Okay, so I've got this quantum mechanics problem here. Let me try to figure it out step by step. Hmm, the state of the system is given by a column matrix with elements (-1, 2, 1). Then there are two observables, P and Q, represented by their respective matrices. The question is about the probability of measuring Q as 0 after measuring P as -1. The options are A to D, with probabilities 2/3, 1/2, 1/6, and 1/3. 

First, I remember that in quantum mechanics, when you measure an observable, the possible outcomes are the eigenvalues of the corresponding operator (matrix). The state of the system after measurement collapses into an eigenvector of the operator, and the probability is the square of the amplitude of that eigenvector in the original state.

So the process here is: measure P first, which gives an eigenvalue of -1, then measure Q. The probability of getting 0 for P and then -1 for Q would involve two steps. Wait, but the problem says "if someone measures Q just after the measurement of P". So the order is P first, then Q. So first, we measure P, get -1, then measure Q. We need the probability of getting -1 for Q after that.

But wait, the question is a bit ambiguous. Is it asking for the probability of getting P=-1 and then Q=-1? Or the probability of getting P=-1 and then Q=0? Wait, the question says "the probability of getting 0 for P and -1 for Q in the respective measurements." Wait, no, let me check again. The wording is: "the probability of getting 0 for P and -1 for Q in the respective measurements." Wait, that's confusing. Wait, the state is given as (-1, 2, 1). Wait, but the measurement of P would give eigenvalues. Let me check the matrix for P.

The matrix P is given as:

First row: 0, 1/√2, 0

Second row: 1/√2, 0, 1/√2

Third row: 0, 1/√2, 0

So it's a 3x3 matrix. Let me write it out:

[0, 1/√2, 0]

[1/√2, 0, 1/√2]

[0, 1/√2, 0]

Hmm. Now, to find the eigenvalues of P. Since the state is given as a vector (-1, 2, 1), but wait, the state vector is in the Hilbert space. Wait, but the state is given as a column vector with elements (-1, 2, 1). Wait, but the standard basis vectors in quantum mechanics are orthonormal, so the state vector must be normalized. Wait, let me check the norm of this vector. The norm squared is (-1)^2 + 2^2 + 1^2 = 1 + 4 + 1 = 6. So the norm is sqrt(6). So the state vector is (1/sqrt(6)) * (-1, 2, 1). Wait, but maybe the problem states the state as a column matrix with elements (-1, 2, 1), so perhaps it's not normalized. But in quantum mechanics, state vectors must be normalized. Hmm, but maybe the problem just gives the vector as it is, and we have to proceed with that. Maybe the normalization is handled elsewhere. Or perhaps the problem is using a different convention. Let's proceed.

First, let's find the eigenvalues of P. Because when we measure P, the possible outcomes are the eigenvalues. Then, the probability of getting a particular eigenvalue is the squared amplitude of the corresponding eigenvector in the state vector. Then, after measuring P, the state collapses to the corresponding eigenvector, and then measuring Q would be based on the new state.

So first step: find eigenvalues of P.

Let me compute the characteristic equation det(P - λI) = 0.

Matrix P - λI:

[-λ, 1/√2, 0]

[1/√2, -λ, 1/√2]

[0, 1/√2, -λ]

Compute determinant:

-λ * [(-λ)(-λ) - (1/√2)(1/√2)] - (1/√2) * [ (1/√2)(-λ) - (1/√2)(0) ] + 0 * something.

So determinant = -λ [λ² - (1/2)] - (1/√2) [ (-λ)/√2 - 0 ].

Simplify:

= -λ³ + (λ)/2 + (1/√2)(λ/√2)

= -λ³ + (λ)/2 + (λ)/2

= -λ³ + λ

So characteristic equation: -λ³ + λ = 0 → λ(-λ² + 1) = 0 → λ = 0, λ = 1, λ = -1.

So eigenvalues of P are 1, -1, 0. Wait, so the possible measurement outcomes for P are 1, -1, 0.

