The Uncertainty principle says that (delta x) (delta p) > hbar/2. So if you measure position exactly then the particle doesn’t have a momentum. How can you measure a particle’s position without also measuring its momentum?
If I catch a ball how could I not know both its position and its momentum? If I have an experiment that catches an electron how could I not know both its position momentum? How could you set up an experiment that only measures one these quantities?
That’s QtM, and I’m rarely uncertain. Wrong perhaps, but not uncertain.
QtM
Baseballs and particles behave very differently. A baseball is a vast collection of particles, and behaves in a classical manner, obeying all of Newton’s Laws of Motion, etc. Observing a baseball involves bouncing light, radar or even sound waves off of it, and making various measurements of the reflections. Whatever type of energy is used to measure the baseball only very slightly affects its trajectory, so little, in fact, that the effect is virtually unmesaurable. To deternime the ball’s position at a given instant only requires one reading, but to determine it’s velocity requires at least two. However, since the first reading didn’t affect the ball in the slightest (as far as our measuring equipment is concerned) a second reading will be just as accurate as the first, so there is no uncertainty (within the limits of the measurements). That is to say, we can determine the position and the velocity of the ball at any given moment to any arbitrary degree of accuracy.
A particle is a bit diferent. Any observation we make invariably affects the particle in some substantial way. For instance, if we try to measure the position of a particle in motion, we must observe some interaction between that particle and either another particle, or a field of some type (such as a magnetic field), but doing so alters it’s velocity. Any attempt to simultaneously measure both quantities will invariably result in measuring only one to any arbitrary degree of certainty. The other quantity will be uncertain. This is, of course, vastly oversimplified, but it illustrates the basic idea. For more information on QM, I highly suggest Alice in Quantumland, by Robert Gilmore. It’s a highly entertaining and educational look at the concepts behind quantum physics.
First, remember that momentum is a vector that depends on kinetic energy and direction of travel. Position is just that, X,Y and Z position at a given time.
I’ve never worked with electron detectors so let me talk about photons instead. The most common method of measuring position is to use an imaging sensor such as film or CCD. When a photon hits the sensor, it is absorbed (destroyed) and the energy converted into electrons which is then measured as an electrical signal, or causes chemical reactions in case of film. So you have measured the position but now the original photon is gone, so you have no idea which direction it was travelling, and how fast.
If instead you want to know the momentum, you use a telescope or camera. A lens focuses light onto a detector so that all light coming from a particular direction falls on the same spot. You also use a color film so you know the energy of the incoming photon. Now you have a system that measures momentum. But you’ve also lost the positional information; all you know is that the light hit somewhere on the lens. If you make the lens smaller you can narrow down the position of the photon, but then diffraction causes the image to get fuzzy so you lose the accuracy of momentum measurement. On an astronomical telescope you don’t care where the photon hit, you just care where it came from, so you use as large a mirror as possible. You can work out the resolution limit by treating light as a wave, or you can just apply the uncertainty principle and get the same number.
Thanks scr4 that makes sense. But when you say
couldn’t you theoretically use the impact on the sensor to determine the momentum?
No, because when a photon is absorbed, the energy depends on the wavelength of the photon, not it’s momentum.
I would say that a baseball can be treated as a quantum object, just like a particle can; the only difference is one of scale. So if delta-x = 10[sup]-7[/sup] m for a particle, we think of that as a “big” uncertainty, but for a baseball, it would be “small”. That doesn’t mean that the uncertainty principle doesn’t apply to a baseball, or a planet, or a galaxy.
Also, a photon’s wavelength and momentum are the same quantity, directly related by lambda = h / p. You can’t measure a photon’s wavelength and position both to arbitrary precision.
As I understand it, The Uncertainly Principle has nothing to do with disturbing a system during the act of observation. That’s called the Measurement Problem.
The Uncertainty Principle states that uncertainty is inherent and cannot be taken away. It is always there, whether we measure the system or not.
Slip is largely correct, though I would argue that the measurment problem is an aspect of the uncertainty principle. However, the uncertainty principle is not a “technology” problem. It is a fundamental aspect of the Universe.
The problem is that the position and momentum operators do not commute. The uncertainty prinicple is, therefore, a necessary consequence of QM.
“The Uncertainty principle says that (delta x) (delta p) > hbar/2. So if you measure position exactly then the particle doesn’t have a momentum. How can you measure a particle’s position without also measuring its momentum?”
