Complex Versus Real Wave Velocities in Quantum Mechanics There's a fantastic quote in Schrodinger's second 1926 paper1  that apparently provides some motivation for the discrete energy levels (I think) that I'm having trouble interpreting:

I would not like to suppress the remark here (and it is valid
  quite generally, not merely for the oscillator), that nevertheless this
  vanishing and becoming imaginary of the velocity of propagation
  is something which is very characteristic. It is the analytical reason
  for the selection of definite proper values, merely through the 
  condition that the function should remain finite. I would like to
  illustrate this further. A wave equation with a real velocity of 
  propagation means just this : there is an accelerated increase in the value
  of the function at all those points where its value is lower than the
  average of the values at neighbouring points, and vice versa. Such an
  equation, if not immediately and lastingly as in case of the equation
  for the conduction of heat, yet in the course of time, causes a levelling 
  of extreme values and does not permit at any point an excessive
  growth of the function. A wave equation with an imaginary velocity
  of propagation means the exact opposite: values of the function above
  the average of surrounding values experience an accelerated increase
  (or retarded decrease), and vice versa. We see, therefore, that a function
  represented by such an equation is in the greatest danger of growing
  beyond all bounds, and we must order matters skilfully to preserve it
  from this danger. The sharply defined proper values are just what
  makes this possible. Indeed, we can see in the example treated in
  Part I (Hydrogen atom). that the demand for sharply defined proper values 
  immediately ceases as soon as we choose the quantity E to be positive, 
  as this makes the wave velocity real throughout all space.

I'm in danger of misinterpreting this claim due to my lack of knowledge of quantum mechanics, so could anyone elaborate on the implications of a wave solution of the Schrodinger equation having a complex velocity implying discrete eigenvalue energy levels, & a positive velocity removing this requirement (if that's what he means)? Thanks

1 Quantization as a Problem of Proper Values, Part II. Annalen der Physik (4), vol. 79, 1926. Available as part of Dreams That Stuff Is Made Of: The Most Astounding Papers of Quantum Physics here, or in Collected Papers on Wave Mechanics from here.
 A: I) Schrödinger is apparently talking about the speed
$$ v ~=~\sqrt{\frac{2(E-V)}{m}} $$
of a non-relativistic particle with energy $E$ in a potential $V$. 
II) If the energy $E<V_0$ is less than the asymptotic value of the potential 
$$V_0~:=~\lim_{|{\bf r}|\to \infty} V({\bf r})$$ 
(corresponding to a bound state), then the wave function solution $\psi$ to the time-independent Schrödinger equation (TISE) generically blows up exponentially in the asymptotic region (corresponding to imaginary speeds $v$) except for special discrete values of $E$ (the bound states), where the solution instead exponentially decays asymptotically, and the wave function is normalizable. 
Consider for simplicity 1D. The TISE is then a linear 2nd order ODE in $\psi$. Locally in the asymptotic region, the general solution is a linear combination of two solutions: an exponentially decaying and an exponentially growing solution. However, an exponentially decaying solution in one asymptotic region will generically correspond to a linear combination with a non-zero exponentially growing component in another region. Only for special discrete values of $E$ (the bound states), it is possible to consistently glue together exponentially decaying solutions in all parts of the asymptotic region.
III) On the other hand, if the energy $E>V$, then the speed $v$ is real and the wave function $\psi$ is oscillatory, and there is no quantization condition on the energy $E$ (corresponding to a continuous energy spectrum).
A: I agree with QMechanic that the velocity in question is the quantity $$v=\sqrt\frac{2(E-V)}{m}.$$ Schrödinger is using terminology that's quite convoluted by modern standards so it is understandable to be confused by this passage. There are two possible regimes for this velocity, and they both affect how the wavefunction behaves. Here Schrödinger is working in the WKB regime - or you can at least get the asymptotics from this regime - in which the wavefunction can be represented as
$$\psi(x)\propto\frac{1}{\sqrt{v(x)}} \exp\left( i \frac{m}{\hbar}\int\! v(x)\,\text dx\right).$$


*

*In the classically allowed regions, which have potential energy $V(x)<E$, the speed is real valued. This means that the wavefunction is oscillatory. Like in the classical case, points with a slower velocity have a higher amplitude because the particle spends more time there: there is an accelerated increase in the value of the function at all those points where its value is lower than the average of the values at neighbouring points, and vice versa.

*The classically forbidden regions have energy $E<V(x)$, and the speed is therefore imaginary there. This means that the exponential is real instead of oscillatory, and it trumps the power-law factor. It will either go down rapidly to zero, or it will exponentially explode, as Schödinger remarks: values of the function above the average of surrounding values experience an accelerated increase (or retarded decrease), and vice versa. An exponential increase is catastrophic, and it rules out any physical interpretation, so it must be avoided. a function represented by such an equation is in the greatest danger of growing beyond all bounds, and we must order matters skilfully to preserve it from this danger. 
What this means is that the phase of the oscillation as the wavefunction crosses from a classically allowed region into a forbidden one must be carefully chosen to ensure that only the decreasing exponential is chosen. 
 Source.
For a bound state, you have classically forbidden regions stretching to infinity on both sides of the real line. This means that you have to carefully control the phase of the oscillation on two ends of it, and this is not trivial. You can start the oscillation from the left with the right phase, but only for certain frequencies (= velocities), and therefore only for certain energies, will this oscillation have the right phase to slide gracefully into a decreasing exponential. This is a condition on the energy for the solutions to the TISE to be physical states, and (at least in vanilla cases) it is only satisfied by a discrete set of energies: this is your quantization condition, the piece of machinery that discretizes the energy levels.
