Quantum Mechanics 1

Using orbital diagrams, determine whether magnesium (atomic number 12) is paramagnetic or diamagnetic.

Fill the orbital diagram for phosphorus, which has 15 electrons.

Draw the orbital diagram of a sulfide ion (S²⁻), considering an atomic number of 16 for sulfur.

Draw the orbital diagram for the aluminum plus three cation (Al³⁺).

Draw the orbital diagram for the cobalt plus two ion (Co²⁺), considering transition metal ion rules.

Using the method of separation of variables, solve the time-independent Schrodinger equation to find the stationary states and corresponding energy levels for a potential V.

Derive the expression for the energy of a particle in a three-dimensional potential well, assuming the lengths of the box are equal, forming a cube.

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.

In time-dependent perturbation theory, the first-order transition amplitude from an initial state $|m\rangle$ to a final state $|n\rangle$ is given by

$c_n^{(1)}(t) = -\frac{i}{\hbar} \int_0^t \langle n | V(t') | m \rangle , e^{i \omega_{nm} t'} , dt',$

where $\omega_{nm} = \frac{E_n - E_m}{\hbar}$ and $V(t)$ is the time-dependent perturbation.

a) Explain the physical meaning of the transition amplitude $c_n^{(1)}(t)$.

b) Write down the expression for the transition probability from state $|m\rangle$ to $|n\rangle$, and explain how it relates to the amplitude.

Suppose an electron of mass $m$ is in an infinite square well of length $L$. At time $t = 0$, the electron is prepared in a triangular state. Build a Python function to find the initial state, ensure it is normalized, and then determine the time-dependent wave function by finding the expansion coefficients.

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.

Given an unperturbed system with known exact eigenvalues and eigenfunctions, apply time-independent perturbation theory to find the first-order correction to the eigenvalue $E_n$ and the eigenfunction $\psi_n$.

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

Imagine you need to solve the quintic polynomial equation $x^5 - x - 1 = 0$.

Consider a particle that can move only in one dimension between two impenetrable walls at $x = 0$ and $x = a$. The potential is zero between the walls and infinite outside. Solve the time-independent Schrödinger equation to find the allowed wave functions and corresponding energies of the particle.

Solve the time-independent Schrödinger equation using the separation of variables method when the potential energy $V$ is a function of $x$.

Calculate the time evolution operator $U$ for a system with a time-independent Hamiltonian using the exponential function.

For a system where the Hamiltonian depends on time, derive the recursive relation for the time evolution operator $U$.

Using the variational method, determine the parameter $\lambda$ that minimizes the energy $E_{\phi}(\lambda)$ for a trial wave function $\phi(\lambda)$ such that $\frac{dE_{\phi}}{d\lambda} = 0$.

Using the Ritz method, determine the ground state energy of a system given a test wave function $\psi_T$ with parameters $a, b, c, d$. Calculate the extremal point of the energy functional by finding values for $a, b, c, d$ that minimize the energy.