Magnetic dipole in magnetic field

• Niles
In summary, the conversation discusses the relationship between classical and quantum mechanics in regards to the energy of a magnetic dipole in an external magnetic field. It is noted that the energy is zero when the dipole and the field are perpendicular and maximal when they are antiparallel. The quantum-mechanical equation for energy is also presented, showing that it is related to the Zeeman shift of an atom. The question then arises on how these two scenarios are related, to which it is explained that in both cases, there is a change in the value of µ, resulting in a change in energy.

Niles

Hi

The energy U of a magnetic dipole in an external magnetic field is given by
$$U = -\mu \cdot B$$
so the energy is zero when they are perpendicular and maximal when they are antiparallel. This makes very good sense intuitively. Quantum-mechanically we have that
$$\mu = -m_Fg_F \mu_B$$
so U becomes
$$U = \mu_Bg_Fm_FB,$$
which is just the Zeeman shift of an atom. My questions is on how these two different scenarios - quantum and classical - are related.

The first relation states that the particle gains energy due to the torque exerted on it by B. However a Zeeman shift of an atom is - how I have understood it - basically not related to that the atom gains enegy. It just means that its internal levels are shifted. So it is not intuitive to me how the magnetic field "imparts" energy onto the particle in the second relation.

I hope my question is clear.

Best,
Niles.

The first relation states that the particle gains energy due to the torque exerted on it by B.
Only if it can rotate in that direction. The transition between the corresponding quantum mechanical energy levels is not really a rotation, but has a similar effect: it is a change of µ, which changes the energy.

mfb said:
Only if it can rotate in that direction. The transition between the corresponding quantum mechanical energy levels is not really a rotation, but has a similar effect: it is a change of µ, which changes the energy.

Thanks. When you say that it is a change of μ, then you are referring to that mF is changed in
$$\mu = -m_Fg_F \mu_B$$
?

Best,
Niles.

As the other two are constant... right ;).

1. What is a magnetic dipole?

A magnetic dipole is a type of magnet with two opposite poles, north and south, separated by a distance. It can be thought of as a tiny bar magnet.

2. How does a magnetic dipole interact with a magnetic field?

A magnetic dipole experiences a torque when placed in a magnetic field, causing it to align itself with the field. This is known as the magnetic moment.

3. What is the equation for magnetic dipole moment?

The equation for magnetic dipole moment is μ = I * A, where μ is the magnetic moment, I is the current flowing through the dipole, and A is the area enclosed by the current loop.

4. How does the strength of a magnetic dipole affect its interaction with a magnetic field?

The strength of a magnetic dipole is directly proportional to its interaction with a magnetic field. A stronger magnetic dipole will experience a greater torque when placed in a magnetic field.

5. How is the direction of a magnetic dipole's magnetic moment determined?

The direction of a magnetic dipole's magnetic moment is determined by the right hand rule, where the thumb points in the direction of the current flow and the fingers curl in the direction of the magnetic field.