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2 votes
0 answers
233 views

Is expectation value of the Hamiltonian always the energy? [duplicate]

There are time dependent & space dependent systems (magnetic fields) and time independent (particle in a box or harmonic oscillator). In the latter the expectation value is the 'average' energy ...
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  • 329
0 votes
1 answer
111 views

Why is the energy function not always equal to total energy? [duplicate]

Why is the energy function $h = \dot{q_i}\frac{\partial L}{\partial \dot{q_i}} - L $ not always equal to total energy $E = T + V$? Here $T$ is Kinetic Energy and $V$ is Potential Energy. I've read ...
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  • 85
0 votes
1 answer
82 views

Hamiltonian = total energy? [duplicate]

How do I figure out if the energy in a Hamiltonian is conserved or not? I have found the conditions for $H=E$ in Goldstein's Analytical Mechanics that the equations defining the generalized ...
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26 votes
15 answers
4k views

What is the fundamental definition of force?

As I pick up more physics I see that the definitions of force commonly provided in books and classrooms are misleading. "A force is a push or pull." This seems to be a "correct" ...
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11 votes
5 answers
11k views

Hamiltonian is conserved, but is not the total mechanical energy

I wondering about the interpretation for the energy difference between the Hamiltonian and the total mechanical energy for systems where the Hamiltonian is conserved, but it is not equal to the total ...
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  • 111
5 votes
2 answers
12k views

Time dependence of Lagrangian and Hamiltonian?

I am reading a online tutorial about Lagrangian mechanics. In one section, it states that if the kinetic term in Lagrangian has no explicit time dependence, the Hamiltonian does not explicitly depends ...
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  • 2,273
5 votes
4 answers
906 views

What is a Hamiltonian of a System?

What is a Hamiltonian of a System? When learning about Hamiltonian for the first time it is an object introduced as Legendre Dual Transform of Lagrangian of the same system. And we learn further that ...
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10 votes
2 answers
5k views

Energy operator

Does the Hamiltonian always translate to the energy of a system? What about in QM? So by the Schrodinger equation, is it true then that $i\hbar{\partial\over\partial t}|\psi\rangle=H|\psi\rangle$ ...
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  • 103
5 votes
1 answer
3k views

Conservation of Hamiltonian vs Conservation of Energy

What is the difference between conservation of the Hamiltonian and conservation of energy?
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  • 738
8 votes
1 answer
1k views

Example where Hamiltonian $H \neq T+V=E$, but $E=T+V$ is conserved

I'm looking for an example of a Hamiltonian $H$, where $H\neq T+V$, but the total energy in the system, $E=T+V$, is still conserved. While I'm at it, I might as well add that I'd be most interested ...
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  • 1,594
5 votes
2 answers
1k views

Generalized definitions of Lagrangian and Hamiltonian functions

When we enter into the scope of Analytical mechanics we usually start with these two primary notions: Lagrangian function & Hamiltonian function And usually textbooks define Lagrangian as $L=T-V$ ...
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4 votes
4 answers
6k views

When Hamiltonian and the total energy are the same

In which condition, the Hamiltonian is the same as the total energy of the system, or say $H=T+V$?
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  • 233
3 votes
1 answer
689 views

Is it obvious that the Hamiltonian observable in Quantum Mechanics should also be the Energy observable?

In Quantum Mechanics, the Hamiltonian observable is defined as the generator of time translations. It's easy to show that if we take this to be the definition of the Hamiltonian, then it is of the ...
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2 votes
2 answers
2k views

Lagrangian and hamiltonian of interaction

How to prove that lagrangian of interaction is equal to hamiltonian of interaction with minus sign? For example, I can't prove it for special case - quantum electrodynamics.
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3 votes
1 answer
1k views

When does Hamiltonian equals to energy of the system?

In classical mechanics, the Hamiltonian is well defined by the Lagrangian. Whereas, energy is a very ambiguous term. We just say $E=T+U$, and usually it equals to Hamiltonian. Does there exist a way ...
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