Quantum Mechanics 1: Approximation Methods

Using the variational method, find an upper bound on the energy of the ground state for a quantum system. If the trial function deviates by an order $\epsilon$, how does this affect the upper bound?

Use the variational method to find the upper bound for the ground state energy of the 1D infinite square well problem.

Using the variational method, find the best possible wave function approximation for a given system where $\Psi(\lambda)$ depends on the parameter $\lambda$. Calculate the expectation value of the energy and find the condition for minimum energy.

For a Hamiltonian partitioned as $H = H_0 + \lambda V$, determine the first order correction to the energy $E_1$ using perturbation theory. Consider the zeroth order Hamiltonian $H_0 \Psi_0 = E_0 \Psi_0$ to have a known solution.

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.

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 quintic polynomial equation $x^5 - x - 1 = 0$.

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.

Find the transmission coefficient using the WKB approximation for the case when the energy is less than the potential.

Integrate over the momentum $p(x)$ and calculate $e^{-2\gamma}$, where $\gamma = \frac{1}{\hbar} \int p(x) \, dx$.