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## 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 B

^{2}or not?

iv. What is the uncertainty of B

^{2}?

part B

: Suppose, instead, |Ψ> is an eigenstate of operator B

^{2}, 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 B

^{2}?

vi. Starting from the expansion given above, write out B

^{2}|Ψ> in the B basis (byoperating with B twice in a row since you know that B

^{2}|Ψ> = B(B|Ψ>).

vii. Write out the eigenvalue equation for B

^{2}in 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

<B

^{2}> = <Ψ| B B |Ψ> = b

_{i}<Ψ| B |Ψ> = b

_{i}

^{2}

ΔB = sqrt (<B

_{2}> - <B>

^{2}) = 0

**The real problem is the rest :**

iii.

I can do nothing else but operate B twice on the state

B

^{2}|Ψ> = b

_{i}

^{2}|Ψ>

so I guess here b

_{i}

^{2}is the corresponding eigenvalue for B

^{2}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 B

^{2}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 B

^{2}is zero since |Ψ> is an eigenstate for B

^{2}.

vi.I tried to operate B on |Ψ>, so I get

B

^{2}|Ψ> = λ |Ψ>.

Then use the expansion:

|Ψ> =Σc

_{i}|b

_{i}>

B

^{2}|Ψ> = Σc

_{i}B

^{2}|b

_{i}>

= Σc

_{i}b

_{i}

^{2}|b

_{i}>

vii.

<b

_{j}| B

^{2}|Ψ> =<b

_{j}| Σc

_{i}b

_{i}

^{2}|b

_{i}>

=c

_{j}b

_{j}

^{2}

= <b

_{j}| λ |Ψ> = λ<b

_{j}|Ψ>

= λ c

_{j}

**Thus, λ = b**

I got this but cannot conclude anything from this.

What does this relate to the results from B operator ?

_{j}^{2}.I got this but cannot conclude anything from this.

What does this relate to the results from B operator ?