Exercise 6.2

For the state vector in Eq. 6.5 to be normalized, the coefficients must satisfy the following condition: \[ \alpha_u^*\beta_u^*\alpha_u \beta_u + \alpha_u^*\beta_d^*\alpha_u \beta_d + \alpha_d^*\beta_u^*\alpha_d \beta_u + \alpha_d^*\beta_d^*\alpha_d \beta_d = 1 \]

We can factor out \(\alpha_u^*\alpha_u\) and \(\alpha_d^*\alpha_d\): \[ \begin{flalign*} \alpha_u^*\alpha_u \underbrace{(\beta_u^*\beta_u + \beta_d^*\beta_d)}_{=1} + \alpha_d^*\alpha_d \underbrace{(\beta_u^*\beta_u + \beta_d^*\beta_d)}_{=1} &= \alpha_u^*\alpha_u + \alpha_d^*\alpha_d \\ &= 1 \end{flalign*} \]

if Eqs. 6.4 are satisfied.