Actually, if you measure the position exactly (delta x = 0) then the particle’s momemtum is not zero, it could be anything (delta p = infinity).
“If I catch a ball how could I not know both its position and its momentum? If I have an experiment that catches an electron how could I not know both its position momentum? How could you set up an experiment that only measures one these quantities?”
Actually, the uncertainty principle just “comes out in the math” for Quantum Mechanics. We can thank ol’ Heisenberg and his blackboard full of equations for that one. But if you want to undestand it physically, don’t think about baseballs-- they’re too big to behave in a quantum fashion. Think electrons. Then think of how you would measure it’s position-- maybe shine a light on it, ie, hit it with a photon. But the photon would greatly disturb the electron’s movement (and hence, momentum). The more light you shine on it (to get a better view of it’s position) the more you will disturb it’s motion. Then if you tried to measure it’s motion (or momentum), it will have been significantly changed from what it was when you determined it’s position.
My memory is a bit rusty, but I’m going to see if I can remember the concept behind the math. In QM, the momentum wave function is the Fourier Transform of the position wave function (and, of course, vice versa). To have an absolute location, the position wave function would have to be a delta function (zero everywhere except at one point where it is infinite). And What is the FT of a delta function-- well it’s a function made up equally of all frequencies. So, going into momentum space, the wave function has no “location”-- it has an equal amplitude at ALL (an infinite numnber of) frequencies, so all possible values for momentum have an equal probability.
Does anyone know if I got that right? I know I skipped a few steps, but is the outline correct? It’s been a LONG time…
I have been confused about the uncertainty principle- maybe someone could clear it up.
Momentum and position- two definitie properties. You can find both of them. Just not at the same time. If it has two properties, shouldn’t you be able to measure both regardless? Say, you make a quantum camera and replay it in slo-mo at precisely 1 billionth the speed. Is there something else to this? Granted, I haven’t taken advanced Chemistry or Physics yet (we sort of breezed through the principle without really going into any depth at all this year).
Think about how a camera works. You shine light on an object, the light reflects back, gets focused by a lens and makes an image. But if you try to do that to an electron, your illumination will knock the electron off course. You can make one image, but now the electron is moving on a different path and you’ll never know its original path.
The amazing and weird thing is that a particle does not possess an exact value of both properties at the same time. In measuring one property exactly, you have wiped out the possibility of there being an exact value for the other property. I.e. you have in a way altered reality, not just altered the possibility of measuring the other value. This is the standard interpretation of QM, which is considered to be proven by experiment.
The fourier transform of the position takes you into k-space which is directly related to momentum as the wavenumber of a wave can give you its momentum.
The way the math works is that whenever you have two operators (such as position and momentum) you can ask, “does it matter what order I take the operations in?” If it does, then you can calculate something called the “commutator” which is the amount by which the “uncertainty” in the two operations must be. In a practical (non-theoretical) sense, this is a lower bound when your operators are measurables (that is, they are Hermitian operators).
So, it happens that the commutator of the position <x| and the momentum operators <-ih_bard/dx| is h_bar/2. That’s where the Uncertainty Principle comes from.
How can this be done? In scr4’s experiments you can only measure one variable at a time. In order to prove the uncertainty relation it would seem you would have to measure both variables for a lot of particles and then do a statistical analysis. How would this be done?
Sacroiliac, the most famous demonstration of this is the [/url=http://www.anu.edu.au/Physics/courses/A07/studentsites/studentsites2001/WAG/DoubleSlit.html]Electron Double Slit Experiment.
There was a “definitive” experiment done by Aspect et al., testing ideas derived from the “EPR paradox”. Einstein didn’t believe in the implications of QM, and (along with Podolsky and Rosen) came up with a thought experiment, involving two correlated particles. Roughly, the idea was that you could measure the momentum (or other attribute) of particle A without disturbing particle B, but would therefore instantly know the momentum of B (since they were correlated), and so could measure the position of B without disturbing its known momentum. This seemed to argue for hidden determinism. It was shown by Bell that if there were such hidden determinism, ie if the particles did possess exact simultaneous values of both attributes being measured, a certain relationship had to hold between the measured values (“Bell’s inequality”). The Aspect experiment tested this, and found that Bell’s inequality was violated. Well, end of story for belief in simple deteriminism, although a (small) crack is left open for more bizarre theories of reality that still include some form of hidden variables.
JS Princeton when I tried that url I got a 404 error document not found.
This is the link JS Princeton meant to post. It works fine.