Exploring the Uncertainty Principle of Quantum Physics

[maxwilli06]

So I am just beginning to emerge myself into quantum physics.

I am reading some things about the uncertainty principle and find myself very confused.
Why exactly is locating a particle in a small region of space make the velocity of that particle uncertain and vice versa?

I'm guessing it's because its really hard to measure the speed of a quantum particle when you slow it down enough to find it's position, and really hard to measure a position when a particle is moving. Is this right? So the more you know about its position, the less you know about its speed, and the more you know about its speed, the less you know about its position.

And why is the shorter the wavelength, the greater the certainty of a position, and the longer is less certain?

 

[PseudoIntellect]

Remember, the only way for an object to be visible is to shine photons on it and let those particles reflect back you or to let it emit photons of its own. In every-day life, our objects are too massive to be affected much by photons bouncing off of them. On the quantum level, electrons are so tiny that shining even a few photons on it will change its position significantly. Shooting more photons at it means you will get more of a reading of its position, but this also means you'll disrupt the nature of its direction/velocity a lot. Shooting fewer photons gives you less of a reading of its position, since the image you get back will be fainter and more obscure. However, you'd be disturbing its direction and velocity by much less. This is why there is a limit to the degree of certainty according to the principle.

A shorter wavelength of a particle means if you have less of a margin for error within wavelengths. Think of a ruler or measuring unit; shorter lengths (millimeters) are more certain and precise. Longer lengths (meters or kilometers) have more margins for error. Think of the wavelength in relationship to the frequency and how many tick marks there are on the ruler.

 

[malawi_glenn]

But does not that tell you that its uncerntanty is due to your experiment and NOT an Intrinsic property of the electron?

The uncertainty principle just arrives from the formalism of Quantum Mechanics, it does not tell you anything about photon scattering etc, just the commutator relations of p and x operators.

 

[Gerenuk]

My biggest advice is to read only little about popular word explanation of quantum physics, but instead read a proper book about the basic mathematical treatment of quantum mechanics. For example Cohen-Tannoudji is quite good.
Words only confuse you and yield more unanswered questions. A simple equations answers them all.

To your question:
A particle can have a distribution of positions. Each position has a probability and a phase however. Mathematically one can transform this information as it is into a distribution of momenta. So the momenta are not free to choose.
One can show that the spread in particle position is inversely proportional to the spread in momenta with that transformation.

The fact that the probabilities have phases is not straightforward to imagine. I'll think about that later...

 

[malawi_glenn]

That is my advice too :-)

 

[maxwilli06]

I was looking at that Cohen-Tannoudji textbook. Unfortunately it's a little out of my price range. Maybe you have a cheaper recommendation, perhaps something under 30$? Or maybe you have and example you could give me that leads to Planck's constant, or some otherway of understanding uncertainty using formulas?

 

[malawi_glenn]

almost any introductory QM book will do, check your library.

also there are millions of free tutorials and lecture notes on the internet.

http://farside.ph.utexas.edu/teaching/qmech/lectures/lectures.html

http://www.nyu.edu/classes/tuckerman/stat.mech/lectures/lecture_12/node1.html

 

[maxwilli06]

thanks for those links. funny that you referred me to an nyu site, since i go there.

 

[malawi_glenn]

I did not know that, enjoy and good luck :-)

If you have any more questions, about this after you have read this, ask :-)