Practice Problems 1

Inner product practice

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

a)

Find $⟨w|w⟩$

b)

Find $⟨c|w⟩$

c)

Let $\hat{A}$ be a hermitian operator (i.e ${\hat{A}}^{†}=\hat{A}$), and $\hat{A}|a⟩=\hslash |a⟩$.
Find $⟨a|\hat{A}|w⟩$

Commutator practice

Let $\hat{a}=\frac{1}{\sqrt{2}}(y+\frac{\partial }{\partial y})$, and ${\hat{a}}^{†}=\frac{1}{\sqrt{2}}(y-\frac{\partial }{\partial y})$. Show that $[\hat{a},{\hat{a}}^{†}]=1$. 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 $|\psi ⟩$: $P(\alpha )=|⟨a|\psi ⟩{|}^2$, where $|a⟩$ is the corresponding eigenvector of eigenvalue $\alpha$.
In the Stern-Gerlach experiment, a subsequent z-axis measurement of state $|+⟩$ results in only $|+⟩$ and not $|-⟩$. Using the postulates, explain why (see notes for subsequent experiments).

b)

Suppose we allow one of the states $|+⟩$ or $|-⟩$ 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 $|+{⟩}_x$ and $|-{⟩}_x$, use postulate 4 to write out 4 equations relating the new states and old states.

c)

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