Quantum Mechanics 1: Time Independent Schrodinger Equation

Using the time-independent Schrödinger equation, find the energy eigenvalues and eigenstates of a system.

Solve the time-independent Schrödinger equation for a potential well where: $V(x) = -V_0$ for $-a < x < a$, and $V(x) = 0$ for $|x| > a$. Consider the bound state where the energy level is less than zero.

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.

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

Solve the Schrödinger equation in one dimension using the method of separation of variables.

Derive the time-independent Schrödinger equation from the time-dependent Schrödinger equation by separating the wavefunction into spatial and temporal components, $\psi(x, t) = \phi(x) \cdot \xi(t)$, and demonstrate each step in the derivation.

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.

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 $x$.