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
An electron in the Coulomb field of a proton is in the following state of coherent superposition of orthonormal states $\psi_{nlm}$
$\Psi = \frac{1}{3}\psi_{100} + \frac{1}{\sqrt{3}}\psi_{210} - \frac{\sqrt{5}}{3}\psi_{320}$
Let $E_1, E_2$, and $E_3$ represent the first three energy levels of the system. A sequence of measurements is done on the same system at different times. Energy is measured first at time $t_1$ and the outcome is $E_2$. Then total angular momentum is measured at time $t_2 > t_1$ and finally energy is measured again at $t_3 > t_2$. The probability of finding the system in a state with energy $E_2$ after the final measurement is $P/9$. The value of $P$ is ______________ (in integer).- $H$ 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:
$Y_0^0 = \sqrt{\frac{1}{4\pi}}$, $Y_1^0 = \sqrt{\frac{3}{4\pi}} \cos\theta$, $Y_1^{\pm 1} = \mp \sqrt{\frac{3}{8\pi}} \sin\theta e^{\pm i\phi}$- The spin $ \vec{S}$ and orbital angular momentum $ \vec{L}$ of an atom precess about $ \vec{J}$, the total angular momentum. $ \vec{J}$ precesses about an axis fixed by a magnetic field $ \vec{B}_1 = 2B_0 \hat{z}$, where $B_0$ is a constant. Now the magnetic field is changed to $ \vec{B}_2 = B_0( \hat{x} + \sqrt{2} \hat{y} + \hat{z})$. Given the orbital angular momentum quantum number $l = 2$ and spin quantum number $s = 1/2$, $ \theta$ is the angle between $ \vec{B}_1$ and $ \vec{J}$ for the largest possible values of total angular quantum number $j$ and its $z$-component $j_z$. The value of $ \theta$ (in degree, rounded off to the nearest integer) is ________
