# Angular momentum quantum number and angle

• leviathanX777
In summary, the possible values for the magnetic quantum number m_l for an electron with an orbital quantum number l = 4 are -4, -3, -2, -1, 0, 1, 2, 3, and 4. The allowed values for the angles between L and the z-axis in this case can be determined by measuring the operator \hat{L_z}^2, which corresponds to the angle between \mathbf{L} and the z-axis. The allowed values for this operator will vary depending on the specific state, but for an electron in the state l = 4, the allowed values can be found by solving the equation \hat{L_z}^2=\hat{

## Homework Statement

What are the possible values for the magnetic quantum number ml for an electron whose orbital quantum number is l = 4? What are the allowed values of the angles between L and the zaxis in this case?

## The Attempt at a Solution

Is Ml equal to four for the first part? Because the angular momentum quantum number is equal to four?

I have no idea for the second part. I don't have the gradient of the magnetic field of the z-component and don't have any distance or velocity values.

leviathanX777 said:

## Homework Equations

How exactly is this equation relevant? The problem is in regard to an electron orbiting a nucleus, not an electron moving through an inhomogeneous magnetic field as in the Stern-Gerlach experiment.
Is Ml equal to four for the first part? Because the angular momentum quantum number is equal to four?

Open your textbook up and read the section on hydrogen like atoms and angular momentum. The allowed values of $m_l$ for any given value of $l$ will be clearly stated in your text.

I have no idea for the second part.
If $\theta$ is the angle between $\mathbf{L}$ and the z-axis, then $L_z\equiv \mathbf{L}\cdot \textbf{k}=|\mathbf{L}|\cos\theta[/tex]. So, in terms of operators, you would expect $$\hat{L_z}^2=\hat{L}^2\cos^2\hat{\theta}$$ Where the hat is to denote that we are talking about operators here. (i.e. $$\hat{\theta}$$ is an operator whose value upon measurement corresponds to the angle between [itex]\mathbf{L}$ and the z-axis)

What are the allowed values when you measure [tex]\hat{L_z}^2[/itex] for an electron in the state $l=4$?

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## 1. What is the angular momentum quantum number (l) and what does it represent?

The angular momentum quantum number (l) is one of the four quantum numbers used to describe the energy level and location of an electron in an atom. It represents the shape of the electron's orbital, with each value of l corresponding to a different orbital shape (s, p, d, f).

## 2. How is the angular momentum quantum number (l) related to the principal quantum number (n)?

The angular momentum quantum number (l) is related to the principal quantum number (n) by the equation l = 0, 1, 2, ..., n-1. This means that for each value of n, there can be up to n-1 possible values of l, representing the different orbital shapes within that energy level.

## 3. What is the maximum number of electrons that can be in an orbital with a given angular momentum quantum number (l)?

The maximum number of electrons that can be in an orbital with a given angular momentum quantum number (l) is 2(2l+1). This means that for an s orbital (l=0), there can be a maximum of 2 electrons, for a p orbital (l=1), there can be a maximum of 6 electrons, and so on.

## 4. How does the angular momentum quantum number (l) affect the energy of an electron in an atom?

The angular momentum quantum number (l) affects the energy of an electron in an atom by determining the energy level and location of the electron within the atom. As the value of l increases, the energy level of the electron also increases. This is due to the fact that higher values of l correspond to more complex and higher energy orbital shapes.

## 5. Can two electrons in the same atom have the same values for both their principal quantum number (n) and angular momentum quantum number (l)?

No, according to the Pauli exclusion principle, no two electrons in the same atom can have the same set of quantum numbers. This means that while two electrons may have the same value for their principal quantum number (n), they must have different values for their angular momentum quantum number (l).