The concept that particles are also waves leads to a number of seeming paradoxes. A wave is not localized. A wave can exist of a region of space. So in the particle picture we can say that the particle is located at coordinate in space (x,y,z), but in the wave picture it is not clear how to describe where the particle is located. In fact, the consequence of this picture is that the particle is found in a region of space. We could say that there is some probability distribution that describes where the particle is. Therefore, we are uncertain about the precise location. Likewise, the issue of velocity is complicated by the difference between what we call the velocity of a wave. The velocity of a wave depends on the product of the wavelength and the frequency. Fundamentally, each of these variables is related to its conjugate variable.

Distance <-> Momentum or wavevector (for a wave)

Time <-> Energy of frequency (for a wave)

Conjugate variables are related by a Fourier transform. A broad distribution in one space (e.g. distance) leads to a narrow distribution in the conjugate space (e.g. momentum). A similar inverse relationship exists for time and energy. These relationships are derived intuitively using a graphical picture in the following segment.