How to deal with an $\alpha$ particle in a $^{238}\,\,U$ nucleus with a "definite" speed? Some books claims that an $\alpha$ particle in a $^{238}\,\,U$ nucleus has a speed of, for example, $c/3$. However, I think it's quite unlikely for such a particle to have well defined momentum or speed because its quantumstate may be the superposition of several states of well defined momenta.
So what does it really mean by the speed $c/3$ ?
Here is an example of such textbook. An exercise from The Physics of Quantum Mechanics by James Binney and David Skinner

The word suppose makes the situation tricky. Yet I don know how to start doing the question. Should I start the question states with well defined momentum?
 A: This is a common semiclassical representation of spontaneous decay: a particle is seen as periodically bouncing on a potential barrier, each bounce providing the opportunity to escape via quantum tunnelling. The frequency of the oscillation, and thus the decay rate, is then given by the particle speed.
See for example how in Modeling Alpha Half-life it is stated that "the alpha emission rate depends upon how many times an alpha particle with this energy inside the nucleus will hit the walls".
See also this answer here in PSE: https://physics.stackexchange.com/a/61790/109928, where it is said that some decay rate is proportional to "the frequency with which a cluster assaults the Coulomb barrier".
As you rightly point out, this is not a satisfactory representation in the sense that it imposes the idea that the trapped particle has a trajectory all along, making the particle speed a useful concept.
Now I found it is surprisingly difficult to find expositions of any other representation of decay, that would be intuitive while non-semiclassical. I still haven't, actually.
See this related question of mine: How can quantum tunnelling lead to spontaneous decay? and its answers.
A: The consistent way to model quantum mechanically the decays is through quantum mechanical tunneling.
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The half-lives of heavy elements which emit alpha particles varies over 20 orders of magnitude, from about a tenth of a microsecond to 10 billion years. This half-life range depends strongly on the observed alpha kinetic energy which varies only about a factor of two; from about 4 to 9 MeV. This extraordinary dependence upon kinetic energy suggests an exponential process, and is modeled by quantum mechanical tunneling through the Coulomb barrier. 

The alpha has definite kinetic energy as it is measured , and thus speed, but as it is in MeV it is nowhere near the speed of light.
Now the tunneling process for the alpha:

(note that in QM tunneling the particle is at the same energy level in and out of the barrier, consistent with shell model solutions for nuclei)

The illustration represents an attempt to model the alpha decay characteristics of polonium-212, which emits an 8.78 MeV alpha particle with a half-life of 0.3 microseconds. The Coulomb barrier faced by an alpha particle with this energy is about 26 MeV, so by classical physics it cannot escape at all. Quantum mechanical tunneling gives a small probability that the alpha can penetrate the barrier. To evaluate this probability, the alpha particle inside the nucleus is represented by a free-particle wavefunction subject to the nuclear potential. Inside the barrier, the solution to the Schrodinger equation becomes a decaying exponential. Calculating the ratio of the wavefunction outside the barrier and inside and squaring that ratio gives the probability of alpha emission. 

Italics mine.
All quantum mechanical measurements build up a probability distribution which can be checked against a quantum mechanical wavefunction, a solution for a potential problem,  for the given boundary conditions and potential.
Now in my opinion, the quote you are giving is an unnecessary mix up of classical and quantum mechanical states and is misleading. The alpha is not bouncing around against barriers, it has a probability distribution to exist within the effective potential well.
The above link continues in methods of calculation.
