Meaning of Expectation Values for <x^2> and <p^2> in Classical Mechanics

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In summary, the conversation discusses the meaning and calculation of quantum mechanical operators and their relationship to observable quantities in classical mechanics. The expectation value <x^2> represents the average value of the square of the position, while <v^2> represents the average value of the square of molecular speeds. The square of the uncertainty in position is represented by \Delta x = \sqrt {<x^2> - <x>^2} and becomes exact as the number of measurements approaches infinity.
  • #1
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Every quantum mechanical operator has an observable in classical mechanics

<x> - position
...
<x^2> - ?
<p^2> - ?

What is the meaning on these expectation values?

v^2 = <x^2> - <x>^2

What is the meaning of this? edit: It looks to me like uncertainty in position. Is it the average uncertainty in position?

Admin Please move me to the right forum, if I'm not in eet.
 
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  • #2
13characters said:
v^2 = <x^2> - <x>^2

This is the square of the uncertainty in position. That is,

[tex]\Delta x = \sqrt {<x^2> - <x>^2}[/tex]
 
  • #3
jtbell said:
This is the square of the uncertainty in position. That is,

[tex]\Delta x = \sqrt {<x^2> - <x>^2}[/tex]

That's what i figures too after some digging in the kinetics section of my chem book

cheers.

i still don't get what <x^2> means. What i found was "the average of the square of molecular speeds" but i still don't entirely get why its operator is X^2.

edit: Also is there a ways to represent it geometrically or graphically
 
  • #4
13characters said:
i still don't get what <x^2> means.

It's simply the average value of the square of the position, in the limit as the number of measurements goes to infinity. If you have N measurements of the position, then

[tex]<x^2> = \frac{1}{N} \sum_{i=1}^N {x_i^2}[/tex]

Actually this is only an approximation. It becomes exact as [itex]N \rightarrow \infty[/itex].

The "average of the square of molecular speeds" would be [itex]<v^2>[/itex], calculated similarly, but with v replacing x.
 
  • #5
jtbell said:
It's simply the average value of the square of the position, in the limit as the number of measurements goes to infinity. If you have N measurements of the position, then

[tex]<x^2> = \frac{1}{N} \sum_{i=1}^N {x_i^2}[/tex]

Actually this is only an approximation. It becomes exact as [itex]N \rightarrow \infty[/itex].

The "average of the square of molecular speeds" would be [itex]<v^2>[/itex], calculated similarly, but with v replacing x.

i get it now.

thanks.
 
Last edited:

What is the meaning of expectation values in classical mechanics?

Expectation values refer to the average value of a physical quantity in a given system. In classical mechanics, this is calculated using the probability distribution of the system's state.

What is the significance of and in classical mechanics?

and represent the expectation values for the position and momentum, respectively, in classical mechanics. They provide information about the spread or uncertainty in the system's position and momentum.

How do you calculate the expectation values for and in classical mechanics?

The expectation value for is calculated by taking the product of the position operator and the wave function, and then integrating over all possible positions. The same process is used for , but with the momentum operator and the wave function.

What do the expectation values for and tell us about the system?

The expectation values for and can tell us about the spread or uncertainty in the system's position and momentum, respectively. They can also provide information about the average position and momentum of the system.

How do expectation values in classical mechanics differ from those in quantum mechanics?

In classical mechanics, expectation values are calculated using the probability distribution of the system's state, while in quantum mechanics, they are calculated using the wave function. Additionally, in quantum mechanics, the uncertainty principle limits the precision with which both position and momentum can be known simultaneously, while in classical mechanics, there are no such limitations.

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