Uncertainty Principle: Velocity, Position and Energy

[Kiley]

If velocity is delta position vs delta time and you know the velocity and change in time exactly why is it impossible to find the exact position of the electron? Same question for energy and position.

 

[Kiley]

Ps. I don't know if the implications of my question are entirely correct, this is just how I interpreted what my book said.

 

[berkeman]

If velocity is delta position vs delta time and you know the velocity and change in time exactly why is it impossible to find the exact position of the electron? Same question for energy and position.

Welcome to the PF.

Please post links to the reading you have been doing about this question, and point to parts of that reading that are confusing you. Thanks.

 

[berkeman]

Like here: https://en.wikipedia.org/wiki/Uncertainty_principle

 

[PeroK]

If velocity is delta position vs delta time and you know the velocity and change in time exactly why is it impossible to find the exact position of the electron? Same question for energy and position.

The Heisenberg Uncertainty Principle (HUP) is a statistical law. It applies to the measurements of, for example, momentum and position on an ensemble of identically prepared particles. If you prepare a large number of particles and measure, say, the momentum (at some time $t$) for half of them and the position (at time $t$) for the other half of them, then you will get a spread of measurements for both momentum and position. If you then take the standard deviation of these measurements, then they obey:
$$\sigma_x \sigma_p \ge \frac{\hbar}{2}$$
Where $\sigma_x, \sigma_p$ are the standard deviations for position and momentum respectively. One interpretation of this is that if you prepare a state with a well-defined momentum, then that state will have a relatively large spread of position measurements; and, vice versa.

The HUP doesn't say anything in particular about any single measurement of position or momentum of a particle.

 

[Kiley]

C. Chemistry: a molecular approach, Tro.
Can you use mechanics equations for this? If not, why not?

 

[Kiley]

The Heisenberg Uncertainty Principle (HUP) is a statistical law. It applies to the measurements of, for example, momentum and position on an ensemble of identically prepared particles. If you prepare a large number of particles and measure, say, the momentum (at some time $t$) for half of them and the position (at time $t$) for the other half of them, then you will get a spread of measurements for both momentum and position. If you then take the standard deviation of these measurements, then they obey:
$$\sigma_x \sigma_p \ge \frac{\hbar}{2}$$
Where $\sigma_x, \sigma_p$ are the standard deviations for position and momentum respectively. One interpretation of this is that if you prepare a state with a well-defined momentum, then that state will have a relatively large spread of position measurements; and, vice versa.

The HUP doesn't say anything in particular about any single measurement of position or momentum of a particle.

Thank you, this is helpful in part.

 

[PeroK]

View attachment 220427 C. Chemistry: a molecular approach, Tro.
Can you use mechanics equations for this? If not, why not?

It's more usual when dealing with quantum particles to express the kinetic energy in terms of momentum:

$T = \frac{p^2}{2m}$

The expression $T = \frac12 mv^2$ looks a little out of place.

Note that what your book doesn't emphasise is that the energy of an electron in an atom has two components: potential energy and kinetic energy. If an electron is in a specific energy eigenstate, therefore, that state does not have a specific kinetic energy or a specific potential energy; but, a specific total energy.

 

[PeroK]

Thank you, this is helpful in part.

PS It's also worth noting that in three dimensions, position and momentum in different directions are compatible, in the sense that:

$\hat{x}$ commutes with $\hat{p_y}$ and $\hat{p_z}$ etc.

Where $\hat{x}$ represents the observable of position in the $x$ direction and $\hat{p_y}, \hat{p_z}$ represent the observables of momentum in the $y, z$ directions respectively.

 

[Kiley]

PS It's also worth noting that in three dimensions, position and momentum in different directions are compatible, in the sense that:

$\hat{x}$ commutes with $\hat{p_y}$ and $\hat{p_z}$ etc.

Where $\hat{x}$ represents the observable of position in the $x$ direction and $\hat{p_y}, \hat{p_z}$ represent the observables of momentum in the $y, z$ directions respectively.

Wow, thank you that's very cool. Are there any textbooks you can recommend specifically about this?

 

[PeroK]

Wow, thank you that's very cool. Are there any textbooks you can recommend specifically about this?

I like Griffiths book on QM. Not everyone on PF would agree with that! But, it's a fairly standard undergrad introduction. But, you might be best to talk to your department about how much QM you are expected to learn.

 

[Kiley]

I like Griffiths book on QM. Not everyone on PF would agree with that! But, it's a fairly standard undergrad introduction. But, you might be best to talk to your department about how much QM you are expected to learn.

Awesome, thank you so much for your help, I was very nervous posting on here, so thank you for not being mean.