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### 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 ...
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 ...
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 ...
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" ...
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 ...
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 ...
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 ...
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 $i\hbar{\partial\over\partial t}|\psi\rangle=H|\psi\rangle$ ...
3k views

### Conservation of Hamiltonian vs Conservation of Energy

What is the difference between conservation of the Hamiltonian and conservation of energy?
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 ...
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 $L=T-V$ ...
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 $H=T+V$?
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 ...
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.
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 ...