Quantum Mechanics 1

Using Bohr's model, predict the wavelength of a photon associated with an electron transition from $n=4$ to $n=2$ in a hydrogen atom.

Derive the radius of a hydrogen-like atom using the given equations.

Calculate the velocity of a hydrogen-like atom in orbit using the provided formulae.

Derive the energy level of a hydrogen-like atom.

Discuss the implications of symmetric and anti-symmetric wave functions in relation to the Pauli Exclusion Principle.

A particle of mass m is confined in a two-dimensional infinite square well potential of side a. The eigen energy of the particle in a state is given as $E = \frac{25 \pi^2 \hbar^2}{m a^2}$. Determine whether this state is four-fold, three-fold, two-fold, or non-degenerate.

Three identical non-interacting particles, each with spin $\frac{1}{2}$ and mass m, are moving in a one-dimensional infinite square well potential. The well has a side length of a and a potential of 0 inside the box and infinite outside. Determine the ground state energy of the system.

Solve the Schrödinger equation for an infinite square well potential and find the energy values and wave function $\psi$.

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.

Using the ladder and number operators, express the Hamiltonian of a quantum harmonic oscillator in terms of these operators.

In the context of the quantum harmonic oscillator, calculate the commutation relation between the ladder operators $a$ and $a^\dagger$.

Given that $n$ is an eigenstate of the number operator $N$, show how the action of the lowering operator $a$ and the raising operator $a^\dagger$ changes the eigenvalue.

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.

Estimate the size of an atom in picometers.

Consider a system with three quantum dots and a mobile electron.

Describe the quantum state of the electron, its representation as a vector, and the condition for normalization.

Verify whether a given function is an eigenfunction for a specified operator by calculating derivatives and checking if the function returns in the form $a \psi$.

What is the Hamiltonian operator in quantum mechanics and how is it represented in the Schrödinger equation?