Quantum Fundamentals: Winter-2026
Extra practice: Due W5 D5

1. Dimensional Analysis of Kets
   1. \(\left\langle {\Psi}\middle|{\Psi}\right\rangle =1\) Identify and discuss the dimensions of \(\left|{\Psi}\right\rangle \).
   2. For a spin-\(\frac{1}{2}\) system, \(\left\langle {\Psi}\middle|{+}\right\rangle \left\langle {+}\middle|{\Psi}\right\rangle + \left\langle {\Psi}\middle|{-}\right\rangle \left\langle {-}\middle|{\Psi}\right\rangle =1\). Identify and discuss the dimensions of \(\left|{+}\right\rangle \) and \(\left|{-}\right\rangle \).
   3. In the position basis \(\int \left\langle {\Psi}\middle|{x}\right\rangle \left\langle {x}\middle|{\Psi}\right\rangle dx = 1\). Identify and discuss the dimesions of \(\left|{x}\right\rangle \).
2. ISW Right Quarter

   A particle is confined in a one-dimensional infinite square well on \(0<x<L\). For each of the first three energy eigenstates (i.e., \(n=1,2,3\)), calculate the probability that a position measurement yields a result in the region \(\frac{3L}{4}<x<L\).