Quantum Mechanics 1: One Dimensional Quantum Systems

Using the transfer matrix, find eigenstates in cases of truly bound states when the first and last layers are infinitely thick with potentials $v_1$ and $v_{n+2}$, and the energy $E$ is below both potentials.

Prove that the energy eigenfunction of bound states in one-dimensional potential is non-degenerate by considering two energy eigenfunctions $\psi_1(x)$ and $\psi_2(x)$ with the same eigenvalue $E$.

A particle of energy $E$ is scattered off a one-dimensional potential $\lambda \delta(x)$, where $\lambda$ is a real positive constant, with a transition amplitude $t_x$. In a different experiment, the same particle is scattered off another one dimensional potential $-\lambda \delta(x)$ with a transmission amplitude $t_{-}$ In the limit $E \to 0$, the phase difference between $t_{+}$ and $t_{-}$ is

1. $\pi / 2$

2. $\pi$

3. 0

4. $3\pi / 2$

A particle is confined into a box with a 1D Delta potential at the center. Determine which energy equation the system satisfies.

Develop a model to predict the energy eigenvalues (E_1, E_2, ...) and wave functions ($\Psi_1$, $\Psi_2$, ...) within a finite quantum well.

Determine the probability distribution for a particle in a one-dimensional potential box for different quantum states $n = 1$, $n = 2$, and the classical limit as $n \to \infty$.

Find the eigenenergies and eigenfunctions of a particle in an infinite square well, also known as the Particle in a Box problem, where the potential $V(x)$ equals infinity everywhere except between $x = 0$ and $x = a$, where it equals zero.

Find the eigenenergies and eigenfunctions of a particle in an infinite square well, also known as the Particle in a Box problem, where the potential $V(x)$ equals infinity everywhere except between $x = 0$ and $x = a$, where it equals zero.

Consider a particle that is traveling towards a barrier with energy $E$ less than the potential barrier $U_0$. Classically, the particle would bounce off, but quantum mechanically, it can tunnel through. Calculate the tunneling probability for the particle using the boundary conditions and wavefunctions for the three regions separated by the potential barrier.

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.