How Does the Uncertainty Principle Affect Momentum Measurement Accuracy?

[tymartin]

Homework Statement

Suppose you want to measure the position of an electron to within 1 angstrom. If you also want to measure its momentum, how accurately can you do this if you don't want to mess up your measurement of its position?

Homework Equations

An angstrom is 1 x 10-10

The Attempt at a Solution

Well the principle states that it's impossible to know both the position and momentum of a particle at the same time. My question is to what degree of accuracy can I measure the position of the electron?

 

[LowlyPion]

http://en.wikipedia.org/wiki/Uncertainty_principle

 

[nlightner]

The Uncertainty Principle, also known as the Heisenberg Uncertainty Principle, states that it is impossible to simultaneously know the exact position and momentum of a particle. This is due to the fact that the act of measuring one of these properties will inevitably disturb the other. In the case of an electron, which is a subatomic particle, this disturbance can be significant even at very small scales.

In order to better understand this principle, let's consider the example given in the homework statement. If we want to measure the position of an electron to within 1 angstrom, which is a very small unit of measurement (1 x 10^-10 meters), the uncertainty principle tells us that the accuracy of our measurement of its momentum will be greatly affected. In other words, the more precisely we try to measure the position of the electron, the less precisely we can measure its momentum.

This is because in order to measure the position of the electron, we need to interact with it in some way, such as by using a photon to "bounce off" the electron and determine its position. However, this interaction also changes the momentum of the electron, making it impossible to know both its position and momentum with 100% accuracy.

To answer the question, if we want to measure the position of the electron to within 1 angstrom, the uncertainty principle tells us that the accuracy of our measurement of its momentum will be on the order of 1 x 10^-10 kg*m/s. This means that if we want to know the momentum of the electron to within 1% accuracy, we would need to measure it to within 1 x 10^-12 kg*m/s. This may seem like a very small difference, but at the subatomic level, it can have significant implications.

In conclusion, the uncertainty principle reminds us that there are fundamental limits to what we can know about the behavior of particles at the subatomic level. It is an important concept in the field of quantum mechanics and has been confirmed by numerous experiments. While it may seem counterintuitive, it is a fundamental aspect of the physical world and has greatly contributed to our understanding of the universe.