Okay, so I need to find the uncertainty in P_z for a given quantum state. The state is given as a column vector (-1/2, 1/√2, -1/2). The operators P_x, P_y, and P_z are represented by their respective matrices. The basis vectors are the eigenvectors of P_z. 

First, I remember that the uncertainty ΔP_z is the square root of the expectation value of P_z squared minus the square of the expectation value of P_z. So, ΔP_z = sqrt⟨P_z²⟩ - ⟨P_z⟩². 

Since the basis vectors are the eigenvectors of P_z, maybe working in that basis would simplify things. Let me check the matrix of P_z. The first row is (ℏ, 0, 0), the second row is (0, 0, 0), and the third row is (0, 0, -ℏ). So, this is a diagonal matrix with ℏ on the first diagonal element and -ℏ on the third. Therefore, the eigenvalues of P_z are ℏ and -ℏ. The eigenvectors corresponding to these eigenvalues are the basis vectors. 

Wait, but the state given is a column vector in the original basis. Hmm. Wait, the basis here might be the standard basis, but the eigenvectors of P_z are probably the basis vectors. Let me confirm. If P_z is diagonal with entries ℏ, 0, -ℏ, then the eigenvectors are e1, e2, e3 where e1 = (1,0,0), e2 = (0,1,0), e3 = (0,0,1). But the given state is (-1/2, 1/√2, -1/2). Let me check if this state can be expressed in terms of the eigenvectors of P_z. 

Alternatively, maybe the basis here is the standard basis, and the eigenvectors of P_z are e1, e2, e3. So, the state is given in this basis. If that's the case, then the expectation values can be computed directly using the matrix elements. 

But since the state is an eigenstate of P_x with eigenvalue -ℏ, maybe there's a connection there. Wait, the problem says the system is found in the eigenstate of P_x corresponding to eigenvalue -ℏ. The explicit form of the state is given as (-1/2, 1/√2, -1/2). So, perhaps I can verify if this state is indeed an eigenstate of P_x. Let me check. 

The matrix for P_x is given as:
Row 1: (0, ℏ/√2, 0)
Row 2: (ℏ/√2, 0, ℏ/√2)
Row 3: (0, ℏ/√2, 0)

So, to find P_x acting on the state vector (a, b, c), it's:

First component: 0*a + (ℏ/√2)*b + 0*c = (ℏ/√2)*b
Second component: (ℏ/√2)*a + 0*b + (ℏ/√2)*c
Third component: 0*a + (ℏ/√2)*b + 0*c = (ℏ/√2)*b

Given the state vector is (-1/2, 1/√2, -1/2). Let's compute P_x times this vector:

First component: (ℏ/√2)*(1/√2) = ℏ/2
Second component: (ℏ/√2)*(-1/2) + (ℏ/√2)*(-1/2) = (ℏ/√2)*(-1) = -ℏ/√2
Third component: (ℏ/√2)*(1/√2) = ℏ/2

So the resulting vector is (ℏ/2, -ℏ/√2, ℏ/2). For this to be an eigenstate with eigenvalue -ℏ, we should have P_x |ψ⟩ = -ℏ |ψ⟩. Let's check:

Left-hand side: (ℏ/2, -ℏ/√2, ℏ/2)
Right-hand side: -ℏ*(-1/2, 1/√2, -1/2) = (ℏ/2, -ℏ/√2, ℏ/2)

Yes, they match. So the state is indeed an eigenstate of P_x with eigenvalue -ℏ. That checks out. 

Now, back to calculating ΔP_z. Since the state is given, we can compute ⟨P_z⟩ and ⟨P_z²⟩. 

First, let's compute ⟨P_z⟩. Since P_z is a diagonal matrix, the expectation value is simply the sum of the diagonal elements multiplied by the squares of the components of the state. Wait, no. Let me recall that for a diagonal operator in the standard basis, the expectation value ⟨P_z⟩ is the sum over |c_i|^2 * eigenvalues. 

Given the state vector is (-1/2, 1/√2, -1/2). The components are the coefficients in the standard basis. The eigenvalues of P_z are ℏ, 0, -ℏ. So, the expectation value ⟨P_z⟩ is:

(-1/2)^2 * ℏ + (1/√2)^2 * 0 + (-1/2)^2 * (-ℏ) 

Calculating each term:

(1/4) * ℏ + (1/2) * 0 + (1/4) * (-ℏ) 

So that's (ℏ/4) - (ℏ/4) = 0. 

So ⟨P_z⟩ = 0. 

Now, ⟨P_z²⟩. Since P_z squared is a diagonal matrix with entries ℏ², 0, ℏ². So, the expectation value of P_z squared is the sum of |c_i|^2 * (eigenvalues of P_z squared). 

