Quantum Mechanics 1: Angular Momentum and Spin

Verify equation 4175, which is the S1 triplet state when adding two spin 1/2 particles, using the Clebsch-Gordan table instead of operators.

Also, verify equation 4176, which is the singlet state for 00.

Verify equation 4176, which is the singlet state for $00$, using the Clebsch-Gordan table instead of operators.

Calculate the commutators of the angular momentum components and demonstrate that they are non-zero.

Using the angular momentum operators, calculate the magnitude of angular momentum $L$ for a given quantum number $l$ and the z-component of angular momentum for a quantum number $m$.

Explain how the Zeeman effect demonstrates the quantization of the z-component of angular momentum in quantum mechanics.

A 10-kilogram disc of radius 3 meters is spinning at 15 radians per second. What is the inertia of the disc?

The angular momentum of a rod changes from 15 to 35 kilograms times meter squared times radians per second in four seconds. What is the average torque on the rod?

A force of 300 newtons acts on a 2.5 meter long rod initially at rest. What is the torque acting on the rod?

What is the final angular momentum of the rod if the force acts on it for eight seconds?

What is the final angular speed of the rod?

Calculate the work done by the force in two ways: by calculating the change in rotational kinetic energy and using the rotational torque multiplied by the angular displacement.

Calculate the final speed of the merry-go-round after a child jumps on it using conservation of angular momentum.

Given a ball shot directly upward, find the angular momentum about point P when the ball is halfway back to the ground after reaching its maximum height.

For a disc rotating like a merry-go-round with a torque that varies, find the angular momentum at three seconds using the expression for torque.

Consider an electron in the spin state $|\chi\rangle = \begin{pmatrix} a \\ 3i \\ 4 \end{pmatrix}$. For part A, determine the normalization constant $a$.

Using the electron spin state $|\chi\rangle = \begin{pmatrix} a \\ 3i \\ 4 \end{pmatrix}$, find the expectation values of the spin in the $x$, $y$, and $z$ directions.

Find the standard deviations (uncertainties) in the spins of the $x$, $y$, and $z$ directions for the given electron spin state $|\chi\rangle = \begin{pmatrix} a \\ 3i \\ 4 \end{pmatrix}$.

Confirm your results are consistent with all three uncertainty principles for the electron spin state $|\chi\rangle = \begin{pmatrix} a \\ 3i \\ 4 \end{pmatrix}$.

Using the algebra of the Pauli matrices, calculate the product of two Pauli operators, given the specific operators involved.

In an experiment like the one Einstein proposed, two particles are entangled. If one particle is measured to have spin up, the other measured in the same direction must have spin down. What happens if their spins were vertical and opposite, but both are measured in the horizontal direction? Explain how this relates to the conservation of angular momentum and the concept of entanglement.