Heisenberg’s Uncertainty Principle. Why, exactly, can you not know both the velocity and position of a particle?

In: 146

Let’s look at a baseball you throw. First, let’s take a picture of a single instant in time. We know exactly where the baseball is, but from that picture we have no idea how fast it’s going (or even the direction).

Now suppose instead of an image, we use a radar gun to measure the speed of the ball. To do this, the ball has to move from one place to another. So now we know the speed, but there is some uncertainty in the position of the ball because our speed measurement isn’t instantaneous (it is an average over some period of time).

You can’t have zero uncertainty in both position & momentum for this system because one value (momentum) requires measurement over some time period and the other value changes during that time period (position).

**TLDR**

Particles are small and small things behave more like waves than large objects. Waves are spread out and so they don’t exist in a precise location, and if they aren’t spread out, then they are less like a wave and move with ambiguous speed.

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**More technical:**

As the size of an object decreases (the limit of this we can consider a “particle”), the wavelike properties of that object become more important.

The wavefunction of a particle tells you where a particle may be with a certain probability, and the wavefunction is a wave.

The least useful wavefunction in determining where a particle may be would be a sine/cosine function that spreads out in all of space with complete uniformity.

Imagine this wave extending throughout all of space. Ok, so the uncertainty in position of this wave is infinity. The particle could be in Hawaii, or Alpha Centauri, or whereever with equal probability.

However, in this case we have a perfect wave which means we have a perfect wavelength (the distance between peaks/crests of the wave)

The wavelength is related in an exact equation to the momentum which gives you the velocity. So if we know the wavelength with high precision, we know the velocity with high precision.

The inverse is if our spread out wave is now condensed into one little crest.

In this case, we have a great understanding of where the wave is (its just in the small region where the crest is), but now since it looks less uniform as a wave, its wavelength and thus velocity is more poorly defined.

Because they’re also waves. They can’t have well defined position and well defined velocity simultaneously.

For example

If the wind is blowing over the ocean, it makes waves with a regular velocity, but it’s everywhere (no position)

If someone cannonballs into the ocean, you can tell where it happened, but it makes a splash rather than a normal wave.

Because small energetic particule like electron or photon are part wave and part matter. Where a wave end or begin is not precise.

At the smallest, quantum scale, velocity and position are not just numbers, like they are for big things like cups and balls. They are a wave function, which means that a particular value might be the most likely value, but other numbers close to it are also pretty likely. This uncertainty can’t be reconciled through more measurements, the more accurately you confine position the less precise your velocity measurements – and vice versa.