Uncertainty principle presentation

[sythrox]

Hi all.

In my physics class I have to give a 15 minute blackboard lecture on a topic in modern physics. I chose the uncertainty principle because I feel there is a tendency (at least at the undergrad level) to just accept what it is without proof or intuition. I would like to go into greater depth with this topic. The problem is I am having a hard time finding good resources. They are either too simple or too complicated. I don't mind the complexity of a proof but I don't want my lecture to be strictly proof orientated. I'm trying to strike a balance between proof and intuition.

Any suggestions folks?

Also, I am entertaining the idea of beginning by presentation with showing how movement/translation through conserved space can be expressed by a series expansion and in turn by a wave function and THEN go into the uncertainty principle but my knowledge is lacking in this area and I don't want to...bite off more than I can chew.

It should also be stated that the class has only had introductory physics prior to this class.

Thanks!

 

[bhobba]

The best way is to state the principle correctly and take it from there.

The correct statement is suppose you have a large number of similarly prepared systems and divide it into two lots . In the first lot you measure position. There is nothing stopping each measurement being as accurately as you like. In the second lot you measure momentum. Again there is nothing stopping doing the measurement as accurately as you like. When you compare the statistical spread (standard deviation) of the two results it must always be as per the uncertainty relations.

Note:
1. The uncertainty principle puts no constraint on how accurately you can measure anything.
2. It is a statistical statement about a large number of similarly prepared systems.
3. It brings home the point that a quantum state and system preparation procedure are basically synonymous and the state applies to a statistical ensemble of such systems.

I have zero idea what you mean by 'movement/translation through conserved space can be expressed by a series expansion and in turn by a wave function'

A wave-function is simply the expansion of the state in terms of the position observable.

Thanks
Bill

 

[sythrox]

Okay. I was unsure weather to include that last bit. Disregard. But I'm confused why the uncertainty principle puts no constraint on how accurately you can measure something. What if were talking only about a single electron?

 

[bhobba]

Okay. I was unsure weather to include that last bit. Disregard. But I'm confused why the uncertainty principle puts no constraint on how accurately you can measure something. What if were talking only about a single electron?

Its right at its foundations. Here are the two axioms of QM:

Axiom 1
Associated with each measurement we can find a Hermitian operator O, called the observations observable, such that the possible outcomes of the observation are its eigenvalues.

Axiom 2 - called the Born Rule
Associated with any system is a positive operator of unit trace, P, called the state of the system, such that expected value of of the outcomes of the observation is Trace (PO).

Axiom 1 says nothing about the accuracy of the measurement - that entirely depends on the accuracy of what you are doing the measurement with.

Think of a slit you are passing an electron through. The slit can be made any width - the narrower it is the more accurate the position the electron will be measured just behind the slit. There is no limit in principle at all to how accurately you can measure any observable in QM.

In fact this is responsible for the double slit experiment:
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

Briefly the quantum explanation is just behind the slit we know its position very accurately hence its momentum is unknown and could be anything. But its kinetic energy remains the same hence the size of the velocity is the same but its scattered in an unknown direction. When you have two slits its obviously a superposition of the state with a single slit and you get an interference pattern as explained in the above paper.

Thanks
Bill

 

[atyy]

Why not use the signal processing approach, where the uncertainty principle is between frequency and time?

To define frequency we need many cycles of a sinusoidal wave. If there are many cycles, the wave is spread out in time.

If the wave is sharply localized in time, it is a superposition of many sinusoids with different frequencuies, so it cannot have a well-defined frequency.

The uncertainty principle in quantum mechanics is exactly analogous, except that time becomes position and frequency becomes momentum.

Why is momentum a "frequency"? That was de Broglie's big discovery of wave particle duality, building on Einstein's intuition that wavelike light can have particle properties.

http://www.fas.harvard.edu/~scidemos/QuantumRelativity/UncertaintyPrinciple/UncertaintyPrinciple.html

http://newt.phys.unsw.edu.au/jw/uncertainty.html

http://chemwiki.ucdavis.edu/Physica...ts_of_Quantum_Mechanics/De_Broglie_Wavelength

 

[atyy]

But I'm confused why the uncertainty principle puts no constraint on how accurately you can measure something. What if were talking only about a single electron?

Quantum mechanics does put constraints on how accurately one can simultaneously measure position and momentum, because of the canonical commutation relations.

However, the uncertainty principle (in the form usually found in textbooks), although also a consequence of the canonical commutation relations, say nothing about the simultaneous measurement of position and momentum. First, quantum mechanics only makes pobabilistic predictions. So if we want to test the predictions of quantum theory, we must always do it on ensembles of particles. In the simplest case, we take an ensemble of particles in which every particle is in the same quantum state. If we accurately measure position on every particle in the ensemble, we will get some distribution of positions. If we accurately measure momentum on every particle in *another* ensemble that is identical to the one we made position measurements on, we will get a distribution of momenta. The uncertainty principle is about the spread in the distribution of positions and the distribution of momenta.

 

[vanhees71]

Quantum mechanics does put constraints on how accurately one can simultaneously measure position and momentum, because of the canonical commutation relations.

This is a dangerous statement, because it's not entirely correct (see bhobbas posting #2). The statement becomes correct when writing

"Quantum mechanics does put constraints on how accurately one can simultaneously measure position and momentum on a single particle, because of the canonical commutation relations."

 

[dextercioby]

And then of course you have the ensemble interpretation which says from the beginning that measurements on a single particle make sense, iff one has an infinity of identically prepared particles and on all of them certain observables are (simultaneously or not) measured. But in the end, it's not QM which puts constraints on how accurately measurements are done (this constraint is of purely mechanical nature, such as any human-built measurement apparatus has a small δ - like a ruler or calipers), but only tells you that there are some observables whose measured values have a statistical spread around the ensemble mean.