How Does a Magnetic Field Influence a 2D Harmonic Oscillator?

In summary, the problem involves finding the Hamiltonian for a charged particle in a 2D system with a magnetic field present. The Hamiltonian is given by H = 1/2m (p - e/c A)^2, where A = 1/2*B x r and p and r have two components. The goal is to express H in terms of B along the z axis, which resembles a 2D harmonic oscillator with an extra term. This can be achieved by using the vector potential A = B x \hat y and following the calculation shown in the solution. Other vector potentials may not give the same result.
  • #1
iamnoy
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Homework Statement


charge e is within 2 dimensions in presence of magnetic field.

H = 1/2m (p - e/c A)^2

A = 1/2* B x r

p and r have two components

Show: H in terms of B along z axis resembles 2D HO (with some extra term)

express H in terms of x, y, p_x, L_y

Homework Equations



L = r x p[/B]

Hamiltonian for HO
dbd8f853d8d1e754b5089385733a4f9a.png


The Attempt at a Solution



I plugged in B into H equation above so that gives:

H = 1/2m [p- e/c (1/2*Bxr)]

but i do not know what do do next. How do I express H in terms of B along z axis?
 
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  • #2
Well, if you use the vector potential [itex] \vec A= B x \hat y [/itex], then you can find the complete calculation here. I'm not sure using other vector potentials, gives the same effect or not!
 

FAQ: How Does a Magnetic Field Influence a 2D Harmonic Oscillator?

What is the Hamiltonian of charge in 2D?

The Hamiltonian of charge in 2D is a mathematical representation of the total energy of a charged particle in a two-dimensional space. It takes into account the kinetic energy and potential energy of the particle in this space.

How is the Hamiltonian of charge in 2D calculated?

The Hamiltonian of charge in 2D is calculated using the position and momentum operators in the x and y directions. It is a function of these operators and the electric potential in the 2D space.

What is the significance of the Hamiltonian of charge in 2D?

The Hamiltonian of charge in 2D is an important concept in quantum mechanics and is used to study the behavior of charged particles in a two-dimensional space. It helps in understanding the energy levels and dynamics of these particles.

How does the Hamiltonian of charge in 2D relate to the Schrödinger equation?

The Schrödinger equation describes the evolution of a quantum system over time, and the Hamiltonian of charge in 2D is a key component of this equation. It represents the total energy of the system and is crucial in predicting the behavior of charged particles in a two-dimensional space.

Can the Hamiltonian of charge in 2D be applied to other systems?

Yes, the concept of the Hamiltonian of charge in 2D can be extended to other systems, such as spin systems or systems with multiple particles. It can also be generalized to higher dimensions, such as 3D or even more complex spaces.

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