • the free wavicle fails this test! Normalizing it is tricky! • no matter how pathological V(x), Y is piecewise continuous in x • let’s check whether Y(x,t) ‘stays’ normalized as time goes by..

7. Finishing the normalization check of the solution to the TDSE • First term is zero because Y has to die off at x = ±∞ • Second term is obviously zero • therefore, probability is conserved • a subtle point is that when a matter wave encounters a barrier that it can surmount, one must consider the probability flux rather than the probability…

8. Elements of the Heisenberg Uncertainty Principle • uncertainty in a physical observable Q is standard deviation sQ • for the familiar example of position x and momentum p: a particle whose momentum is perfectly specified is an infinitely long wave, so its position is completely unknown: it is everywhere! • a particle which is perfectly localized, it turns out, must be made of a combination of wavicles of every momentum in equal amounts, so knowledge of its momentum is lost once it is ‘trapped’ • Heisenberg showed that the product of the uncertainties could not be less than half of Planck’s constant: • it is amusing to confirm this inequality for well-behaved Y • there is also an energy-time HUP of the same form, and an angular momentum-angular position one of the same form • we’ll derive this soon much more rigorously