Quantum Mechanics 1: Quantum Harmonic Oscillator

Using the harmonic oscillator model, calculate the energy levels of a diatomic molecule given the spring constant $K$ and the reduced mass $\mu$.

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 $a^\dagger$.

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 raising and lowering operators (A+ and A-), determine the action of these operators on the eigenstates of the quantum harmonic oscillator.

Explain the properties and significance of raising and lowering operators in the quantum harmonic oscillator.

Solve the Schrödinger equation with the harmonic oscillator potential energy, $\frac{1}{2} kx^2$, to find the wave function $\psi(x)$. Consider potential solutions like $\sin(ax)$, polynomials (e.g., $ax^3$, $ax^4$), or exponential functions ($e^{ax}$).

Determine the allowed energies and wave function for a particle undergoing harmonic motion.

Using the Schrödinger equation, find the energies for a quantum harmonic oscillator, and relate them to the classical harmonic oscillator.

Consider a Hamiltonian which is a harmonic oscillator with a time-dependent coefficient in front of $x^2$. The system starts in the ground state at t = -b1fty, and at this time the Hamiltonian is just the ordinary harmonic oscillator Hamiltonian with frequency b1mega_0. For small values of the parameter $K$, estimate the probability that the system will have made a transition to some other state by t = b1fty.

Normalize the quantum harmonic oscillator wave function given by $|\psi\rangle = 2|0\rangle + \sqrt{5}i|1\rangle + \frac{4}{i}|4\rangle$.