But the question mentions measuring P and getting -1. So we need to find the eigenvectors corresponding to eigenvalue -1 for P.

So let's find eigenvectors for P when λ = -1.

We solve (P - (-1)I) v = 0.

Matrix P + I:

[1, 1/√2, 0]

[1/√2, 1, 1/√2]

[0, 1/√2, 1]

We need to find the null space of this matrix.

Let me write the equations:

1*v1 + (1/√2)*v2 + 0*v3 = 0

(1/√2)*v1 + 1*v2 + (1/√2)*v3 = 0

0*v1 + (1/√2)*v2 + 1*v3 = 0

From the third equation: (1/√2)v2 + v3 = 0 → v3 = - (1/√2) v2.

From the first equation: v1 + (1/√2)v2 = 0 → v1 = - (1/√2) v2.

Substitute into the second equation:

(1/√2)*(-1/√2 v2) + v2 + (1/√2)*v3 = 0

Compute each term:

(1/√2)*(-1/√2 v2) = - (1/2) v2

v2 remains as is.

(1/√2)*v3 = (1/√2)*(-1/√2 v2) = - (1/2) v2

So sum:

- (1/2) v2 + v2 - (1/2) v2 = (-1/2 + 1 - 1/2) v2 = (0) v2 = 0.

So the second equation is automatically satisfied. So the solution is v1 = - (1/√2) v2, v3 = - (1/√2) v2. So the eigenvectors are scalar multiples of ( -1/√2, 1, -1/√2 ). Let's check:

Let me take v2 = √2, then v1 = -1, v3 = -1. So the eigenvector is (-1, √2, -1). Let's verify if this satisfies the equations.

First equation: 1*(-1) + (1/√2)(√2) + 0*(-1) = -1 + 1 + 0 = 0. Correct.

Second equation: (1/√2)(-1) + 1*(√2) + (1/√2)(-1) = (-1/√2) + √2 - (1/√2) = (-2/√2) + √2 = (-√2) + √2 = 0. Correct.

Third equation: (1/√2)(√2) + 1*(-1) = 1 -1 = 0. Correct. So the eigenvector is (-1, √2, -1). But in our state vector, we have (-1, 2, 1). Let's see if this state is aligned with the eigenvector.

Wait, the state vector given is (-1, 2, 1). Let's compute the inner product between this state and the eigenvector (-1, √2, -1).

Inner product = (-1)(-1) + 2*(√2) + 1*(-1) = 1 + 2√2 -1 = 2√2. The norm squared of the state vector is (-1)^2 + 2^2 + 1^2 = 1 +4 +1=6. The norm of the eigenvector is sqrt( (-1)^2 + (√2)^2 + (-1)^2 ) = sqrt(1 +2 +1) = sqrt(4)=2. So the probability amplitude is (2√2) / (sqrt(6)*2) )= (2√2)/(2√6) )= √2/√6 = 1/√3. Wait, because the inner product is 2√2, and the norm of the state is sqrt(6), and the norm of the eigenvector is 2. So the probability is |(2√2)|^2 / (6 * 4) )? Wait, no. Wait, the probability is |<v|ψ>|^2, where |ψ> is the state vector, and |v> is the eigenvector. So |<v|ψ>|^2 = |( (-1)(-1) + √2 *2 + (-1)*1 ) / (norm of v * norm of ψ)|^2. Wait, but the inner product is (1 + 2√2 -1) = 2√2. The norm of v is 2, norm of ψ is sqrt(6). So the probability is (2√2)^2 / (2^2 * (√6)^2) )= (8) / (4 *6) )= 8/24 = 1/3. Wait, that's 1/3. So the probability of measuring P=-1 is 1/3. Wait, but the question asks for the probability of getting 0 for P and -1 for Q. Wait, maybe I got confused.

Wait, the question is: "the probability of getting 0 for P and -1 for Q in the respective measurements?" Wait, but P's eigenvalues are 1, -1, 0. So getting 0 for P. Then after that, measuring Q, which has eigenvalues... Let's check Q's matrix.

