Quantum Mechanics 1

Explain why the kinetic energy term in the Hamiltonian is written as $\frac{p^2}{2m}$, knowing that $p$ is the momentum operator and $m$ is the mass of the system.

Consider two quantum operators $A$ and $B$. Determine whether $[A, B] = 0$ or not, where $[A, B]$ is the commutator of $A$ and $B$.

Determine whether the kinetic energy operator and the momentum in the X direction operator commute.

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.

Given a matrix $A$ and a vector $x$, find an eigenvector of $A$, where the eigenvalue is the factor by which the eigenvector is stretched when the matrix is applied.

Given a quantum system, such as the spin of an electron, determine the eigenstates and eigenvalues when a measurement is made on the system.

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

What functions are there that I can take the derivative of and get back a constant times the original function?

An electron in a hydrogen atom drops from energy level 5 to energy level 2. What is the energy of the emitted Photon as this electron completes this quantum jump?

Consider two 1s electrons in helium. We need to write their total wave function, considering they are indistinguishable, and apply the properties of fermions and bosons. Determine the valid wave function configuration for both fermions and bosons.

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

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.

We have a force of 300 newtons used to stretch a horizontal spring with a 0.5 kilogram block attached by 0.25 meters. The block is released from rest and undergoes simple harmonic motion. Calculate the spring constant, the amplitude, the maximum acceleration, the mechanical energy of the system, the maximum velocity, and the velocity when x is 0.15 meters.

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

Prove that the Hermitian adjoint of a scalar is the complex conjugate.

Prove that the hermitian adjoint of a product of operators is calculated by switching their order and taking the individual hermitian adjoints.

Prove that the eigenvalues of any Hermitian operator must be real numbers.

The total energy of the electron in a hydrogen atom is -0.278 eV. In part A, what is the maximum value of the magnitude of the orbital angular momentum of the electron? In part B, what is the minimum value for the magnitude of the orbital angular momentum of the electron? In part C, what are the greatest and second greatest possible values of the z component of the orbital angular momentum of the electron?

Explain why atoms are stable and why electrons do not collide with the nucleus.