The Stern-Gerlach experiment demonstrates the quantization of angular momentum. When atoms with a magnetic moment pass through a non-uniform magnetic field (Stern-Gerlach apparatus), they are deflected based on the orientation of their magnetic moment.
Stern-Gerlach Splitting Principle
The number of distinct beams an atom splits into is determined by the possible spatial orientations of its angular momentum vector. For an atom with total angular momentum quantum number j, the projection quantum number mj can take values from -j to +j in integer steps. The total number of possible values for mj, and thus the number of beams (N), is given by the formula:
$ N = 2j + 1 $
Calculating Angular Momentum Quantum Number (j)
The magnitude of the angular momentum (J) is related to the quantum number j by the equation:
$ J = \sqrt{j(j+1)}\hbar $
We are given that the magnitude of the angular momentum is $ J = \sqrt{12}\hbar $. Equating the two expressions:
$ \sqrt{j(j+1)}\hbar = \sqrt{12}\hbar $
Squaring both sides and removing $\hbar$:
$ j(j+1) = 12 $
Rearranging into a quadratic equation:
$ j^2 + j - 12 = 0 $
Factoring the quadratic equation:
$ (j+4)(j-3) = 0 $
Since the angular momentum quantum number j must be non-negative, we have:
$ j = 3 $
Determining the Number of Beams
Now, we use the value of j to find the number of beams (N) using the formula $ N = 2j + 1 $:
$ N = 2(3) + 1 $
$ N = 6 + 1 $
$ N = 7 $
Therefore, the beam of atoms splits into 7 distinct beams.
Important Questions from Angular Momentum Operators Eigenvalues Clebsch Gordan
- Consider two particles with angular momenta $j_1 = 2\hbar$ and $j_2 = \hbar/2$. If the expression
$$|j = 5/2, m = 3/2\rangle = \begin{cases} c_1|j_1 = 2, m_1 = 1\rangle|j_2 = 1/2, m_2 = 1/2\rangle + \\ c_2|j_1 = 2, m_1 = 2\rangle|j_2 = 1/2, m_2 = -1/2\rangle \end{cases}$$
gives an eigenstate of the total angular momentum of the two particles, using standard notation. Which of the following is true?
(Hint: $\hat{J}_{\pm}|j, m\rangle = \sqrt{j(j + 1) - m(m \pm 1)} |j, m \pm 1\rangle$) - A system of three non-identical spin $\frac{1}{2}$ particles has the Hamiltonian $H = \frac{A}{\hbar^2} (\vec{S}_1 + \vec{S}_2) \cdot \vec{S}_3$, where $\vec{S}_1, \vec{S}_2$ and $\vec{S}_3$ are the spin operators of particles labelled $1,2$ and $3$ respectively and $A$ is a constant with appropriate dimensions. The set of possible energy eigenvalues of the system is
- $H$ is the Hamiltonian, $\vec{L}$ the orbital angular momentum and $L_z$ is the $z$-component of $\vec{L}$. The $1s$ state of the hydrogen atom in the non-relativistic formalism is an eigen function of which one of the following sets of operators?
- In the vector model of angular momentum applied to atoms, what is the minimum angle in degrees (in integer) made by the orbital angular momentum vector and the positive $z$ axis for a $2p$ electron?
A particle has wavefunction
$\psi(x,y,z) = N ze^{-\alpha(x^2+y^2+z^2)}$,
where $N$ is a normalization constant and $\alpha$ is a positive constant. In this state, which one of the following options represents the eigenvalues of $L^2$ and $L_z$ respectively?
Some values of $Y_l^m$ are:
$Y_0^0 = \sqrt{\frac{1}{4\pi}}$, $Y_1^0 = \sqrt{\frac{3}{4\pi}} \cos\theta$, $Y_1^{\pm 1} = \mp \sqrt{\frac{3}{8\pi}} \sin\theta e^{\pm i\phi}$
