Breaking Down the Uncertainty Principle

[einsteinoid]

I'm currently in an intro chemistry class and a lot of the material is review from the multiple physics classes I have taken. Most of it comes back easily, including how to use the uncertainty principle to solve the small array of homework questions thrown at me but when trying to grasp a deeper understanding of the principle there is something I can't seem to figure out and it isn't mentioned in detail in the chemistry book.

I understand, conceptually, that the closer the uncertainty of x gets to zero, the more the uncertainty of v would approach infinity but algebraically, I'm not sure how he came up with the formula, more specifically where he got the 4pi from.

I recall from calculus that 4pi would be two rotations around a circle. This lead me to think that perhaps 4(pi) could be representative of the period of the wave exhibited by the particle--assuming the particle to have both particle and wavelike properties. If this were true, putting the period of the wave in the denominator (as it is in the uncertainty principle) would be equivalent to multiplying by the reciprocal of the period (the frequency).

However the frequency would be unknown, would it not? So the period would have to be unknown as well.

Here I go confusing myself again, i'll quit rambling now.

In summary, where does the 4(pi) come from in the denominator of Heisenberg's uncertainty principle?

 

[Feldoh]

The might be a bit hard to explain for an intro chemistry class. Physically it doesn't really mean anything important. In fact physicists typically define a new value:

$$\hbar = \frac{h}{2 \pi}$$ which makes the uncertainty $$\sigma_x \sigma_p \geq \frac{\hbar}{2} = \frac{h}{4 \pi}$$

The Heisenberg Uncertainty Principle is simply a mathematical limit on how well we can measure two certain quantities. In fact there are as many uncertainty relations as quantities we can measure.

The value "h-bar" divided by 2 is simply the relation between position and momentum.

 

[victorphy]

I think you can take a look at some books on QM,like Sakurai's.

 

[Fredrik]

The Heisenberg Uncertainty Principle is simply a mathematical limit on how well we can measure two certain quantities.

This is an accurate description of the original version of the uncertainty principle, but the modern version has a very different interpretation. The original uncertainty principle predates quantum mechanics. It provided some of the motivation for the mathematical structure of QM, but it's not actually a part of the theory. The modern uncertainty principle is a mathematical theorem in QM, and as I said, its interpretation is very different. See this thread for more.

 

[Gerenuk]

In summary, where does the 4(pi) come from in the denominator of Heisenberg's uncertainty principle?

For that you probably would have to look at one of them mentioned mathematical derivations. The 2pi comes essentially from the Fourier transform, one could say.

The uncertaintly principle isn't an additional law in quantum mechanics, but rather follows from the structure that you are allowed to use to describe physics. Basically, one can only specify the position distribution of the electron. What velocity it has follows directly from this mathematically, so you are not free to chose its velocity anymore. Playing around with this contraint, one can find that either you have a single position and a mix of momenta OR you have a mix of positions and a single momentum. You cannot make both to be a single value.

 

[Feldoh]

This is an accurate description of the original version of the uncertainty principle, but the modern version has a very different interpretation. The original uncertainty principle predates quantum mechanics. It provided some of the motivation for the mathematical structure of QM, but it's not actually a part of the theory. The modern uncertainty principle is a mathematical theorem in QM, and as I said, its interpretation is very different. See this thread for more.

The more generalized version is exactly the one you have derived. I don't think you read what I was saying very carefully.

 

[Fredrik]

The more generalized version is exactly the one you have derived. I don't think you read what I was saying very carefully.

I don't think you read what I said very carefully. The two versions of the uncertainty principle that I was talking about isn't a "generalized" and a "specific" result. I'm distinguishing between mathematical theorems derived from the axioms of QM and a heuristic result that was found before the axioms of QM had even been written down for the first time.

The Heisenberg Uncertainty Principle is simply a mathematical limit on how well we can measure two certain quantities.

As I tried to tell you before, this is the correct interpretation of the old heuristic result, but if it's supposed to be an interpretation of the theorem I derived, it's just wrong. QM says that a particle doesn't have a well-defined position and a well-defined momentum at the same time, so it's not just about how well we can measure those quantities.

 

[Gerenuk]

True, but it still includes the fact that there is a mathematical limit to how well we can measure quantities. Regardless of why it happens it still does happen, does it not? We still cannot measure incompatible observables to arbitrary precision.

It is correct to say "includes the fact".
You should just be aware of that the reason for these limits is not the measurements, but just that there are physical contraints that these quantities cannot exist together regardless of whether you measure or not.

 

[einsteinoid]

For that you probably would have to look at one of them mentioned mathematical derivations. The 2pi comes essentially from the Fourier transform, one could say.

Ahhh, this is more what I was looking for.

Thanks for all of the input, everyone.