Okay, let's tackle this quantum mechanics problem step by step. So the question is about finding the expectation value of a certain operator for a spin-half particle in a given state. Hmm, I remember that expectation values are calculated using the inner product of the state with the operator acting on the state. Let me recall the formula: the expectation value ⟨O⟩ = ⟨ψ| O |ψ⟩. Right, so I need to compute this for the operator 10σ_z + 5σ_x, where σ_z and σ_x are the Pauli matrices.

First, let's parse the given state. The particle is in a linear superposition 0.5|↑⟩ + sqrt(3)/2|↓⟩. So the coefficients are 0.5 for spin-up and sqrt(3)/2 for spin-down. I should check if this is a normalized state. The norm should be |0.5|^2 + |sqrt(3)/2|^2 = (0.25) + (3/4) = 1. Yes, so it's already normalized. Good, no need to normalize it again.

Now, the operator in question is 10σ_z + 5σ_x. Since expectation value is linear, I can split this into 10⟨σ_z⟩ + 5⟨σ_x⟩. So I need to compute the expectation values of σ_z and σ_x separately and then combine them.

Starting with σ_z. The Pauli matrix σ_z is a diagonal matrix with entries [1, -1]. So in the basis |↑⟩ and |↓⟩, σ_z is [[1, 0], [0, -1]]. The expectation value ⟨σ_z⟩ for the given state is the probability of being in |↑⟩ times 1 plus the probability of being in |↓⟩ times (-1). The probabilities are |0.5|^2 = 0.25 and |sqrt(3)/2|^2 = 3/4. So ⟨σ_z⟩ = 0.25*(1) + (3/4)*(-1) = 0.25 - 0.75 = -0.5. Then 10 times that is 10*(-0.5) = -5.

Next, the σ_x term. The Pauli matrix σ_x is the off-diagonal matrix [[0, 1], [1, 0]]. The expectation value ⟨σ_x⟩ is calculated as ⟨ψ| σ_x |ψ⟩. For a state |ψ⟩ = a|↑⟩ + b|↓⟩, σ_x acting on |ψ⟩ would give a|↓⟩ + b|↑⟩. Then the inner product ⟨ψ| σ_x |ψ⟩ is the conjugate of a times the conjugate of b times 1 (from the off-diagonal terms). Wait, maybe I should compute it more carefully.

Alternatively, the expectation value ⟨σ_x⟩ can be found using the matrix representation. Let's write the state as a column vector. The state |ψ⟩ is [0.5; sqrt(3)/2]. Then σ_x is [[0, 1], [1, 0]]. So σ_x |ψ⟩ is [sqrt(3)/2; 0.5]. Then ⟨ψ| σ_x |ψ⟩ is the conjugate transpose of |ψ⟩ multiplied by σ_x |ψ⟩. Since all coefficients here are real, it's just the transpose. So the transpose of [0.5, sqrt(3)/2] times [sqrt(3)/2; 0.5] is 0.5*sqrt(3)/2 + (sqrt(3)/2)*0.5. Let's compute that: 0.5*sqrt(3)/2 = sqrt(3)/4, and the other term is the same. So total is sqrt(3)/4 + sqrt(3)/4 = sqrt(3)/2 ≈ 0.866. So ⟨σ_x⟩ is approximately 0.866. Then multiplying by 5 gives 5*(0.866) ≈ 4.33.

Now, adding the two contributions: -5 (from σ_z) + 4.33 (from σ_x) gives approximately -0.67. Wait, but looking at the options, the closest is C. -0.7. But let me check my calculations again to make sure I didn't make a mistake.

Wait, maybe I made a mistake in calculating ⟨σ_x⟩. Let me redo that part. The state is |ψ⟩ = 0.5|↑⟩ + sqrt(3)/2 |↓⟩. The matrix representation of σ_x is [[0,1],[1,0]]. So σ_x |ψ⟩ is 0.5|↓⟩ + sqrt(3)/2 |↑⟩. Therefore, ⟨σ_x⟩ = ⟨ψ| σ_x |ψ⟩ = (0.5⟨↑| + sqrt(3)/2 ⟨↓|) (0.5|↓⟩ + sqrt(3)/2 |↑⟩). Multiplying these out: 0.5* sqrt(3)/2 ⟨↑|↑⟩ + 0.5*(sqrt(3)/2) ⟨↑|↑⟩ + sqrt(3)/2 *0.5 ⟨↓|↓⟩ + sqrt(3)/2 * sqrt(3)/2 ⟨↓|↑⟩. Wait, maybe I should compute it term by term.

