Quantum Mechanics 1: Operators and Eigenvalues

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.

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.

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

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.

Given that $n$ is an eigenstate of the number operator $N$, show how the action of the lowering operator $a$ and the raising operator $a^\dagger$ changes the eigenvalue.

Verify whether a given function is an eigenfunction for a specified operator by calculating derivatives and checking if the function returns in the form $a \psi$.

What is the Hamiltonian operator in quantum mechanics and how is it represented in the Schrödinger equation?

Using the derived formula, determine the rate of change of position for a quantum mechanical particle, given the expression $H = \frac{p^2}{2m} + U(x)$ for the Hamiltonian.

In this problem, we're dealing with a situation where the magnetic field is at some angle relative to the z-axis, but we've written the Hamiltonian in the spin up spin down basis. Part A: Obtain the matrix representation of $H$ in the spin up spin down basis. Part B: Find the eigenvalues and normalized eigenvectors of $H$. Part C: Verify that the normalized eigenvectors of $H$, denoted as $E_1$ and $E_2$, satisfy the Dirac completeness relation.