Hamiltonian formalism and partition function

In summary, in Hamiltonian formalism, the generalized coordinates are represented by q_i and the conjugate moments by p_i. For a dipole in a given magnetic field, the Hamiltonian is -μBcosθ, where θ is the angle between the dipole moment μ and the magnetic field B. To compute the partition function for this system, either θ or cosθ can be considered as a generalized coordinate, with the associated conjugate momentum being related to angular momentum.
  • #1
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TL;DR Summary
is it possible to find a (q,p) couple for a dipole in a magnetic field?
In hamiltonian formalism we have the generalized coordinates ##q_i## and the conjugates moments ##p_i##.
For a dipole in a give magnetic field ##B## the Hamiltonian is ##H=-\mu B cos \theta## where ##\theta## is the angle between ##\vec \mu## and ##\vec B##.
Can i consider ##\theta## or ##cos \theta## as a generalized coordinate? if yes what is the associated conjugate momentum ##P_\theta##?
I ask this question because i'd like to compute the partition function for a dipole in a magnetic field starting from ##\frac 1 {\hbar ^f} \int dr^fdp^f exp(-\beta H(r_1...r_f,p_1...p_f))##
 
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  • #2
Either ##\theta## or ##\cos\theta## could be considered as a generalised coordinate. The corresponding canonical momentum would be related to angular momentum.
 
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FAQ: Hamiltonian formalism and partition function

1. What is Hamiltonian formalism?

Hamiltonian formalism is a mathematical framework used to describe the dynamics of a physical system. It is based on the concept of the Hamiltonian, which is a mathematical function that describes the total energy of a system.

2. How is Hamiltonian formalism different from Lagrangian formalism?

Hamiltonian formalism and Lagrangian formalism are two different mathematical approaches used to describe the dynamics of a physical system. While Lagrangian formalism is based on the concept of the Lagrangian, which is a mathematical function that describes the kinetic and potential energy of a system, Hamiltonian formalism is based on the concept of the Hamiltonian, which describes the total energy of a system.

3. What is the partition function in Hamiltonian formalism?

The partition function is a mathematical function used in statistical mechanics to calculate the thermodynamic properties of a system. In Hamiltonian formalism, the partition function is defined as the sum of all possible states of a system, weighted by the Boltzmann factor, which takes into account the energy of each state.

4. How is the partition function related to the Hamiltonian of a system?

In Hamiltonian formalism, the partition function is related to the Hamiltonian of a system through the Hamiltonian operator. The partition function can be expressed as the trace of the exponential of the negative Hamiltonian operator divided by the Boltzmann constant.

5. What are the applications of Hamiltonian formalism and partition function?

Hamiltonian formalism and partition function have various applications in physics, chemistry, and engineering. They are used to describe the dynamics of complex systems, such as molecules, atoms, and particles, and to calculate their thermodynamic properties. They are also used in quantum mechanics, statistical mechanics, and other areas of theoretical physics.

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