Quantum Mechanics 1

If you have a merry-go-round with an inertia of 500 and it's spinning with an angular speed of 2 radians per second, and you place a box on the merry-go-round increasing the inertia from 500 to 600, find the new angular speed of the merry-go-round.

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.

Determine the physical significance of the quantum numbers $l$ and $m$ with respect to angular momentum in a hydrogen atom.

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.

Using the variational method, find an upper bound on the energy of the ground state for a quantum system. If the trial function deviates by an order $\epsilon$, how does this affect the upper bound?

Use the variational method to find the upper bound for the ground state energy of the 1D infinite square well problem.

Using the variational method, find the best possible wave function approximation for a given system where $\Psi(\lambda)$ depends on the parameter $\lambda$. Calculate the expectation value of the energy and find the condition for minimum energy.

For a Hamiltonian partitioned as $H = H_0 + \lambda V$, determine the first order correction to the energy $E_1$ using perturbation theory. Consider the zeroth order Hamiltonian $H_0 \Psi_0 = E_0 \Psi_0$ to have a known solution.

Visualize the wavefunctions of hydrogen orbitals using Bohmian trajectories and describe how these visualizations can help in understanding atomic orbitals better.

Any p orbital can accommodate up to how many electrons?

The principal quantum number is related to what aspect of the orbital?

What is the correct set of quantum numbers for the valence electron of rubidium (Z=37)?

Which of the following sets of quantum numbers represents an impossible arrangement?

Using the transfer matrix, find eigenstates in cases of truly bound states when the first and last layers are infinitely thick with potentials $v_1$ and $v_{n+2}$, and the energy $E$ is below both potentials.

Prove that the energy eigenfunction of bound states in one-dimensional potential is non-degenerate by considering two energy eigenfunctions $\psi_1(x)$ and $\psi_2(x)$ with the same eigenvalue $E$.

Solve the problem of a quantum particle moving in a central potential by separating the solution into radial and angular parts, using spherical coordinates.

Find the wave functions and the allowed energies for an infinite spherical well where the potential $V(r)$ is defined as 0 for $r < a$ and infinity for $r > a$.