Hamiltonian particle problem

In summary, the superposition of arbitrary initial state in eigenstates of the Hamiltonian operator can be represented by the integral of b(k) times the eigenstates, where k is the wave number. The dimension of b(k) is √length, as it represents the probability amplitude of the particle in a given state. This concept is discussed in chapter 5 of the book "Quantum Mechanics" by Liboff.
  • #1
najima
34
0

Homework Statement



if we have the particle ins free its hamiltonian has a continuous spectrum of eigen enegies and superposition of arbitrary initial state in eigenstates φ_k of H( hamiltonian oprator) becomes ∫_(-∞)^∞▒〖b(k) φ_k dk〗,what is the dimemension of b(k) (lb(k)l^2 is a probability density)? where k is wave nomber.

Homework Equations

quantum mechanics by liboff chapter 5



The Attempt at a Solution

lb(k)l^2 dimensions is 1/k because ∫_(-∞)^∞▒〖b(k)^2dk〗=1
 
Physics news on Phys.org
  • #2
, so b(k) must have dimensions of 1/√k. This means that the dimension of b(k) is 1/√(length^-1), which is equivalent to √length. This dimension is often referred to as the square root of length, and it is commonly used in quantum mechanics to describe the probability amplitude of a particle in a given state. Therefore, the dimension of b(k) is √length.
 

1. What is the Hamiltonian particle problem?

The Hamiltonian particle problem is a mathematical framework used to describe the motion of a particle in a conservative system. It is based on the principles of classical mechanics and uses the Hamiltonian function to determine the particle's position and momentum at any given time.

2. What is the Hamiltonian function?

The Hamiltonian function, also known as the Hamiltonian, is a mathematical function that represents the total energy of a particle in a system. It is the sum of the particle's kinetic energy and potential energy, and is used to describe the particle's motion in a conservative system.

3. How is the Hamiltonian particle problem solved?

The Hamiltonian particle problem can be solved using Hamilton's equations of motion, which are a set of differential equations that describe the evolution of the particle's position and momentum over time. These equations can be solved using various mathematical techniques, such as numerical methods or perturbation theory.

4. What is the difference between Hamiltonian mechanics and Lagrangian mechanics?

Hamiltonian mechanics and Lagrangian mechanics are two different mathematical approaches to solving mechanical problems. While Lagrangian mechanics is based on the principle of least action, Hamiltonian mechanics is based on the conservation of energy. In Hamiltonian mechanics, the Hamiltonian function is used to determine the particle's motion, while in Lagrangian mechanics, the Lagrangian function is used.

5. What are some real-world applications of the Hamiltonian particle problem?

The Hamiltonian particle problem has a wide range of applications in physics, engineering, and other fields. It is commonly used to study the motion of particles in classical mechanics, as well as in quantum mechanics. It is also used in fields such as celestial mechanics, fluid dynamics, and electromagnetism to model and predict the behavior of systems with multiple interacting particles.

Similar threads

  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
5
Views
1K
  • Advanced Physics Homework Help
Replies
2
Views
778
Replies
2
Views
1K
  • Quantum Physics
Replies
10
Views
2K
  • Advanced Physics Homework Help
Replies
2
Views
2K
Replies
1
Views
683
  • Advanced Physics Homework Help
Replies
3
Views
2K
Replies
2
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
1K
Back
Top