Atomic $J$ vs $B_1$ Angle Calculation
This solution calculates the angle $ \theta$ between the total angular momentum $ \vec{J}$ and the magnetic field $ \vec{B}_1$. The calculation uses the state corresponding to the maximum possible values of $j$ and $j_z$, with $ \vec{B}_1$ defining the quantization axis.
Maximum $j$ and $j_z$ Values
- Given orbital angular momentum quantum number $ l=2 $ and spin quantum number $ s=1/2 $.
- The maximum total angular momentum quantum number is $ j_{max} = l+s = 2 + \frac{1}{2} = \frac{5}{2} $.
- For the state with $ j=5/2 $, the maximum $z$-component quantum number $m_j$ (corresponding to the largest $j_z$) is $ m_{j, max} = +j = +\frac{5}{2} $.
Relation: $ \theta$ and Angular Momentum
The angle $ \theta$ between $ \vec{J}$ and the $z$-axis (direction of $ \vec{B}_1$) is found using the projection of $ \vec{J}$ onto the $z$-axis: $ \langle J_z \rangle = |\vec{J}| \cos \theta $ Substituting $ \langle J_z \rangle = m_j \hbar $ and $ |\vec{J}| = \sqrt{j(j+1)}\hbar $, we derive: $ \cos \theta = \frac{m_j \hbar}{\sqrt{j(j+1)}\hbar} = \frac{m_j}{\sqrt{j(j+1)}} $
Numerical Calculation of $ \theta$
Using the maximum values $ j = 5/2 $ and $ m_j = 5/2 $: $ \cos \theta = \frac{5/2}{\sqrt{\frac{5}{2}(\frac{5}{2}+1)}} = \frac{5/2}{\sqrt{\frac{5}{2} \cdot \frac{7}{2}}} = \frac{5/2}{\sqrt{35/4}} = \frac{5/2}{\sqrt{35}/2} = \frac{5}{\sqrt{35}} $ Simplify and find $ \theta$: $ \cos \theta = \frac{5\sqrt{35}}{35} = \frac{\sqrt{35}}{7} $ $ \theta = \arccos\left(\frac{\sqrt{35}}{7}\right) $ Using a calculator, $ \frac{\sqrt{35}}{7} \approx 0.84515 $. $ \theta \approx \arccos(0.84515) \approx 32.32^\circ $
Final Rounded Angle
Rounding to the nearest integer, the angle is $ \theta \approx 32^\circ $. This result is consistent with the range provided.
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?
- An atom with non-zero magnetic moment has an angular momentum of magnitude $\sqrt{12}\hbar$. When a beam of such atoms is passed through a Stern-Gerlach apparatus, how many beams does it split into?
- 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:
$Y_0^0 = \sqrt{\frac{1}{4\pi}}$, $Y_1^0 = \sqrt{\frac{3}{4\pi}} \cos\theta$, $Y_1^{\pm 1} = \mp \sqrt{\frac{3}{8\pi}} \sin\theta e^{\pm i\phi}$
