Solve

Prove that every cyclic group of order $n$ is isomorphic to $\mathbb{Z}/n\mathbb{Z}$.

Solve the Schrödinger equation for a particle confined in a one-dimensional infinite potential well of width $L$.

Prove that $H$ is a normal subgroup of $G$ if and only if $gHg^{-1} = H$ for all $g \in G$.

Use Lagrange's Theorem to prove that in a group of prime order $p$, every non-identity element generates the group.

Derive the Heisenberg uncertainty relation $\Delta x \Delta p \geq \hbar/2$ using commutator algebra.

Prove that $f(x) = x^2$ is continuous on $\mathbb{R}$ but not uniformly continuous on $\mathbb{R}$.