Quantum Mechanics 1

Explain the concept and significance of quantum numbers in determining the arrangement of electrons in an atom.

Describe the process of determining the electron configuration of an atom using the Aufbau principle, considering a neutral chlorine atom as an example.

Find the four quantum numbers $n, l, m_l, m_s$ that correspond to a specific electron configuration. For example, determine these quantum numbers for the 2p5 electron.

Using the Dirac theory, show that the fields satisfy the Heisenberg equation of motion.

Find the eigenenergies and eigenfunctions of a particle in an infinite square well, also known as the Particle in a Box problem, where the potential $V(x)$ equals infinity everywhere except between $x = 0$ and $x = a$, where it equals zero.

Find the eigenenergies and eigenfunctions of a particle in an infinite square well, also known as the Particle in a Box problem, where the potential $V(x)$ equals infinity everywhere except between $x = 0$ and $x = a$, where it equals zero.

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.

Bell's theorem experiment: There are two spin detectors, each capable of measuring spin in one of three directions. Measurement directions are selected randomly and independent of each other. Entangled particle pairs are sent to the detectors, recording whether the measured spins are the same or different. Given these conditions, determine the expected frequency of different spin results if the particles contain hidden information. How does this frequency compare with actual experimental results?

Solve the quantum harmonic oscillator using ladder operators to find the eigenfunctions in terms of the ground state wavefunction.

Using ladder operators, derive the expressions for the wavefunctions of the quantum harmonic oscillator in terms of Hermite polynomials.

Given a block of mass $m$ on a frictionless table, attached to a spring with spring constant $k$, derive the equation of motion for the block and find the general solution for its trajectory as it oscillates in simple harmonic motion.

Using the small angle approximation, reduce the motion of a pendulum around its equilibrium position to a simple harmonic oscillator.

Using the Schrödinger equation, explain how an electron's wave function evolves over time considering its kinetic and potential energy components.

What are the n and l values for an electron in the 3d sublevel?

What are the four quantum numbers that correspond to the 3d8 electron?

When $n = 3$ and $L = 0$, calculate the total number of orbitals for the subshells $n = 3, L = 0$ and $n = 3, L = 2$.

What set of quantum numbers matches the orbital shown in the picture?

Given a wave function $\Psi(x)$, determine the probability of finding an electron in a specific region by calculating the square of the absolute value of $\Psi(x)$.

Explain the connection between musical superposition in instruments and quantum superposition in qubits.

Using the derived formula, determine the rate of change of position for a quantum mechanical particle, given the expression $H = \frac{p^2}{2m} + U(x)$ for the Hamiltonian.