Quantum Mechanics 1

Given a wave function $\psi = A \sin(2x)$ for $x \in [-\pi, \pi]$ and $\psi (x) = 0$ otherwise, find the value of $A$ such that the wave function is normalized.

Given a ket state $|\psi\rangle = 3|\uparrow\rangle + 2i|\downarrow\rangle$, normalize this state.

Normalize the quantum harmonic oscillator wave function given by $|\psi\rangle = 2|0\rangle + \sqrt{5}i|1\rangle + \frac{4}{i}|4\rangle$.

Normalize the wave function $\psi(x) = \sin(2 \pi x)$ for a particle in a box of length 1.

Normalize the wave function $\psi(x) = x(L-x)$ for a particle in a box of length $L$.

Find the transmission coefficient using the WKB approximation for the case when the energy is less than the potential.

Integrate over the momentum $p(x)$ and calculate $e^{-2\gamma}$, where $\gamma = \frac{1}{\hbar} \int p(x) \, dx$.

Sketch out what you expect the wave functions to look like from the given potential profile.