Connection between Planck's Constant and Action?

[Mercy]

Hello,

I've noticed that Planck's constant h and the action of a particle S both share the units of Joule-second. I was wondering if there were a connection between the two, but my Modern Physics textbook (Harris) doesn't say anything about it. Wikipedia's definition of Planck's constant says there is a connection but doesn't really go into detail; I was hoping someone here could describe the connection and its implications.

Cheers

 

[DuckAmuck]

h has the units of angular momentum, so does action.

It's sort of like Torque and Energy having the same units, even though they aren't the same thing, although they are related.

Torque of 1 Nm applied over 1 radian requires 1 Joule of energy.

 

[Mercy]

Yes I know they are related by units but I am wondering what their physical/qualitative relationship is. I know that angular momentum is quantized via nћ and I know that linear momentum is defined by h / λ, but I want to know what is the connection to action.

 

[DuckAmuck]

Classically, they seem related by a dyadic multiplication.
The way dyadics work is by directly multiplying vectors.
It so happens that when you do that you get the dot and cross products.
[tex] XY = -X\cdot Y + X\times Y [/tex]

So if you have a vector [tex] \vec{\Gamma} = \alpha \frac{d\vec{r}}{dt} + \beta \vec{r} [/tex]
and you use dyadic multiplication, you can get:
[tex] \vec{\Gamma}\vec{\Gamma} = \alpha^2 \frac{d\vec{r}}{dt}^2 + \beta^2 \vec{r}^2 - 2\alpha\beta \frac{d\vec{r}}{dt} \cdot \vec{r} + 2\alpha\beta \frac{d\vec{r}}{dt} \times \vec{r} [/tex]

The first and last are the action and the angular momentum of a free particle, if α is in units of sqrt(mass*time), and β is in units of sqrt(mass/time).
Γ is basically a measure of distance for a moving object, like x + vt, with some extra factors out front.

 

[Mercy]

Oh interesting I haven't heard of dyadics before. Do those middle two terms have any physical interpretation? The third one looks like energy but I'm not sure what the second one is.
Do you have any reference materials where I could read more about this?

 

[DuckAmuck]

The second term looks like the potential for a harmonic oscillator (spring), and the third term looks like air resistance. The first term is kinetic energy times time (action of a free particle).

No, I don't have any materials, this is just my attempt at explaining the relationship between action and angular momentum.

As for dyadics, they are well documented on wikipedia.

 

[nasu]

This is not a well formulated question.
First, a particle does not "have action". The action is associated with a possible trajectory of a particle not with a state. It is an integral between two states.

And second, action and Plank's constant are two different categories. One is a physical quantity and the other is a constant.
Is like asking if there is any connection between c and velocity of a particle. They both have the same units, don't they?

 

[DuckAmuck]

This is not a well formulated question.
First, a particle does not "have action". The action is associated with a possible trajectory of a particle not with a state. It is an integral between two states.

And second, action and Plank's constant are two different categories. One is a physical quantity and the other is a constant.
Is like asking if there is any connection between c and velocity of a particle. They both have the same units, don't they?

I guess you could say the action in question here is the action of a system consisting of one particle. The question then becomes something like "what is the relationship between that action and the particle's angular momentum?"

 

[Mercy]

Nasu, the fact that one is a constant and one is a variable doesn't mean they aren't related in some way.

In the case of c and v, yes they share units but there are also physical relationships between the two such as 'an object's speed can never reach c', and of course the relationship between c and v has very important implications in special relativity, such as length contraction and time dilation.

I'm interested in knowing if there's any such relation between h and S.

If you have nothing to contribute to the question rather than saying 'it's poorly formulated' then please don't respond in the first place.

edit: sorry i don't mean to sound dickish but if the answer were obvious to me then I wouldn't be asking the question

 

[nasu]

I guess you could say the action in question here is the action of a system consisting of one particle. The question then becomes something like "what is the relationship between that action and the particle's angular momentum?"

Action between what two points?
To say that a particle "has action" is like saying that it has work or heat.

 

[nasu]

Nasu, the fact that one is a constant and one is a variable doesn't mean they aren't related in some way.

In the case of c and v, yes they share units but there are also physical relationships between the two such as 'an object's speed can never reach c', and of course the relationship between c and v has very important implications in special relativity.

I'm interested in knowing if there's any such relation between h and S.

If you have nothing to contribute to the question rather than saying 'it's poorly formulated' then please don't respond in the first place.

It seems that you are not looking for a physical relationship but rather one on different levels.
Then you can also say that h and action have the same units. Would you call this a relationship? If yes, then your question was answered.
Or both "h" and "action" are sequences of letters in the Latin alphabet.

To try to ask a poorly formulated question is a meaningless task. You should start with by making the question meaningful before jumping to "answers".
Of course, this is for physics-related questions. For philosophic one you are right, it doesn't matter. :)

 

[Mercy]

It's hard to know what is and what isn't a meaningful question when the answer could be beyond the scope of what I've learned.
I could analogously ask "what is the relationship between the heat of a system and the system's specific heat capacity". One is a constant and one isn't a state variable, although that doesn't mean there isn't a physical connection between the two (Q = mcΔT).

So I don't agree that the question is inherently poorly formulated, although I think we're going in circles here.

----
The first sentence on Wikipedia's page for Planck's Constant is "The Planck constant is a physical constant that is the quantum of action, central in quantum mechanics."
- so it's implied there is a connection, I just want to know if there is more to their connection than the statement "it is the quantum of action".