Energy & Time, Momentum & Position

In summary, the energy-momentum 4-vector of SR is related to the energy-time and momentum-position relationships in QM uncertainty. However, there is no physical connection between them.
  • #1
LarryS
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In the Energy-Momentum 4-Vector of SR, energy is associated with the time coordinate and momentum is associated with the 3 spatial coordinates. Is this association mathematically related to the energy-time and momentum-position relationships in QM uncertainty? Thanks in advance.
 
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  • #2
no, they are related to the Space-time equations in Einsteins theory of special relativity.

i.e the proper length:
[tex] x^\mu x_\mu = \Delta s ^2 =\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2 - c^2 \Delta t^2}[/tex]
which is an invariant under Lorentz transformations, comes from the contravariant 4.vector:
[itex] x^\mu = (x,y,z,t) [/itex] together with the metric: [itex] g^{\mu\nu}= \text{diag}(1,1,1,-1) [/itex]

Same with momentum 4-vector squared, one obtains the rest-mass:
m^2 = E^2 - p^2
 
  • #3
There seems to be some sort of relationship

[tex]\Delta p \Delta x[/tex]

[tex](E, cp_i) \rightarrow p_\mu \ , \ E/c=p_0[/tex]

[tex](t, cx_i) \rightarrow x_\mu \ , \ x_0=ct[/tex]

Does this generalize to this?

[tex]\Delta E \Delta t[/tex]

There are also the relationships

[itex]p = \hbar k [/itex] relating p to x as
[tex]E = \hbar \omega [/tex] relates E to t.

p:x::E:t

But these are just yet mathematical games without any physics.
 
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  • #4
Phrak said:
There seems to be some sort of relationship, malawi.

[tex]\Delta p \Delta x[/tex]

[tex]E, cp_i \rightarrow p_\mu \ , \ E/c=p_0[/tex]

[tex]t, cx_i \rightarrow x_\mu \ , \ x_0=ct[/tex]

Does this generalize to this?

[tex]\Delta E \Delta t[/tex]

There are also the relationships

[itex]p = \hbar k [/itex] relating p to x as
[tex]E = \hbar \omega [/tex] relates E to t.

But these are just yet mathematical games without any physics.

Yes, there is not 'physical' connection in that sense, and mathematically the uncertainty relations comes from commutator relations. One does not say, "hey, let's take p_0 and x_0 and combine them into an uncertainty relation"..
There is no mathematical way to go from 4-vectors to uncertainty relation here, one just identify the zeroth component of the four vector of x and p, to be the same things that are involved in the uncertainty relations, this is by accident.

But on the deeper level, Energy is the generator of time evolution, and momentum is the generator of spatial translations. From this, one can go to special theory of relativity, and one can go to Quantum mechanics. Classical mechanics is the starting point for both, here lies the connection between SR and QM.
 
  • #5
How about that both relativity and QM come from wave equations?
 
  • #6
atyy said:
How about that both relativity and QM come from wave equations?

explain :-)

SR was realized by einstein from EM waves, but the theory is not really founded on it.
 
  • #7
After mulling this over a bit, I googled "energy-time uncertainty principle".

At the end of a Wikipedia article

http://en.wikipedia.org/wiki/Uncertainty_principle" [Broken]

is something about it, including the uncertainty as to what the meaning of [itex]\Delta T[/itex] should be.

Another article by Baez, 2000, may be a good one, that seems to confer.

http://math.ucr.edu/home/baez/uncertainty.html" [Broken]

I only scanned the both of them.

Yet I seem to recall a relationship between metastable time (delta T) of an electron in an orbital as being inversely proportional to the energy bandwidth (delta E).
 
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  • #8
malawi_glenn said:
explain :-)

SR was realized by einstein from EM waves, but the theory is not really founded on it.

Let's see, I was thinking something like this.

From a wave equation we get an uncertainty principle automatically.

If the wave equation is relativistic, or comes from underlying relativistic fields, then we get 4-vectors like (w,kx,ky,kz).

It's true we can think of Minkowski spacetime as fundamental. Alternatively, if we have the dynamical laws of physics (standard model) in one inertial frame, we can infer Lorentz covariance and Minkowski spacetime.

But actually thinking this way, the uncertainty principle and 4-vectors aren't really related, since we can have non-relativistic wave equations like Schrodinger's, and the uncertainty relations will still hold, without there being any 4-vectors.
 

1. What is the relationship between energy and time?

The relationship between energy and time is complex and can vary depending on the specific context. In general, energy is the ability to do work and is measured in joules (J). Time, on the other hand, is a measure of duration and is typically measured in seconds (s). In physics, the concept of time is often used to describe changes in energy over a period. For example, power, which is the rate at which energy is transferred or used, is measured in joules per second (J/s) or watts (W). In other words, energy and time are closely related because time is often used to describe the changes in energy, such as how quickly energy is being transferred or used.

2. How does momentum relate to position?

Momentum is a key concept in physics and is defined as the product of an object's mass and velocity. In other words, it is a measure of an object's motion. On the other hand, position refers to an object's location in space. The two are related in the sense that an object's momentum can affect its position. For example, a moving object with a high momentum will be harder to stop and will travel a greater distance before coming to a stop compared to an object with a lower momentum. In this way, momentum and position are related because momentum can influence an object's position in space.

3. What is the principle of conservation of energy?

The principle of conservation of energy is a fundamental law in physics that states that energy cannot be created or destroyed, only transferred or converted from one form to another. This means that the total amount of energy in a system will remain constant over time. For example, if a ball is rolling down a hill, it will convert its potential energy (due to its position on the hill) into kinetic energy (due to its motion). The total amount of energy in the system (ball + Earth) will remain the same, as energy is simply being transferred between different forms.

4. How is time dilation related to momentum?

Time dilation is a phenomenon in which time appears to pass slower for an object that is moving at high speeds. This is a consequence of Einstein's theory of relativity, and it is related to momentum because an object's momentum increases as its speed increases. As an object approaches the speed of light, its momentum increases exponentially, and time dilation becomes more significant. In other words, the faster an object moves, the more time dilation will occur, and the more significant the object's momentum will be.

5. What is the uncertainty principle and how does it relate to energy and position?

The uncertainty principle is a fundamental concept in quantum mechanics that states that it is impossible to know both the exact position and momentum of a particle at the same time. This means that the more precisely we know a particle's position, the less precisely we can know its momentum, and vice versa. This principle is related to energy and position because energy is a property of a particle's momentum, and by knowing its position with more certainty, we are sacrificing our knowledge of its momentum and therefore its energy. This principle has significant implications for understanding the behavior of particles on a quantum level.

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