Quantum Mechanics 1: Foundations of Quantum Mechanics

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.

We have a force of 300 newtons used to stretch a horizontal spring with a 0.5 kilogram block attached by 0.25 meters. The block is released from rest and undergoes simple harmonic motion. Calculate the spring constant, the amplitude, the maximum acceleration, the mechanical energy of the system, the maximum velocity, and the velocity when x is 0.15 meters.

Explain the ultraviolet catastrophe and how Planck's law resolved it.

Describe the photoelectric effect and how it proves the particle nature of light.

Discuss the concept of wave-particle duality in light.

Explain why the measurement postulate in quantum theory is problematic and why re-interpreting it does not remove the problem.

In a triangle, given angles of 55 and 95 degrees, find the missing angle x.

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?

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

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?

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

Explain the phenomenon of quantum tunneling and its application in quantum communication using x-ray beams from the Advanced Photon Source.

Given the time evolution operator $U = e^{-i H t}$ where $H$ is the Hamiltonian, prove that if $H$ is Hermitian, then $U$ is a unitary operator, i.e., $U U^{\dagger} = U^{\dagger} U = 1$. Additionally, prove the vice-versa: if $U$ is a unitary operator, then $H$ must be Hermitian.

Imagine you need to solve the quadratic equation $x^2 - 1 = 0$.