Linear combination (10-20 min)

We introduce our `vector notation' (Dirac notation) from section 3, where the normalized wave functions $\psi_n(x)$ are denoted as $\vert n\rangle$, because they are vectors in a Hilbert space. In this problem, the $\vert n\rangle$ shall correspond to the normalized wave functions of the one-dimensional harmonic oscillator of frequency $\omega$. The $\vert n\rangle$ form an orthogonal system; we write the scalar product as

$$\langle n\vert m\rangle \equiv \int_{-\infty}^{\infty}dx \psi^*_n(x)\psi_m(x)= \delta_{n,m}.$$ (33)

1. Consider the state

$$\vert\phi\rangle = a\vert 1\rangle + b \vert 3\rangle,\quad a,b, \in C.$$ (34)

Which condition must the coefficients $a$,$b$ fulfill in order that $\vert\phi\rangle$ is normalized? Write the normalization condition in the `abstract, elegant form', using

$$\langle\phi\vert = a^*\langle 1\vert + b^* \langle 3\vert,$$ (35)

as $1=\langle \phi\vert \phi \rangle =...$

2. What is the probability to find the energy values $E_1$ and $E_3$ in an energy measurement of a system in the state $\vert\psi\rangle$ ?

3. Calculate the expectation value of the energy in the state $\vert\phi\rangle$ for general $a$ and $b$ and for $a=b=1/\sqrt{2}$.