# Why are wave packets constructed so that the maximum probability occurs at K0 (the average wave number)?

The definition of a wave packet I have been given is that it is "a superposition of many plane waves, with wave numbers grouped around an average value $k_0$".

I was told that, for a particle we want, its wave packet to be such that it has a maximum probability at $k_0$ - but I was under the impression that you integrate your probability density function with respect to position, not wave-number. Have I misunderstood the concept of a wave-packet? Could someone actually explain to me what they are?

edit: I'm asking about why we are considering values of the wave-number when determining the maximum probability of localising a particle in space and not the values of x(along the x-axis, if we think about it in one dimension). How would we relate the position of a wave and its wavenumber together? If you can do that

For a general wavepacket A type of uncertainty principle exists for general wave equations: similar to the Heisenberg uncertainty principle , HUP, for quantum mechanics. Interpreting the wave packet as a probability distribution, if you control k, through the HUP you are also localizing x .

• So is it like the probability function is the essentially the "wave function" of the particle and you use the probability function (which is y squared in the case of your photo) to determine where the particle is in space at time t or is it y is the actual equation for your particle and y itself determines how it moves? – BigWig May 8 '18 at 10:29
• it is a probability envelope in quantum mechanical systems – anna v May 8 '18 at 10:33