The matrix Q is:

First row: 1, 0, 0

Second row: 0, 0, 0

Third row: 0, 0, -1

So the eigenvalues are 1, 0, -1. So possible outcomes for Q are 1, 0, -1.

But the question is about getting 0 for P and -1 for Q. Wait, but if we first measure P and get 0, then the state collapses to the eigenvector of P corresponding to 0. Then, we measure Q. So we need the probability of getting -1 for Q in that case.

But wait, the initial state is (-1, 2, 1). So first, we measure P. The possible outcomes are 1, -1, 0. The probability of each is determined by the projection of the state onto the respective eigenvectors. Then, if we get 0, the state collapses to the eigenvector for P=0, and then we measure Q. The probability of getting -1 for Q in that case is the squared amplitude of the component of the new state in the eigenvector of Q corresponding to -1.

But the question is asking for the probability of getting 0 for P and then -1 for Q. Wait, but the wording is a bit ambiguous. Wait, the question says: "the probability of getting 0 for P and -1 for Q in the respective measurements." So does that mean P=0 and Q=-1? Or does it mean the measurements are 0 for P and -1 for Q? The wording is a bit unclear, but I think it's P=0 and then Q=-1. So the joint probability of first measuring P=0 and then Q=-1.

But wait, in quantum mechanics, when you measure P first, the state collapses to an eigenstate of P, and then you measure Q. So the joint probability is the probability of P=0 times the probability of Q=-1 given that the state is the P=0 eigenstate. But the question is phrased as "probability of getting 0 for P and -1 for Q in the respective measurements". So it's the probability that P is 0 and then Q is -1. So it's the product of the two probabilities: probability of P=0 times the conditional probability of Q=-1 given P=0.

But first, let's confirm the steps:

1. Measure P: possible outcomes 1, -1, 0. Probability of each is calculated by projecting the state onto the respective eigenvectors.

2. If outcome is 0, then the state collapses to the eigenvector of P=0. Then, measure Q. The probability of getting -1 for Q is the squared amplitude of the component of the state vector in the eigenvector of Q corresponding to -1.

So the total probability is P(P=0) * P(Q=-1 | P=0).

So first, compute P(P=0):

We need the probability of P=0, which is |<v0|ψ>|², where |v0> is the eigenvector of P with eigenvalue 0. Wait, earlier when we found the eigenvector for P=0, we had (-1, √2, -1). Let's confirm that again.

Wait, when λ = 0, the matrix P - 0I = P. So the equations are:

0*v1 + (1/√2)v2 + 0*v3 = 0 → (1/√2)v2 = 0 → v2 = 0.

(1/√2)v1 + 0*v2 + (1/√2)v3 = 0 → (1/√2)(v1 + v3) = 0 → v1 + v3 = 0.

0*v1 + (1/√2)v2 + 0*v3 = 0 → same as first equation, v2=0.

So v2=0, and v1 = -v3. So the eigenvectors are of the form (v1, 0, -v1). Let's take v1=1, then v3=-1. So the eigenvector is (1, 0, -1). Wait, earlier I thought it was (-1, √2, -1), but that was for λ=-1. Wait, no, for λ=0, the eigenvector is (1, 0, -1). Let me check if this satisfies P*v = 0*v.

First component: (0*1) + (1/√2*0) + (0*(-1)) = 0. Correct.

Second component: (1/√2*1) + (0*0) + (1/√2*(-1)) = (1/√2 -1/√2) = 0. Correct.

Third component: (0*1) + (1/√2*0) + (0*(-1)) = 0. Correct. So the eigenvector for λ=0 is (1, 0, -1). Let's check the normalization. The norm squared is 1² +0² + (-1)² = 2. So the normalized eigenvector is (1/√2, 0, -1/√2).

