Probability of Finding System in a State Given a Particular Basis

[The Head]

Homework Statement

Note: I am going to use |a> <a| to denote ket and bra vectors

The components of the state of a system| ω1> in some basis |δ1>, |δ2>, |δ3> are given by
<δ1|ω1> = i/sqrt(3), <δ2|ω1> = sqrt(2/3), <δ3|ω1> = 0

Find the probability of finding the system in the state |ω2> whose components in the same basis are
<δ1|ω2> = (1+i)/sqrt(3), <δ2|ω2> = sqrt(1/6), <δ3|ω2> = sqrt(1/6)

Homework Equations

P(δn)= |<δn|ω2>|^2/(<ω2|ω2>)

The Attempt at a Solution

I am actually rather confused just on how to start this problem. I am more familiar with the examples where I am given the matricies of observable and the components of a state function. Then I know that you have to find the eigenvectors corresponding to an eigenvalue and to use those to find the probability of an energy level.

So here we are dealing with two states being represented in a basis and I don't know how to weed through this to get any real information on either of these states or the basis.

Thank you for reading!

 

[TSny]

Since |δ1>, |δ2>, |δ3> form a basis, it must be possible to express |ω1> in terms of the basis vectors. So, there exist numbers a₁ ,a₂, and a₃ such that

|ω1> = a₁|δ1> + a₂|δ2> + a₃|δ3>.

Can you identity the values of a₁, a₂, and a₃?

Similarly for the state |ω2>.

 

[The Head]

So I am assuming to find a1 I simply perform <δ1|ω1>, and I would get a1=i/sqrt(3), and similarly a2=sqrt(2/3), a3=0, and then I could put |ω1> into column form:
(a1)
(a2)
(a3)

(that is my attempt at a column matrix).

I could do a similar thing for |ω2> to get coefficients b1=(1+i)/sqrt(3), b2=1/sqrt(6)=b3.

To calculate the probability of being in |ω2>, what I am tempted to do is now perform the operation |<ω2|ω1>|^2/<ω1|ω1>, which yields 5/9, if I calculated correctly. But I am not entirely confident in the strategy. I go back and forth with this, but I feel like the ω terms are the eigenvectors representing two different eigenvalues/states. But at the same time, I feel like I am treating ω1 as my initial state vector (like ψ(0)), and then projecting a specific state, ω2 in this case, onto ψ.

So, is the strategy seem like it is correct, or did I go fishing? Also, did anything I say in the previous paragraph resemble the reality of the situation?

Thank you!

 

[TSny]

Your calculation looks correct to me. I think 5/9 is the answer.

|ω1> and |ω2> are just two possible quantum states of the system. They do not need to be eigenstates of any particular operator. If the system is in state |ω1> then the probability of measuring the system to be in state |ω2> is just |<ω2|ω1>|² assuming normalized states. The "inner product" <ω2|ω1> is similar to finding the scalar product $\vec{a}\cdot\vec{b}$ of two ordinary vectors in 3D space. You can pick any orthonormal set of basis vectors $\hat{i}$, $\hat{j}$, $\hat{k}$, and express $\vec{a}$ and $\vec{b}$ as

$\vec{a} = a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$
$\vec{b} = b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$

and find $\vec{a}\cdot\vec{b} = a_1b_1+a_2b_2+a_3b_3$

Similarly, if we have any orthonormal basis |δ1>, |δ2> ,|δ3>,

|ω1> = a₁|δ1> + a₂|δ2> +a₃|δ3>
|ω2> = b₁|δ1> + b₂|δ2> +b₃|δ3>

and <ω2|ω1> = b₁^* a₁ + b₂^* a₂ + b₃^* a₃

 

[The Head]

Much clearer now...I appreciate you clearing that up for me!