# Would we be able to be consistent in our theory if we were to only assume the particle nature of matter with the uncertainty principle

As far as I understood, the uncertainty principle is direct consequence of the wave - particle duality of matter; however, would we be able to be consistent in our theory if we were to only assume the particle nature of matter with the uncertainty principle . I mean for example, in Germer’s experiment, If we say that the electron behaves like a particle and we cannot determine its position and velocity precisely at the same time, we can say that the electron collides with the lattice with different angle and hence scatters. (of course then we could not explain why the intensity of the scattering in different angles have different values, but that is not the point in here. The point is whether we would be consistent in our theory or not).

• "Wave - particle duality" is a vague idea that does not have any exact consequences, and is more of a tribute to the historical genesis of quantum mechanics and a pedagogical bridge to the classical one. Quantum particles do not behave as either waves or particles, the governing laws are different from both, so "electron behaves like a particle" is nothing but a limited metaphor for special situations. Heisenberg's principle, on the other hand, is a precise inequality derivable from the laws of quantum mechanics. – Conifold Mar 21 '18 at 4:18

In short, yes.

To elaborate on the comment by @Conifold above:

History: The historical development of QM was guided by the wave metaphor of Debroglie and the Schrodinger wave equation was developed in analogy with experience with classical waves. It is true that many quantum phenomena can be explained and easily visualised using the wave language (eg diffraction or interference), but this is now understood to be mainly of heuristic value, and it works in very limiting cases (eg single particle states for which the wavefunction looks like a function in space. In general the wavefunction lives in configuration space, eg think of a two particle state $\psi(x_1,x_2)$), and is a probability amplitude. In the modern formalism you can use "particle" (or 'quantum particle) to describe eg an electron, but in general it will obey quantum laws rather than classical ones.

Uncertainty Principle: Heisenbergs original argument imagined a measurement process which disturbed the state of the system, resulting in his uncertainty relation $\delta x \delta p \sim h$. However this relation is conceptually different from that which is derived from the formalism of quantum theory $\Delta x \Delta p \ge {\hbar / 2}$, which states that the product of uncertainties (standard deviations) of the noncommuting operators is bounded below. This ("Heisenberg-Robertson Inequality) states that if you prepare a quantum state, then you cannot simultaneously make the variances of the $x$ and $p$ variables as small as possible in that state. There is no "wave-particle duality" here, but see below.

Fourier Transform: A theorem of classical mathematics says that if you have a function $f(x)$ in x-space and its associated Fourier transform $g(k)$ in k-space then you cannot simultaneously make both $f(x)$ and $g(k)$ highly localised in their respective spaces. Eg If $f(x)$ is time-varying signal of short duration, then $g(k)$ would be a very broadly distributed function in frequency space. This is a purely classical result, but if you introduce $\hbar$ through $p=\hbar k$, then it turns into the Heisenberg-Robertson inequality, and then in this case you can have your "wave description" of quantum phenomena (a "wave" in position space which is dual to a "wave" in momentum space etc).

So, coming back to your electron: It obeys quantum laws, which are intrinsically probabilistic. We cant determine where each electron will end up in a scattering experiment, but if we send in many electrons we can calculate their distribution and that statistical pattern looks like a wave-interference pattern. Eg. If you do a double slit experiment, sending in one electron at a time, you will see single pings on the screen ("particle"), which overtime build up to the statistical interference pattern ("wave").

In the mainstream quantum mechanical formalism the Heiseneberg uncertainty principle comes from the commutation relations of operators .

For non-relativistic energies there exists a theory, called Bohmian mechanics which reproduces the mathematics and the probabilistic values having particles accompanied by "pilot waves", which generate the probabilities and uncertainties. As it is a matter of preference whether the mainstream formalism or the "pilot wave" formalism holds really, your question can be answered in the affirmative.

For relativistic energies the pilot wave theory is not able to keep up with observations, and thus is not useful for particle physics studies. There are a minority of theoretical studies pursuing this path, trying to reconcile it with special relativity.