# Magnetic Moments of Hydrogen Atom in 2T Magnetic Field

• Brewer
JMT, the conversation is about the Hydrogen atom in a magnetic field of 2T and the calculation of the magnetic moments and energy splitting for the proton and electron in the ground state. The g-factor and m_j quantum number are discussed, with j being the total angular momentum quantum number, and m_j being the m quantum number associated with j. After understanding these concepts, the remainder of the question becomes straightforward.
Brewer

## Homework Statement

Consider the Hydrogren atom in a magnetic field of 2T. If the atom is in the ground state (orbital angular momentum L=0)
a) Write down the magnetic moments of the proton and the spinning electron
b) What is the splitting of the energy of the ground state in eV due to the electron?
c) Same question for the proton.

## Homework Equations

$$\mu = -g\frac{e\hbar}{2m_e}m_j$$
$$E = -\mu B$$

## The Attempt at a Solution

Well generally I think that its ok - its not that hard to do, except for the g-factor, and the $$m_j$$ bit of the magnetic moment equation.

I've got that $$g = 1 + \frac{j(j+1) - l(l+1) + s(s+1)}{2j(j+1)}$$, but I'm a little confused at how to use this.

I can gleam from my textbook and notes that s=spin and in this case s=1/2, and I think that l=0, but then what is j? I think its something to do with both s and l, but I can't work out what it is.

Following from that what's this $$m_j$$ component thing. In all the examples in textbooks it just seems to disappear, and I don't follow why.

After this - the whole question is a doddle, so any hints would be welcomed.

Thanks

Brewer said:

## Homework Statement

Consider the Hydrogren atom in a magnetic field of 2T. If the atom is in the ground state (orbital angular momentum L=0)
a) Write down the magnetic moments of the proton and the spinning electron
b) What is the splitting of the energy of the ground state in eV due to the electron?
c) Same question for the proton.

## Homework Equations

$$\mu = -g\frac{e\hbar}{2m_e}m_j$$
$$E = -\mu B$$

## The Attempt at a Solution

Well generally I think that its ok - its not that hard to do, except for the g-factor, and the $$m_j$$ bit of the magnetic moment equation.

I've got that $$g = 1 + \frac{j(j+1) - l(l+1) + s(s+1)}{2j(j+1)}$$, but I'm a little confused at how to use this.

I can gleam from my textbook and notes that s=spin and in this case s=1/2, and I think that l=0, but then what is j? I think its something to do with both s and l, but I can't work out what it is.
"j" is the quantum number associated to the total angular momentum. The value of j ranges from |l-s| to l+s in steps of 1 (here |l-s| means the absolute value of l-s). In your case, l=0 and s=1/2 so j may only take the value j=1/2.
Following from that what's this $$m_j$$ component thing. In all the examples in textbooks it just seems to disappear, and I don't follow why.

After this - the whole question is a doddle, so any hints would be welcomed.

Thanks
$m_j$ is simply the m quantum number associated to j, so it may range from -j to +j in steps of 1. If j=1/2 as in your example, there are two possible m_j and two possible energies (depending is the angular momentum is "aligned" with the B field or opposite to it).

Patrick

## 1. What is a magnetic moment?

A magnetic moment is a measure of the strength and direction of a magnetic field generated by a particle or object.

## 2. How is the magnetic moment of a hydrogen atom calculated?

The magnetic moment of a hydrogen atom can be calculated using the Bohr magneton formula, which takes into account the spin of the electron and its orbital angular momentum.

## 3. What is the significance of a 2T magnetic field in relation to the hydrogen atom's magnetic moment?

A 2T magnetic field is a strong magnetic field that can cause the hydrogen atom's magnetic moment to align with the field, making it easier to study and measure.

## 4. How does the magnetic moment of a hydrogen atom change in a 2T magnetic field?

In a 2T magnetic field, the magnetic moment of a hydrogen atom increases, as the electron's spin and orbital angular momentum align with the field, resulting in a stronger magnetic field.

## 5. What are the practical applications of studying the magnetic moments of hydrogen atoms in a 2T magnetic field?

Studying the magnetic moments of hydrogen atoms in a 2T magnetic field can provide insight into the behavior of atoms in strong magnetic fields and has applications in fields such as nuclear magnetic resonance imaging and quantum computing.

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