# Joint distribution of position and momentum

• atyy
In summary, the quantum mechanics literature seems to be divided on whether there is a joint distribution of position and momentum. Some argue that there is, while others argue that this is not the case. Bohmian mechanics, an interpretation of QM, suggests that there is in fact a joint distribution, but it is not the same as the classical distribution.
atyy
In quantum mechanics, there doesn't seem to be a joint distribution of position and momentum.

But in Bohmian mechanics there is.

But Bohmian mechanics is quantum mechanics, so what is the error in my reasoning?

atyy said:
In quantum mechanics, there doesn't seem to be a joint distribution of position and momentum.
Isn't this what they call the Wigner distribution function?

Bill_K said:
Isn't this what they call the Wigner distribution function?

No, it isn't. The Wigner distribution is the closest thing, but in general it has negative bits, so it isn't a true probability distribution. In special cases like the free Gaussian wave packet, the Wigner distribution is positive, which explains why in those cases one can think of classical trajectories.

atyy said:
The Wigner distribution is the closest thing, but in general it has negative bits, so it isn't a true probability distribution.
Heh, well, quantum probability is not the same thing as classical probability anyway.

The "joint" distribution stuff is one of the places where differences arise: Cox's 4th axiom is not applicable as-is, since the notion of ##A \& B## is problematic in QM. So one must use a time-ordering instead.

atyy said:
But Bohmian mechanics is quantum mechanics, so what is the error in my reasoning?

Not quite - its an interpretation of QM.

It has hidden variables not in standard QM and they have properties different to the usual quantum properties eg the particle has both a definite momentum and position. But because of that pesky pilot wave that guides it you can't determine what it is - that's why its hidden.

For more details check out:
http://philsci-archive.pitt.edu/3026/1/bohm.pdf

Thanks
Bill

atyy said:
In quantum mechanics, there doesn't seem to be a joint distribution of position and momentum.

But in Bohmian mechanics there is.
There seem to be several ways to define an "actual" momentum in dBB but it doesn't seem possible to match this actual momentum with the outcome of momentum measurements. From http://arxiv.org/abs/quant-ph/0408113 (p13): "Insisting on the belief that Newtonian momentum (energy, angular momentum) measurements reveal the momentum (energy, angular momentum) leads to the orthodox view of quantum mechanics."

I don't know any details. I just googled it because I've asked myself similar questions before.

1 person
kith said:
There seem to be several ways to define an "actual" momentum in dBB but it doesn't seem possible to match this actual momentum with the outcome of momentum measurements. From http://arxiv.org/abs/quant-ph/0408113 (p13): "Insisting on the belief that Newtonian momentum (energy, angular momentum) measurements reveal the momentum (energy, angular momentum) leads to the orthodox view of quantum mechanics."

I don't know any details. I just googled it because I've asked myself similar questions before.

I think that must be the answer.

## What is the joint distribution of position and momentum?

The joint distribution of position and momentum is a probability distribution that describes the likelihood of finding a particle at a certain position and with a certain momentum. It is a fundamental concept in quantum mechanics and is used to understand the behavior of particles on a microscopic level.

## How is the joint distribution of position and momentum related to Heisenberg's uncertainty principle?

The joint distribution of position and momentum is related to Heisenberg's uncertainty principle, which states that it is impossible to know both the exact position and momentum of a particle simultaneously. The joint distribution quantifies this uncertainty by showing the range of possible values for both position and momentum.

## What is the difference between the joint distribution of position and momentum and the probability distribution of position or momentum alone?

The joint distribution of position and momentum takes into account both the position and momentum of a particle, while the probability distribution of position or momentum alone only considers one of these variables. The joint distribution gives a more complete picture of the particle's behavior and can reveal relationships between position and momentum.

## How is the joint distribution of position and momentum calculated?

The joint distribution of position and momentum is calculated using mathematical formulas, such as the Schrödinger equation in quantum mechanics. This equation takes into account the wave function of a particle, which describes its position and momentum at any given time.

## What insights can be gained from analyzing the joint distribution of position and momentum?

By analyzing the joint distribution of position and momentum, scientists can gain insights into the behavior and properties of particles at the quantum level. This can help us understand fundamental principles of physics and make predictions about the behavior of particles in different scenarios.

Replies
6
Views
2K
Replies
10
Views
2K
Replies
18
Views
2K
Replies
15
Views
1K
Replies
3
Views
797
Replies
3
Views
1K
Replies
12
Views
1K
Replies
17
Views
2K
Replies
24
Views
1K
Replies
2
Views
1K