Quantum Mechanics 1: Quantum States and Wavefunctions

Consider a system with three quantum dots and a mobile electron.

Describe the quantum state of the electron, its representation as a vector, and the condition for normalization.

Determine which of the three wave graphs is correct based on the boundary condition $\psi = 0$ at the ends.

What is the percent contribution of the second physical state $i_2$ to the overall resonance hybrid in a molecule with given resonance contributors?

Given a wave function $\Psi(x)$, determine the probability of finding an electron in a specific region by calculating the square of the absolute value of $\Psi(x)$.

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 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 $L$.

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