Hamiltonian as the generator of time translations

In summary, the conversation discusses the concept of the Hamiltonian ##H## being the generator of time translations in literature. The question arises as to why this is the case and where this statement comes from. The possibility of this being derived from the observation that ##\frac{dF}{dt}=\lbrace F,H\rbrace+\frac{\partial F}{\partial t}## for a given function ##F(q,p)## is discussed, with the conclusion that Noether's theorem may provide a deeper understanding. Susskind's classical mechanics course on YouTube, specifically lectures 4 and 8, are recommended as helpful resources.
  • #1
Frank Castle
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In literature I have read it is said that the Hamiltonian ##H## is the generator of time translations. Why is this the case? Where does this statement derive from?

Does it follow from the observation that, for a given function ##F(q,p)##, $$\frac{dF}{dt}=\lbrace F,H\rbrace +\frac{\partial F}{\partial t}$$ In particular, if ##F## is not explicitly dependent on time, then $$\frac{dF}{dt}=\lbrace F,H\rbrace $$ Or is there more to it?
 
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  • #2
Ah poisson brackets!

I think what you are looking for is Noethers theorem.

Susskinds classical mechanics course on youtube, i think lecture 4 (symmetries) and 8 (poisson) will help you.
 

FAQ: Hamiltonian as the generator of time translations

1. What is a Hamiltonian in physics?

A Hamiltonian is a mathematical operator that represents the total energy of a physical system. It takes into account the kinetic and potential energies of all particles within the system.

2. How is Hamiltonian related to time translations?

In physics, time translation refers to the change in a physical system over time. The Hamiltonian is the generator of this time evolution, meaning that it dictates the rate at which the system changes over time.

3. What is the role of Hamiltonian in quantum mechanics?

In quantum mechanics, the Hamiltonian plays a crucial role in the Schrödinger equation, which describes the time evolution of a quantum system. It represents the total energy of the system and determines the probability of a particle being in a certain state at a given time.

4. Can the Hamiltonian change over time?

Yes, the Hamiltonian can change over time in some systems. This is especially true for complex systems where the potential energy may vary with time, causing the Hamiltonian to also change.

5. How is the Hamiltonian represented in mathematical equations?

The Hamiltonian is typically represented by the symbol H and is written as a sum of the kinetic and potential energy terms, H = T + V. In quantum mechanics, it is often expressed in terms of operators, with the Schrödinger equation being written as HΨ = iħ ∂Ψ/∂t.

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