Okay, let's tackle this quantum mechanics problem step by step. Hmm, the question is about finding the probabilities of measuring an electron in certain eigenstates and the average value of an operator. Let me start by understanding what's given.

First, the state |α> is given as proportional to (1+i)|up> + (2-i)|down>. Since |up> and |down> are eigenstates of the z-spin operator, they form an orthonormal basis. So the first thing I need to do is normalize this state. Normalization means finding the coefficient A such that the sum of the squares of the absolute values of the coefficients equals 1.

Let me compute the squared magnitudes. The coefficients are (1+i) and (2-i). The magnitude squared of (1+i) is |1+i|² = (1)² + (1)² = 2. Similarly, |2 - i|² is (2)² + (-1)² = 4 + 1 = 5. So the total squared magnitude is 2 + 5 = 7. Therefore, the normalization factor A is 1/sqrt(7). So the normalized state is |α> = [ (1+i)/sqrt(7) ] |up> + [ (2 - i)/sqrt(7) ] |down>.

Next, the problem asks for the probability of measuring the particle in each of the eigenstates of the operator with matrix representation A, where A_ij = ħ/2 if i ≠ j, and 0 otherwise. Wait, that matrix must be a 2x2 matrix since it's for spin states. Let me write it out. If it's 2x2, then the diagonal elements are 0, and the off-diagonal elements are ħ/2. So the matrix A is:

[ 0      ħ/2  ]
[ ħ/2    0    ]

This operator is actually the Pauli X matrix (spin flip operator) multiplied by ħ/2. Wait, the Pauli X matrix is [[0,1],[1,0]], so multiplying by ħ/2 would make it [[0, ħ/2], [ħ/2, 0]]. So yes, that's the operator.

The eigenstates of this operator correspond to the eigenvectors of the Pauli X matrix. The eigenvalues of Pauli X are ±1, so here they would be ±ħ/2. The eigenvectors for Pauli X are (1,1) and (1,-1), up to normalization. So the eigenstates are (|up> + |down>)/sqrt(2) and (|up> - |down>)/sqrt(2). So the eigenstates of the operator A are these two states.

Now, to find the probabilities of measuring the system in these eigenstates, we need to compute the squared magnitudes of the projections of |α> onto each eigenstate.

First eigenstate: (|up> + |down>)/sqrt(2). Let's write |α> in terms of |up> and |down>:

|α> = (1+i)/sqrt(7) |up> + (2 - i)/sqrt(7) |down>

Projection onto the first eigenstate:

< eigenstate | α > = [ (1/sqrt(2)) conjugate(1) + (1/sqrt(2)) conjugate(1) ] Wait, no. Let me correct that. The eigenstate is (|up> + |down>)/sqrt(2). The inner product is:

(1/sqrt(2)) [ <up| + <down| ] |α>

Which is (1/sqrt(2)) [ (1+i)/sqrt(7) + (2 - i)/sqrt(7) ]

= (1/sqrt(2)) [ (1+i + 2 -i)/sqrt(7) ]

The i terms cancel out: (1 + 2)/sqrt(7) = 3/sqrt(7)

So the inner product is 3/(sqrt(2)*sqrt(7)) = 3/sqrt(14)

The probability is the square of the magnitude: (3/sqrt(14))² = 9/14 ≈ 0.6429.

Now the second eigenstate: (|up> - |down>)/sqrt(2). Similarly, compute the inner product:

(1/sqrt(2)) [ <up| - <down| ] |α>

= (1/sqrt(2)) [ (1+i)/sqrt(7) - (2 - i)/sqrt(7) ]

= (1/sqrt(2)) [ (1+i -2 +i)/sqrt(7) ]

= (1/sqrt(2)) [ (-1 + 2i)/sqrt(7) ]

The magnitude squared is [ (-1 + 2i) / (sqrt(2)*sqrt(7)) ] times its conjugate.

Wait, the magnitude squared is | (-1 + 2i)/sqrt(14) |² = ( (-1)^2 + (2)^2 ) / 14 = (1 +4)/14 = 5/14 ≈ 0.3571.

So the probabilities are approximately 0.6429 and 0.3571. Looking at the options, let's check the probabilities first. The options are:

A. 0.54, 0.46

B. 0.64, 0.36

C. 0.28, 0.72

D. 0.61, 0.29

So 0.64 and 0.36 (option B) matches the first two probabilities. Wait, but wait, the probabilities sum to 1, so 0.64 + 0.36 = 1, which works. So option B has 0.64 and 0.36, which matches our calculation. But let me verify again.

Wait, the first projection gave 9/14 ≈ 0.6429, which is approximately 0.64, and the second is 5/14 ≈ 0.3571 ≈ 0.36. So option B's probabilities are 0.64 and 0.36. So that's the first part done.

Now the second part: the average value of the operator. The average is the expectation value <A> = <α| A |α>.

Since the operator A is diagonal in the eigenbasis of the Pauli X matrix, but actually, no, the operator A is the Pauli X scaled by ħ/2. The expectation value can be found by summing the eigenvalues multiplied by their probabilities. Since the eigenvalues are ±ħ/2, and the probabilities are 9/14 and 5/14, the average is:

(9/14)*(ħ/2) + (5/14)*(-ħ/2) = [9 -5]/14 * ħ/2 = 4/14 * ħ/2 = (2/7)*(ħ/2) = ħ/7.

So the average value is ħ/7. Looking at the options, option B says "hbar / 7", which matches. So option B is the correct answer.

Wait, but let me check the average calculation again. The eigenvalues are ħ/2 and -ħ/2. The probabilities are 9/14 and 5/14. So (9/14)*(ħ/2) + (5/14)*(-ħ/2) = (9 -5)/14 * ħ/2 = 4/14 * ħ/2 = 2/7 * ħ/2 = ħ/7. Yes, that's correct.

Therefore, the probabilities are 0.64 and 0.36, and the average is ħ/7. So the correct answer is option B.
</think>