Quantum mechanics: simultaneous eigenstates for operators

[Tailong]

Homework Statement

Suppose that a state |Ψ> is an eigenstate of operator B, with eigenvalue b_i.

Homework Equations

i. What is the expectation value of B?
ii. What is the uncertainty of B?
iii. Is |Ψi an eigenstate of B² or not?
iv. What is the uncertainty of B²?

part B
: Suppose, instead, |Ψ> is an eigenstate of operator B², with eigenvalue λ.
Without loss of generality, any state can be expanded in eigenstates of B: |Ψ> =
Σc_i|b_i>. (You can still assume that |bi> is eigenstates of operator B, so B|b_i> = b_i |b_i>)

v. What is the uncertainty for B² ?

vi. Starting from the expansion given above, write out B² |Ψ> in the B basis (byoperating with B twice in a row since you know that B²|Ψ> = B(B|Ψ>).

vii. Write out the eigenvalue equation for B²in this basis.
Taking the inner product of both sides of the equation with <bj|, you can
extract information about the cj coefficients and the bj eigenvalues, given your
knowledge of λ. Write this equation and explain in words what conclusions
you can draw about the results you might get if you measured B.

The Attempt at a Solution

By the expectation formula:

i.
<B> = <Ψ| B |Ψ> = bi <Ψ|Ψ> = bi
since for eigenstates, they are normalized, the inner product is 1.

ii.
For eigenstates, the uncertainty is simply 0.
Also we can do this mathematically as
<B²> = <Ψ| B B |Ψ> = b_i <Ψ| B |Ψ> = b_i²
ΔB = sqrt (<B₂> - <B>²) = 0

The real problem is the rest :

iii.
I can do nothing else but operate B twice on the state
B² |Ψ> = b_i ² |Ψ>
so I guess here b_i² is the corresponding eigenvalue for B² here?

I feel like the answer is wrong, since if it is, the answer for part iv is zero too.
That looks bad.
So, I tried to do this in another way like
B B |Ψ> = b_i B |Ψ>
so let B|Ψ> = is also an eigenstate for operator B
but |χ> = B|Ψ> = b_i|Ψ> is with same basis of |Ψ>.
So B² share a simultaneous eigenstate |Ψ> with B......
And then I get lost of my target.
Am I thinking too much ?

iv. I cannot solve this since I cannot solve part iii.v. Uncertainty for B² is zero since |Ψ> is an eigenstate for B².

vi.I tried to operate B on |Ψ>, so I get
B²|Ψ> = λ |Ψ>.
Then use the expansion:
|Ψ> =Σc_i|b_i>

B² |Ψ> = Σc_i B² |b_i>
= Σc_i b_i² |b_i>

vii.
<b_j| B² |Ψ> =<b_j| Σc_i b_i² |b_i>
=c_jb_j²
= <b_j| λ |Ψ> = λ<b_j |Ψ>
= λ c_j

Thus, λ = b_j².
I got this but cannot conclude anything from this.
What does this relate to the results from B operator ?

 

[BvU]

Hello, mr. T, and welcome to PF :)

i, ii, ii, iv: I can't do better than that either. B is just a complex number.
Now, in v. it becomes more interesting. There are two solutions for ##\lambda = b^2## ...