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Hello Tiago,

That's a good question. Heisenberg's Uncertainty Principle tells you that you
can't measure
position and velocity at the same time. This is commonly explained in the
following way: "You
want to measure the position of an electron, so you shoot a photon at it. This
measurement
changes its velocity, so you can't measure it accurately later." You've asked:
why can't you
*calculate* what how the first measurement will change the velocity, and
combine
that with the second measurement?

That would be perfectly true for, say, billiard balls. Unfortunately, the
simple explanation of
Heisenberg---"the position measurement makes the velocity change"---is
*wrong*.
A better statement would be "a position measurement *erases* the
velocity", or "a
position measurement makes the velocity *uncertain*." The explanation is
a bit
complicated, so bear with me.

The underlying principle of Quantum Mechanics is something called a
"wavefunction". This is a
sort of mathematical description of any particle. You can't see the
wavefunction itself, for some
reason: in a way, you can only ask it questions. "What is your velocity, O
Wavefunction? What is
your position? What is your axis of rotation?" But there are two problems:
the wavefunction
won't give you back the same answer every time. It only has a certain
*probability*
of giving a certain answer. That's very strange, but that's what we observe.
Secondly, after you
ask it the first question, it will stick to that answer---we say it has
"collapsed into a single
state"---but the process of collapsing makes all the *probabilities*
change for every
other question you could ask.

Here's a simple example, in an imaginary world where the math is easy to type out. Imagine that, for any particle, there are only three possible velocities---let's call them v1, v2, and v3. Let's say there are three possible positions, x1, x2, and x3. Now we'll invent a bunch of different wavefunctions, W. (Remember, I'm making this all up, but this system has the same properties as a real quantum particle.). The thing we need to know about the wavefunction is, "What is the probability that, if you ask the wavefunction for its velocity, you get v1/v2/v3? What is the probability that the "position" question gives x1,x2,or x3?" Here's a table of the possible answers:

Wavefunction | probability for measuring v | probability for measuring x | ||||
---|---|---|---|---|---|---|

Wavefunction name | v1 | v2 | v3 | x1 | x2 | x3 |

Wa | 33% | 33% | 33% | 100% | 0 | 0 |

Wb | 33% | 33% | 33% | 0 | 100% | 0 |

Wc | 33% | 33% | 33% | 0 | 0 | 100% |

Wd | 100% | 0 | 0 | 33% | 33% | 33% |

We | 0 | 100% | 0 | 33% | 33% | 33% |

Wf | 0 | 0 | 100% | 33% | 33% | 33% |

Is that clear? What that table means is, if you have a particle in wavefunction Wf, you can try to measure its velocity, and you'll get v3 100% of the time, but if you try measure its position, you're equally likely to get x1, or x2, or x3.

So let's start with an arbitrary particle, which again I will invent for this example. Rather than being a perfect Wd wavefunction, let's pretend that we have a mixture of all of these wavefunctions: 10% Wa, plus 65% Wc, plus 25% We. (This "mixture" of wavefunctions sort of means that you've got a 10% chance of picking from the Wa answer set, a 25% chance of getting the Wc answer set, etc. That's not quite true in the real world---there are complex numbers involved, so some things cancel, and it's a bit messy---but let's pretend it is true in this simple example.) But we don't care what the initial state is, right? We just want to measure the position and velocity, right? Let's try it! Suppose that the particle starts off with some *completely unknown* wavefunction. You measure its velocity and get the answer "v3". (Actually, "v2" is the most likely answer, as you can confirm yourself, but v3 will happen regularly.) Now---well, maybe you want to re-check the velocity. You can measure it again--- it's still v3. Measure it a hundred times, you'll always get the answer v3. That's because, as I said before, once you do the first velocity measurement, the particle "collapses" into a new wavefunction where there's only one possible velocity. In this example, our initial wavefunction has collapsed into a pure Wf state. (It doesn't matter that there was no Wf in the original function; after the measurement, this state is sort of created out of the bits of the others.)

OK, now you want to do the second measurement. Go ahead---but, oh no! For a
pure Wf state,
there's an *equal* chance of observing x1, x2, or x3 when you try to find
the
position. In other
words, the position is now *entirely uncertain*. The original
wavefunction's position
information---the fact that it was most likely to come up x3---was erased when
you measured
the velocity.

If you had measured the position *first*, you remember, you'd have been
querying
the original
10% Wa+65% Wc+25% We wavefunction ... you'd have gotten x3 most of the time, and
x1 less
often, and x2 only rarely. Let's suppose that you had done this: you measured
the position first,
you got the answer x3 (which is the most likely answer), and then you tried to
measure the
velocity. Your initial x-measurement caused the wavefunction to "collapse"
into a pure Wc state
... and now the velocity is completely unknown; you'll get v1, v2, and v3 with
equal probability
every time you repeat this experiment. So it's really true that the first
measurement of v
*erases* information about x, and vice-versa.

That's it for the toy model. In the real world, there are an infinite number of possible wavefunctions, which can give any value of x and any value of v. These wavefunctions can add up to give you a particle with a well-defined position (i.e., a wavefunction with a very narrow range of likely position-answers) or a particle with a well-defined velocity (i.e., with a narrow range of possible velocity-answers.) But there's no way to do both at the same time. The well- defined-position-wavefunctions will give a wide range of probable velocities, and the well- defined-velocity wavefunctions will give a wide range of probable positions. You can build an in-between wavefunction, with a sort-of-tight position and a sort-of-tight velocity, but Heisenberg's Uncertainty Principle tells you exactly how close you can get. There just isn't a wavefunction you can write down, according to the quantum mechanics equation, which gives narrow answers to both position and velocity.

You could go ahead and do the sequential velocity/position experiment you
describe, starting
with the same wavefunction every time. You will find that, when you repeat the
experiment, the
*second* step---the speed measurement in your example---will give a
different
answer every
time, no matter how consistent the position measurements were. The more
tightly you
constrain the first measurement, the more spread you will see in the second one.
(If you
*average together* the answers from all of your measurements, you'll get
an answer
which
reproduces classical mechanics----a bouncing quantum particle will, on average,
behave like a
classical billiard ball---but quantum behavior governs the probabilities and the
variance in the
measurements.)

If you want to learn more, it's useful to study the Stern-Gerlach experiment. This is a very clear setup in which you measure an atom's spin; if you try to measure the spin in the up-down direction, you erase any information about the sideways direction, and vice versa.

Hope this helps,

-Ben

PS. Another good way to understand position/momentum interaction is by thinking about listening to a sound. Supposing that a note is played on a violin, and someone asks you: at what time was the note loudest? What was the note's frequency (tone)? If the violinist plays a very long, sustained tone, it'll be easy to figure out its frequency---but you might have trouble pinning down the exact time that the volume peaked. If the violinist plays a very short note, it's easy to say what time it occurred, but hard to identify the frequency. In between, a medium-length note might allow you to get a reasonable (but not excellent) frequency and a reasonable (but not excellent) timing measurement. This is very similar to the situation with wavefunctions: a velocity measurement is basically a frequency measurement, which is most accurate when applied to a long sine wave. A position measurement is most accurate when applied to a very sharp wavepacket, sometimes called a "delta function", with a wide range of probable frequencies inside it.

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