$\newcommand{\ket}[1]{|#1\rangle}$ $\newcommand{\bra}[1]{\langle #1|}$

Practice Problems 1

Inner product practice

Let $\ket w = \frac{1}{3}\ket a + \frac{3}{5}\ket b - \frac{4}{5}i\ket c$. Assume $\ket a$, $\ket b$, and $\ket c$ are all orthonormal.

a)

Find $\bra ww\rangle$

b)

Find $\bra cw\rangle$

c)

Let $\hat A$ be a hermitian operator (i.e $\hat A^\dagger = \hat A$), and $\hat A \ket a= \hbar \ket a$.
Find $\bra a \hat A \ket w$

Commutator practice

Let $\hat{a}=\frac{1}{\sqrt{2}}\left(y+\frac{\partial}{\partial y}\right)$, and $\hat{a}^{\dagger}=\frac{1}{\sqrt{2}}\left(y-\frac{\partial}{\partial y}\right)$. Show that $[\hat a, \hat a^\dagger] = 1$. As you will see later in the course, this is the heart of solving the quantum harmonic oscillator problem elegantly.

Walking through Stern-Gerlach

Adapted from University of Washington, Physics 248A

a)

Recall that postulate 4 tells us a way of calculating the probability of measuring an eigenvalue in a state $\ket \psi$: $P(\alpha) = |\bra a \psi \rangle|^2$, where $\ket a$ is the corresponding eigenvector of eigenvalue $\alpha$.
In the Stern-Gerlach experiment, a subsequent z-axis measurement of state $\ket +$ results in only $\ket +$ and not $\ket -$. Using the postulates, explain why (see notes for subsequent experiments).

b)

Suppose we allow one of the states $\ket +$ or $\ket -$ to be subjected to an x-axis Stern-Gerlach apparatus. We would surprisingly find that the original state ‘split’ into two states again. Denoting the new states $\ket +_x$ and $\ket -_x$, use postulate 4 to write out 4 equations relating the new states and old states.

c)

Since $\ket +$ and $\ket -$ form a complete basis in our system, we can write any other kets in our system as a linear combination of them. Particularly, $$\ket +_x = a \ket + + b\ket -$$ $$\ket -_x = c \ket + + d \ket -$$ Determine the coefficients a, b, c, and d.