How Are Hamiltonian and Lagrangian Related in Quantum Mechanics?

In summary: The Lagrangian density is a mathematical quantity that is used in many branches of physics, most notably in classical mechanics and general relativity. In classical mechanics, it is a function of the field values and their derivatives in some region of space. In general relativity, it is a function of the field values and their derivatives in a certain metric space.
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TimeRip496
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5
"The hamiltonian runs over the time axis while the lagrangian runs over the trajectory of the moving particle, the t'-axis."
What does the above statement means? Isnt hamiltonian just an operator that corresponds to total energy of a system? How is hamiltonian related to lagrangian intuitively?

Besides what is lagrangian density intuitively and mathematically? Is it equal to lagrangian?
 
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In the future, it's best to provide a reference to a quote so that folks can easily look at the source and context of the quote.
I had to do a Google search and guessed this was the source.

Concerning the statement of the quote...
The interpretation doesn't seem to be a standard interpretation... but it seems interesting and might be worthy of further study.
I can't say I understand enough of that interpretation to give a summary of the idea. (Do a search for ... hans de vries largrangian ).

Since you refer to the Hamiltonian as an operator, your context seems to be quantum mechanics or quantum field theory, which appears to be the context of the quote. (See Ch 22 of the document that comes up in the Google search.)

In classical mechanics, the Hamiltonian and Lagrangian are related by a Legendre Transformation.
I don't have yet an "intuitive" explanation of that relationship... beyond saying it's an important transformation of variables. (part of a backburner project)

While the largrangian is used in particle mechanics (with few degrees of freedom),
the Lagrangian density is used in field theory (with many more degrees of freedom).
Rather than being a function of configurations and velocities,
it is a function of the field values and their derivatives in some region of space.
Crudely speaking, the Lagrangian density is in some sense the Lagrangian-per-unit-volume.

Possibly useful:
http://en.wikipedia.org/wiki/Lagrangian
 

Related to How Are Hamiltonian and Lagrangian Related in Quantum Mechanics?

1. What is the difference between Hamiltonian and Lagrangian?

The Hamiltonian and Lagrangian are two mathematical methods used to describe the motion of a system. The main difference between them is that the Hamiltonian approach uses energy as its fundamental concept, while the Lagrangian approach uses the concept of action. Both methods are equivalent and can be used to obtain the same equations of motion for a system.

2. How are Hamiltonian and Lagrangian related to classical mechanics?

The Hamiltonian and Lagrangian methods are both used in classical mechanics to describe the motion of a system. They are based on the principles of conservation of energy, momentum, and angular momentum. These methods are particularly useful for systems with multiple particles and complex interactions.

3. What is the Hamiltonian function?

The Hamiltonian function is a mathematical function that represents the total energy of a system in terms of its position and momentum variables. It is denoted by H and is the sum of the system's kinetic and potential energies. In Hamiltonian mechanics, the dynamics of a system are described by the evolution of this function over time.

4. How is the Lagrangian used in systems with constraints?

The Lagrangian method is particularly useful for systems with constraints, as it allows for the incorporation of these constraints into the equations of motion. The constraints are expressed through the use of Lagrange multipliers, which are added to the Lagrangian function. This allows for the determination of the system's equations of motion while taking into account any constraints.

5. What are some real-life applications of Hamiltonian and Lagrangian?

The Hamiltonian and Lagrangian methods have many applications in physics and engineering. They are used in fields such as mechanics, astrophysics, quantum mechanics, and control theory. Some specific applications include the study of planetary orbits, the motion of particles in magnetic fields, and the design of control systems for spacecraft and robots. These methods provide powerful tools for analyzing and predicting the behavior of complex systems.

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