So, | -1/2 |^2 * ℏ² + |1/√2 |^2 * 0 + | -1/2 |^2 * ℏ² 

Calculating each term:

(1/4) * ℏ² + (1/2) * 0 + (1/4) * ℏ² 

Adding these up: (1/4 + 1/4) ℏ² = (1/2) ℏ² 

Therefore, ⟨P_z²⟩ = (1/2) ℏ² 

Then, the uncertainty ΔP_z is sqrt(⟨P_z²⟩ - ⟨P_z⟩²) = sqrt( (1/2 ℏ²) - 0 ) = sqrt(ℏ² / 2) = ℏ / sqrt(2) 

Wait, but looking at the options, option C is ℏ/√2, which is the same as this result. So the answer should be C. 

But wait, let me double-check. The state is given as a column vector. Let me confirm the calculation again. 

The state vector is ψ = (-1/2, 1/√2, -1/2). The inner product ⟨P_z⟩ is ψ† P_z ψ. Since P_z is diagonal, the multiplication is straightforward. 

Alternatively, since P_z is diagonal with entries ℏ, 0, -ℏ, then ⟨P_z⟩ = (-1/2)^2 * ℏ + (1/√2)^2 * 0 + (-1/2)^2 * (-ℏ) = (1/4)ℏ - (1/4)ℏ = 0. Correct. 

⟨P_z²⟩ is ψ† P_z² ψ. Since P_z² is diagonal with entries ℏ², 0, ℏ². So ⟨P_z²⟩ = (1/4)ℏ² + (1/2)*0 + (1/4)ℏ² = (1/2)ℏ². 

So ΔP_z = sqrt( (1/2)ℏ² ) = ℏ / sqrt(2). 

Therefore, the correct answer is C. 

But wait, the answer options are given as A. ℏ/2, B. √2 ℏ, C. ℏ/√2, D. ℏ. So yes, C is the correct option. 

Alternatively, maybe I made a mistake in assuming the basis is the standard basis. Let me check. The problem states that the basis vectors here are the eigenvectors of P_z. So if the basis is the standard basis, then P_z's eigenvectors are e1, e2, e3. But if the basis is different, perhaps the state is expressed in a different basis. Wait, the problem says "the basis vectors here are the eigenvectors of P_z". So that would be the standard basis, since P_z is diagonal in that basis. 

Therefore, the state is expressed in the standard basis, which is the eigenbasis of P_z. So the calculation as above holds. 

Alternatively, if the basis were not the standard basis, but the problem states that the basis vectors here are the eigenvectors of P_z, which are standard basis vectors. Therefore, the calculation is correct. 

Another way to think: Since the basis is the eigenbasis of P_z, the state is expressed in terms of these eigenstates. So the state ψ is a linear combination of the eigenstates of P_z. The coefficients are the amplitudes in the eigenbasis. 

So the state ψ can be written as a|+> + b|-> where |+> is the eigenstate with eigenvalue ℏ, and |-> is the eigenstate with eigenvalue -ℏ. 

Looking at the state vector (-1/2, 1/√2, -1/2), it's in the standard basis. So to express it in terms of the eigenstates of P_z (which are the standard basis vectors), the coefficients are just the components. 

But wait, the eigenstates of P_z are e1, e2, e3. So the state ψ is already in the eigenbasis. Therefore, the coefficients are the expansion in the eigenbasis. 

Therefore, the expectation value of P_z is the weighted sum of the eigenvalues with the squares of the coefficients. 

As calculated before, the expectation value is 0. 

Similarly, ⟨P_z²⟩ is the sum of the squares of the coefficients multiplied by the squares of the eigenvalues. 

Which gives (1/4)(ℏ²) + (1/2)(0) + (1/4)(ℏ²) = (1/2)ℏ². 

Therefore, the uncertainty is sqrt(1/2 ℏ²) = ℏ/sqrt(2). 

So the correct answer is C. 

I think this is correct. Let me check once more. 

Another approach: The state is given as (-1/2, 1/√2, -1/2). The P_z operator in the standard basis is diagonal with entries (ℏ, 0, -ℏ). 

So to compute ⟨P_z⟩, it's the sum over |c_i|² * eigenvalue_i. 

Components: 

First component: (-1/2)^2 = 1/4, multiplied by ℏ: 1/4 ℏ 

Second component: (1/√2)^2 = 1/2, multiplied by 0: 0 

Third component: (-1/2)^2 = 1/4, multiplied by -ℏ: -1/4 ℏ 

Total: 1/4 ℏ - 1/4 ℏ = 0. Correct. 

⟨P_z²⟩ is sum |c_i|² * (eigenvalue_i)^2 

First component: 1/4 * ℏ² 

Second component: 1/2 * 0 

Third component: 1/4 * ℏ² 

Total: (1/4 + 1/4) ℏ² = 1/2 ℏ² 

Thus, ΔP_z = sqrt(1/2 ℏ²) = ℏ/sqrt(2). 

Yes, that's correct. So the answer is C. 

I think this is solid. I don't see any mistakes here.
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