Quantum Mechanics 1

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.

A 500-kilogram merry-go-round with a radius of 10 meters is moving at a speed of 0.5 radians per second. A child jumps on it four meters away from the central axis of rotation. What is the inertia of the merry-go-round and then find the inertia of the child?

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.

Determine which of the three wave graphs is correct based on the boundary condition $\psi = 0$ at the ends.

Determine the probability distribution for a particle in a one-dimensional potential box for different quantum states $n = 1$, $n = 2$, and the classical limit as $n \to \infty$.

Explain Aufbau's principle, Hund's rule, and Pauli's exclusion principle and their application in filling electron orbitals.

Arrange or distribute five electrons around the nucleus of the Boron atom considering the Pauli Exclusion Principle, Hund's Rule of Maximum Multiplicity, and Aufbau Principle.

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.

What is the percent contribution of the second physical state $i_2$ to the overall resonance hybrid in a molecule with given resonance contributors?