Exploring Quantum Physics - Week 2 Question 1

Question:

Suppose we have a state $| \psi \rangle$ that is a linear combination of two other states $| \psi \rangle = a | 1 \rangle + b | 2 \rangle$, where $a$ and $b$ are non-zero, but otherwise unspecified, and $| 1 \rangle$ and $| 2 \rangle$ are orthonormal ($\langle 1 | 2 \rangle = \langle 2 | 1 \rangle = 0$, $\langle 1 | 1 \rangle = \langle 2 | 2 \rangle = 1$). What is $\langle 1 | \psi \rangle$ ?

Solution:

This is in fact easy, we have:

$\begin{eqnarray*} \langle 1 | \psi \rangle &=& \langle 1 | (a | 1 \rangle + b | 2 \rangle) \rangle \\ &=& \langle 1 | a | 1 \rangle + \langle 1 | b | 2 \rangle \\ &=& a \langle 1 | 1 \rangle + b \langle 1 | 2 \rangle \\ &=& a \end{eqnarray*}$

So that is it - consider this is a warm up for week 2.