Wait, actually, when you multiply (a↑ + b↓) with (c↑ + d↓), you get a*c ⟨↑|↑⟩ + a*d ⟨↑|↓⟩ + b*c ⟨↓|↑⟩ + b*d ⟨↓|↓⟩. For Pauli matrices, since σ_x |↑⟩ = |↓⟩ and σ_x |↓⟩ = |↑⟩, the matrix elements are 0 except for the off-diagonal ones. So the expectation value is (0.5)(sqrt(3)/2) + (sqrt(3)/2)(0.5). Because ⟨↑|σ_x|↓⟩ = 1 and ⟨↓|σ_x|↑⟩ = 1. So each term is 0.5 * sqrt(3)/2 * 1 + sqrt(3)/2 * 0.5 * 1. Wait, but actually, when you compute the inner product, it's the sum of the products of the components. Let me think again.

Alternatively, the expectation value of σ_x for a spin state can also be found using the formula ⟨σ_x⟩ = 2 Re(ψ_↑ ψ_↓*). Wait, no, maybe not. Let me recall that for a state |ψ⟩ = α|↑⟩ + β|↓⟩, the expectation value ⟨σ_x⟩ is |α|^2 Re(σ_x) + cross terms. Wait, maybe another approach. The expectation value is given by ⟨ψ| σ_x |ψ⟩. Let's compute this explicitly.

The state |ψ⟩ is [0.5, sqrt(3)/2]^T. Then σ_x |ψ⟩ is [sqrt(3)/2, 0.5]^T. Then the inner product is (0.5)(sqrt(3)/2) + (sqrt(3)/2)(0.5) = 0.5*sqrt(3)/2 + sqrt(3)/2*0.5 = sqrt(3)/4 + sqrt(3)/4 = sqrt(3)/2 ≈ 0.866. So that's correct. So ⟨σ_x⟩ is sqrt(3)/2.

Therefore, 5 * sqrt(3)/2 ≈ 5 * 0.866 ≈ 4.33. Then the total expectation value is 10*(-0.5) + 5*(0.866) = -5 + 4.33 = -0.67. So approximately -0.67. The options are A.1.65, B.0.85, C.-0.7, D.-1.4. So the closest is C. -0.7. But wait, maybe I made a mistake in the calculation of ⟨σ_z⟩.

Wait, let's check ⟨σ_z⟩ again. The σ_z matrix is [[1,0],[0,-1]]. So ⟨σ_z⟩ = |0.5|^2 * 1 + |sqrt(3)/2|^2 * (-1) = 0.25*1 + 0.75*(-1) = 0.25 - 0.75 = -0.5. That's correct. So 10*(-0.5) = -5. Then 5*sqrt(3)/2 ≈ 5*0.866 ≈ 4.33. So total expectation value is -5 + 4.33 ≈ -0.67. So -0.67 rounds to -0.7, which is option C. But wait, let me check if the problem states "up to one decimal place". So -0.67 is approximately -0.7 when rounded to one decimal place. Therefore, the answer should be C.

Wait, but let me make sure I didn't make a mistake in the calculation. Let's recalculate step by step.

First, compute ⟨σ_z⟩: 0.5^2 *1 + (sqrt(3)/2)^2 * (-1) = 0.25 + (3/4)*(-1) = 0.25 - 0.75 = -0.5. Correct.

Then ⟨σ_x⟩: as computed, it's sqrt(3)/2 ≈ 0.866. Then 5 * sqrt(3)/2 = 5*(0.866) ≈ 4.33. So total is -5 +4.33 = -0.67. Rounded to one decimal place is -0.7. So option C.

Alternatively, maybe the problem expects an exact value. Let's compute exactly:

sqrt(3)/2 is approximately 0.8660254. 5*(sqrt(3)/2) = (5/2)*sqrt(3). So exact value is 10*(-1/2) + 5*(sqrt(3)/2) = -5 + (5/2)sqrt(3). Let's compute this exactly:

(5/2)sqrt(3) ≈ (2.5)(1.73205) ≈ 4.330125. So -5 +4.330125 ≈ -0.669875. Rounded to one decimal place is -0.7. So that's correct.

Therefore, the answer should be C. -0.7.
</think>