Solving a quantum harmonic oscillator using quasi momentum

In summary, the paper introduces a new method for solving the quantum harmonic oscillator by utilizing the concept of quasi momentum, which has been previously used in classical systems but is now being applied to quantum systems in the context of the Quantum Spectral Curve and the spectrum of N=4 SYM. However, there is currently no detailed discussion of this technique in the literature.
  • #1
Prathyush
212
16
In the paper below I've seen a new method to solve the quantum harmonic oscillator
Introduction to the Spectrum of N=4 SYM and the Quantum Spectral Curve

It is done using the concept of quasi momentum defined as
$$p = - i \frac{d(\log \psi)}{dx}$$
See pg 7,8

Is this well know? is it discussed somewhere in detail?
 
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  • #2
The concept of quasi momentum has been studied in the literature for several decades, but it has been mainly used to study classical systems. It is a relatively new development to apply such a concept to quantum systems, particularly in the context of the Quantum Spectral Curve and the spectrum of N=4 SYM. To the best of our knowledge, there is no detailed discussion of this technique in the literature.
 

FAQ: Solving a quantum harmonic oscillator using quasi momentum

1. What is a quantum harmonic oscillator?

A quantum harmonic oscillator is a model used in quantum mechanics to describe the behavior of a particle that is subject to a restoring force proportional to its displacement from its equilibrium position. It is a fundamental model that is used to understand the behavior of many physical systems, such as atoms, molecules, and solid-state materials.

2. What is quasi momentum in quantum mechanics?

Quasi momentum is a concept in quantum mechanics that is used to describe the momentum of a particle in a periodic potential. It is a useful tool for understanding the behavior of particles in a crystal lattice or other periodic systems.

3. How is quasi momentum used to solve a quantum harmonic oscillator?

In order to solve a quantum harmonic oscillator, one must first express the Hamiltonian (the operator that describes the total energy of the system) in terms of quasi momentum. This allows for the use of the Bloch theorem, which states that the wavefunction of a particle in a periodic potential can be written as a product of a plane wave and a periodic function. By using this theorem, the problem can be reduced to solving a one-dimensional Schrödinger equation, which is much simpler than the original problem.

4. What are the applications of solving a quantum harmonic oscillator using quasi momentum?

The solution to the quantum harmonic oscillator using quasi momentum has many practical applications. It can be used to understand the behavior of electrons in a crystal lattice, which is important for understanding the properties of semiconductors and other solid-state materials. It is also useful for studying the behavior of atoms and molecules, as well as for developing new technologies such as quantum computers.

5. Are there any limitations to using quasi momentum to solve a quantum harmonic oscillator?

While using quasi momentum can greatly simplify the solution to a quantum harmonic oscillator, it does have some limitations. It is only applicable to systems with periodic potentials, so it cannot be used for more general systems. Additionally, it assumes that the particle is non-interacting, which may not always be the case in real-world systems.

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