# Two Orthogonal States (Homework 1, Physics 325)

Consider two states
$|\psi1>=|\phi_1>+4i|\phi_2>+5|\phi_3>$

$|\chi>=b|\phi_1>+4|\phi_2>-3i|\phi_3>$

where $|\phi_{1,2,3}>$ are orthonormal, and $b$ is a scalar constant. Find the value of $b$ for which $|\psi>$ and $|\chi>$ are orthogonal. Are these vectors normalized? If not, normalize them, i.e. find such

factors different for each vector that after multiplication by them the vectors do become

normalized.

Since all $|ϕ_n>$ states are orthonormal it means that

$<ϕ_i |ϕ_j>=δ_{ij}=$

$=1 \text{ ,for i=j}$

$=0 \text{ ,for i? j}$

Therefore

$<ψ|χ>=b<ϕ_1 | ϕ_1>+16i<ϕ_2 |ϕ_2>+5(-3i)<ϕ_3 | ϕ_3>=b+16i-15i=b+i$

For $ψ$ and $χ$ to be orthonormal it means

$<ψ|χ> =0$   which means $b+i=0$  or $b=-i$

For $ψ$ and $χ$ to be normalized it means  $<ψ|ψ>=1$   and $<χ|χ> =1$. For the given states one has

$<ψ|ψ>=1+(4i)^2+5^2=1-16+25=10$     and
$<χ|χ>=(-i)^2+4^2+(-3i)^2=-1+16-9=6$

So that the normalized states are

$|ψ>= 1/√10 (1*│ϕ_1>+4i*|ϕ_2> +5*| ϕ_3>)$    and
$|χ>=1/√6(-i*|ϕ_1>+4*|ϕ_2>-3i*| ϕ_3>)$