Section 1
Quantum States
Let us assume that the particle is confined to $\left\{x_0, x_1, \ldots, x_5\right\}$ and the current state vector is$$|\psi\rangle=[2-i, 2 i, 1-i, 1,-2 i, 2]^T .$$What is the likelihood of finding the particle at position $x_3$ ?
Let $|\psi\rangle$ be $\left[c_0, c_1, \ldots, c_{n-1}\right]^T$. Check that multiplying $|\psi\rangle$ by any complex number $c$ will not alter the calculation of probabilities.
Do the vectors $[1+i, 2-i]^T$ and $[2+2 i, 1-2 i]^T$ represent the same state?
Normalize the ket$$|\psi\rangle=[3-i, 2+6 i, 7-8 i, 6.3+4.9 i, 13 i, 0,21.1]^T .$$
(a) Verify that the two state vectors $\left[\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right]^T$ and $\left[\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right]^T$ are each of length 1 in $\mathbb{C}^2$. (b) Find the vector on the unit ball of $\mathbb{C}^2$ representing the superposition (addition) of these two states.
Let the spinning electron's current state be $|\psi\rangle=3 i|\uparrow\rangle-2|\downarrow\rangle$. Find the probability that it will be detected in the up state.
Normalize the ket given in Equation (4.25).
Check that the set $\left\{\left|x_0\right\rangle,\left|x_1\right\rangle \ldots,\left|x_{n-1}\right\rangle\right\}$ is an orthonormal basis for the state space of the particle on the line. Similarly, verify that $\{|\uparrow\rangle,|\downarrow\rangle\}$ is an orthonormal basis of the one-particle spin system.
$$\text { Calculate the bra corresponding to the ket }|\psi\rangle=[3+i,-2 i]^T \text {. }$$
Calculate the amplitude of the transition from $\frac{\sqrt{2}}{2}[i,-1]^T$ to $\frac{\sqrt{2}}{2}[1,-i]^T$.