How is adiabatic invariant proved in a simple dynamic system?

In summary, adiabatic invariance in a simple dynamic system refers to the conservation of a physical quantity despite changes in the system's parameters or external conditions. It is closely related to the first law of thermodynamics and can be proven with the assumption of an isolated, steady state system undergoing slow and reversible changes. This is mathematically proven by using the laws of thermodynamics and the equations of motion to derive a conserved quantity. Adiabatic invariance can also be observed in real-world systems, such as the Earth's atmosphere and the behavior of charged particles and chemical reactions.
  • #1
maverick6664
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I'm reading a book on quontum mechanics in japanese (Quontum Mechanics by Shinichiro Tomonaga) and am stuck in proving the action variable "J" is constant in a one dimensional cyclic movement. i.e.

The action variable "J" created by the trajectory of

H(p(t),q(t),a(t/T)) = E(t)

doesn't change. This trajectory won't make a closed region when a(t/T) changes, but when a(t/T) is fixed or changes very slowly the trajectory is assumed to be closed.

Will anyone give me good online references on it, or recommend nice English books on Quontum Mechanics (not so thick or thin) ? Japanese books don't look nice to me. I'm good at math and have knowledge on electromagnetic mechanics and special/general theories of relativity, but not quontum mechanics.

Thanks in advance!
 
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  • #2
To prove that the action variable "J" is constant in a one dimensional cyclic movement, it is necessary to first understand the definition of the action variable. The action variable is a measure of the energy expended by a particle in a given period of time. It is associated with the total energy of a system over a single period of time, and is related to the momentum and angular momentum of the particle. The action variable "J" can be defined as:J = Integral(0 to T) (p(t)dq(t)-H(p(t),q(t),a(t/T))dt)Where p(t) is the momentum of the particle, q(t) is the position of the particle, a(t/T) is the angular velocity of the particle, and H(p(t),q(t),a(t/T)) is the Hamiltonian of the system. To prove that the action variable J is constant in a one dimensional cyclic movement, it is necessary to demonstrate that the integral of the Hamiltonian over a single period of time is always equal to the total energy of the system. This can be shown by using the Euler-Lagrange equation, which states that the integral of the Lagrangian over a single period of time is equal to the total energy of the system. The Lagrangian for this system is given by:L(p,q,a) = p(t)dq(t)-H(p(t),q(t),a(t/T))Using the Euler-Lagrange equation, we can show that the integral of the Lagrangian over a single period of time is equal to the total energy of the system:Integral(0 to T) (p(t)dq(t)-H(p(t),q(t),a(t/T))dt) = E(t)where E(t) is the total energy of the system. Therefore, the action variable J is constant in a one dimensional cyclic movement. For further information on the action variable and quontum mechanics, you may wish to consult an English language textbook on the subject
 
  • #3


The adiabatic invariant is a fundamental concept in the study of dynamical systems, particularly in quantum mechanics. It states that the action variable, denoted as "J," remains constant in a system that undergoes slow, cyclic changes. In other words, the value of J is conserved as the system evolves over time.

To prove this in a simple dynamic system, we must first understand the concept of action. In classical mechanics, action is defined as the integral of the Lagrangian over time, or the area under the curve of the system's trajectory on a phase space diagram. In quantum mechanics, action is related to the Hamiltonian and is given by the expression J = (2πħ)n, where n is the number of complete cycles in the system's motion.

To prove that J is constant in a one-dimensional cyclic movement, we can use Hamilton's equations of motion and the principle of least action. The Hamiltonian, H, is defined as the total energy of the system, which includes both kinetic and potential energy. Therefore, the Hamiltonian can be written as H = T + V, where T is the kinetic energy and V is the potential energy.

Using Hamilton's equations of motion, we can show that the derivative of the action variable J with respect to time is equal to the derivative of the Hamiltonian with respect to the cyclic coordinate a(t/T). This can be written as:

dJ/dt = (∂H/∂a) x (da/dt)

Since a(t/T) is a cyclic coordinate, its derivative with respect to time is zero. This means that dJ/dt = 0, and therefore, J is a constant.

In simple terms, this means that as the system undergoes slow and cyclic changes, the value of J remains unchanged. This is because the Hamiltonian, and thus the total energy of the system, remains constant during these changes.

As for online references and English books on quantum mechanics, there are many resources available. Some recommended books include "Introduction to Quantum Mechanics" by David J. Griffiths and "Principles of Quantum Mechanics" by R. Shankar. These books provide a comprehensive and accessible introduction to the subject.

Additionally, there are numerous online resources such as lecture notes, video lectures, and interactive simulations that can help you understand the concept of adiabatic invariance and other fundamental concepts in quantum mechanics. Some recommended resources include MIT OpenCourseWare
 

1. What is the concept of adiabatic invariance in a simple dynamic system?

Adiabatic invariance in a simple dynamic system refers to the property of a system where a certain physical quantity, such as energy or momentum, remains constant despite changes in the system's parameters or external conditions.

2. How does adiabatic invariance relate to the laws of thermodynamics?

The concept of adiabatic invariance is closely related to the first law of thermodynamics, which states that the total energy of an isolated system remains constant. Adiabatic invariance can be seen as a consequence of this law in simple dynamic systems.

3. What are the key assumptions for proving adiabatic invariance in a simple dynamic system?

The key assumptions for proving adiabatic invariance in a simple dynamic system include the system being isolated, the system being in a steady state, and the system being subject to slow and reversible changes.

4. How is adiabatic invariance mathematically proven in a simple dynamic system?

The mathematical proof of adiabatic invariance in a simple dynamic system involves using the laws of thermodynamics, specifically the first and second laws, along with the equations of motion for the system. This results in the derivation of a conserved quantity, which is the adiabatic invariant.

5. Can adiabatic invariance be observed in real-world systems?

Yes, adiabatic invariance can be observed in various real-world systems, such as the Earth's atmosphere, the motion of charged particles in a magnetic field, and the behavior of certain chemical reactions. These systems demonstrate the conservation of energy or other physical quantities despite changes in their surroundings.

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