Landau levels and bohr model

In summary: Wb.In summary, the mistake in the energy levels for n=2,4,6 etc. in the derived energy of an electron in a magnetic field appears because the magnetic energy is not taken into account in the calculations. The correct energy intervals should be n\hbar \omega, where \omega is the angular frequency of the rotating electron and \hbar is the reduced Planck constant.
  • #1
Gavroy
235
0
hi

i thought that if i try to derive the energy of an electron in a magnetic field, this could be done with the assumptions of the bohr model.

L=n h/(2π)

mv²/r=qvB => mvr=qBr²=>n h/(2π)=qBr²

E=p²/(2m)=q²B²r²/(2m)=n h/(2π)qB/(2m)

so i get the energy for the first level, but all transitions are wrong, as n=2,4,6 etc. should be forbidden, but i do not get this condition.

so my question is: why does this mistake appear?
 
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  • #2
Gavroy said:
hi

i thought that if i try to derive the energy of an electron in a magnetic field, this could be done with the assumptions of the bohr model.

L=n h/(2π)

mv²/r=qvB => mvr=qBr²=>n h/(2π)=qBr²

E=p²/(2m)=q²B²r²/(2m)=n h/(2π)qB/(2m)

so i get the energy for the first level, but all transitions are wrong, as n=2,4,6 etc. should be forbidden, but i do not get this condition.

so my question is: why does this mistake appear?

Probably, you forget about the "magnetic energy", though the electron (charge= q ) is rotating under the magnetic field. I rearrange your equations here.

According to the Bohr model. the orbital length is an integer (n) times de Broglie's wavelength (=h/mv),
So this fact leads to your first equation of the angular momentum (L).

[tex]2\pi r = n \times \frac{h}{mv} \quad \to \quad L = mvr = n \times \frac{h}{2\pi} = n \hbar[/tex]

The centrifugal force is equal to Lorentz force (= qvB), as shown in your second equation.

[tex] \frac{mv^2}{r} = qvB [/tex]

Using these equations, the angular frequency (= w) of the rotating electron and kinetic energy (K) are

[tex]\omega = 2\pi \times f = 2\pi \times \frac{v}{2\pi r} = \frac{v}{r} = \frac{qB}{m}[/tex]

[tex]K = \frac{1}{2}mv^2 = \frac{n\hbar qB}{2m} = \frac{1}{2}n\hbar \omega[/tex]

Loretz force causes the magnetic moment (= u ) (of rotating electron), which direction is opposite to the external magnetic field,

[tex]\mu = I\pi r^2 = \frac{qv}{2\pi r}\cdot \pi r^2 =\frac{qmvr}{2m} = \frac{qn\hbar}{2m}[/tex]

So the magnetic energy (V) is "plus", as follows,

[tex]V = \mu\cdot B = \frac{n\hbar qB}{2m} = \frac{1}{2}n\hbar \omega[/tex]

As a result, the energy intervals are hw, as follows,

[tex]E = V + K = n\hbar \omega[/tex]

By the way, using Maxwell equation and the above equations, the magnetic flux included in the circular orbit is

[tex]\pi r^2 B = \pi \frac{m^2v^2}{q^2 B} = \frac{h}{2q} \times n[/tex]

where h/2q is "magnetic flux quantum"
 

1. What are Landau levels?

Landau levels refer to the quantized energy levels that an electron can occupy in a magnetic field. These levels were first proposed by Russian physicist Lev Landau in 1930.

2. How are Landau levels calculated?

The energy levels for Landau levels are calculated using the Landau quantization rule, which states that the energy levels are evenly spaced and proportional to the magnetic field strength. The formula is given by E = (n + 1/2)ħωc, where n is the quantum number and ωc is the cyclotron frequency.

3. What is the significance of Landau levels?

Landau levels have significant implications in the study of quantum mechanics and condensed matter physics. They explain the behavior of electrons in a magnetic field and play a crucial role in the understanding of phenomena such as the Integer Quantum Hall Effect.

4. What is the Bohr model?

The Bohr model, also known as the Bohr-Sommerfeld model, is a simplified model of the atom proposed by Danish physicist Niels Bohr in 1913. It describes the structure of the atom as a central nucleus surrounded by electrons in specific energy levels.

5. How does the Bohr model relate to Landau levels?

The Bohr model is a precursor to the more advanced theory of Landau levels. It provides a simplified understanding of the behavior of electrons in an atom, which is similar to the behavior of electrons in Landau levels in a magnetic field. Both models use the concept of quantization to explain the discrete energy levels of electrons.

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