Observables as a function of energy

In summary: Reilly Atkinson says: A time-independant observable is a constant of motion if it commutes with the hamiltonian. That may be what they mean with a conserved observable.By definition, an constant of motion is an observable (and therefore an operator) which (a) is not an explicit function of time and (b) commutes with the Hamiltonian. A conservative system is a system in which the Hamiltonian is not an explicit function of time.
  • #1
zitek
2
0
The question is:
Prove that, for a 1-D harmonic oscillator, every conserved observable is a function of the energy. Find, for the 3-D harmonic oscillator, some conserved observable not a function of the energy and angular momentum.

My first problem with this question is I'm not sure what a "conserved observable" is. I know that the state of a 1-D harmonic oscillator can be expressed by only one position component and one velocity component, and from that you can get a function of energy, but I don't know how to write a proof for this.

Thanks.
 
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  • #2
EASY, in 1D, energy(KE or PE, not total, total energy is constant) is a function of one variable (velocity, position...etc, let's say that variable called q) ONLY, so E=E(q), or q=q(E)
however, in 3D, energy(KE or PE) is a function of 3 variable, (3 components for velocity, position...etc) therefore E=E(p,q,r)...and p=p(E,q,r), knowing E alone can't determent p...
 
  • #3
What else could it be? Is p conserved? Is x conserved? What else is there?
Regards,
Reilly Atkinson
 
  • #4
A time-independant observable is a constant of motion if it commutes with the hamiltonian. That may be what they mean with a conserved observable.
 
  • #5
Galileo said:
A time-independant observable is a constant of motion if it commutes with the hamiltonian. That may be what they mean with a conserved observable.
By definition, an constant of motion is an observable (and therefore an operator) which (a) is not an explicit function of time and (b) commutes with the Hamiltonian. A conservative system is a system in which the Hamiltonian is not an explicit function of time.

The reason its called a "constant of motion" is because the expectation value of the observable is not a function of time. I.e. if A is a constant of motion then d<A>/dt = 0.

For your question to make sense you'd have to assert that all constants of motion are functions of the Hamiltonian. I assume that whomever made that assertion is thinking about the physical quantity associated with the observable. In any case I don't believe that assertion is true. For example, there are two observables I know of which have nothing to do with energy. One is spin and the other position. Each is an observable and thus there is a operator for each and corresponding physical quantity. Each has nothing to do with energy in general. The energy and spin of a particle are unrelated when, for example, the particle is an electron moving in the absence of a EM field.

Pete
 
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  • #6
zitek said:
I know that the state of a 1-D harmonic oscillator can be expressed by only one position component and one velocity component, ...
What is a "velocity component"? :confused:

Velocity is pretty much undefined in quantum mehanics. Do you mean the ratio of momentum to mass (some people like to define a velocity operator. But I think that they're just plain crazy. :tongue: )?

Pete
 
  • #7
I think the original poster of this question/thread needs to clarify if the question has any connection with QM. I suspect it does not since it was posted in the classical physics section. So this may be purely a classical mechanics problem and thus all the connection with "commutation" with the Hamiltonian may be irrelevant (but then again, people on PF have posted questions in the wrong section all the time).

If this is a classical mechanics question, then a "conserved" quantity as defined within the Lagrangian/Hamiltonian mechanics is simply a quantity in which the time derivative is zero. In that formulation, there are only two "observables", the generalized coordinate q and the generalized momentum p (and the associated derivatives).

Zz.
 
  • #8
Oy! What was I thinking! Yep. You're right. For some reason I thought he was asking about quantum mechanics. I guess it was term "observerable" since I've never heard of that term being used outside quantum mechanics before.

Pete
 

1. What are observables as a function of energy?

Observables as a function of energy refer to physical quantities that can be measured or observed as they vary with changes in energy levels.

2. Why are observables studied as a function of energy?

Studying observables as a function of energy allows scientists to understand the relationship between different physical quantities and how they change with energy. This can provide insight into the underlying principles and laws of physics.

3. How are observables measured as a function of energy?

Observables are typically measured through experiments or observations that involve varying the energy of a system and recording the corresponding values of the observable. Mathematical models and equations can also be used to calculate and predict the behavior of observables as a function of energy.

4. What types of observables are commonly studied as a function of energy?

Some common observables that are studied as a function of energy include mass, velocity, momentum, and energy itself. However, any physical quantity that can be measured or observed can also be studied as a function of energy.

5. How does the behavior of observables change with energy?

The behavior of observables can vary greatly with changes in energy levels. Some observables may exhibit a linear relationship with energy, while others may show more complex patterns. The specific behavior of an observable as a function of energy depends on the underlying physical laws and mechanisms that govern it.

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