So the state vector is (-1, 2, 1). Let's compute the inner product with the eigenvector (1, 0, -1). The inner product is (-1)(1) + 2*0 +1*(-1) = -1 +0 -1 = -2. The norm of the state vector is sqrt(6), and the norm of the eigenvector is sqrt(2). So the probability amplitude is -2 / (sqrt(6)*sqrt(2)) )= -2 / sqrt(12) = -2/(2*sqrt(3)) )= -1/sqrt(3). The probability is (1/sqrt(3))² = 1/3.

Wait, so the probability of measuring P=0 is 1/3. Then, the state collapses to the eigenvector (1, 0, -1) normalized. Wait, the eigenvector (1, 0, -1) has norm sqrt(2), so the normalized eigenvector is (1/√2, 0, -1/√2).

Now, after measuring P=0, the state is (1/√2, 0, -1/√2). Now, we need to measure Q. The operator Q has eigenvalues 1, 0, -1. The matrix Q is diagonal, so its eigenvalues are on the diagonal. The eigenvectors are the standard basis vectors. So the eigenvectors for Q=1 is (1,0,0), for Q=0 is (0,1,0), and for Q=-1 is (0,0,1).

So the state after P=0 is (1/√2, 0, -1/√2). Let's compute the inner product with each eigenvector of Q.

For Q=1: (1/√2)(1) + 0*0 + (-1/√2)(0) = 1/√2. The probability is |1/√2|² = 1/2.

For Q=0: (1/√2)(0) + 0*1 + (-1/√2)(0) = 0. Probability is 0.

For Q=-1: (1/√2)(0) + 0*0 + (-1/√2)(1) = -1/√2. The probability is | -1/√2 |² = 1/2.

Wait, so if we measure Q after P=0, the probabilities are 1/2 for Q=1, 0 for Q=0, and 1/2 for Q=-1. So the probability of getting Q=-1 is 1/2.

Therefore, the total probability of P=0 and then Q=-1 is P(P=0) * P(Q=-1|P=0) = (1/3) * (1/2) = 1/6. But wait, that's not one of the options. The options are 2/3, 1/2, 1/6, 1/3. Oh, 1/6 is option C. But wait, let me check again.

Wait, the state after P=0 is (1/√2, 0, -1/√2). When measuring Q, the possible outcomes are 1, 0, -1 with probabilities 1/2, 0, 1/2. So the probability of Q=-1 is 1/2. So the joint probability is 1/3 * 1/2 = 1/6. So the answer should be C. But let me check again.

Wait, but maybe I made a mistake in the eigenvectors. Let me verify the eigenvectors again.

For P=0, solving (P - 0I)v = 0. The matrix is:

[0, 1/√2, 0]

[1/√2, 0, 1/√2]

[0, 1/√2, 0]

So the equations are:

(1/√2)v2 = 0 → v2 = 0

(1/√2)v1 + (1/√2)v3 = 0 → v1 + v3 = 0

(1/√2)v2 = 0 → v2 =0

So v2=0, v1 = -v3. So eigenvectors are of the form (v1, 0, -v1). So the eigenvector is (1, 0, -1), which has norm sqrt(2). So normalized, it's (1/√2, 0, -1/√2). Correct.

The state vector is (-1, 2, 1). The inner product with (1, 0, -1) is (-1)(1) + 2*0 +1*(-1) = -1 -1 = -2. The norm of the state is sqrt(6), so the probability amplitude is -2 / sqrt(6). The probability is (2/sqrt(6))² = 4/6 = 2/3. Wait, wait a second! Wait, I think I made a mistake here. Because the inner product is -2, but the probability is |<v|ψ>|². The state vector is (-1, 2, 1). The eigenvector is (1, 0, -1). So the inner product is (-1)(1) + 2*0 +1*(-1) = -1 +0 -1 = -2. The norm of the state is sqrt{ (-1)^2 + 2^2 +1^2 } = sqrt(1+4+1) = sqrt(6). The norm of the eigenvector is sqrt(1^2 +0 + (-1)^2 ) = sqrt(2). So the probability is | -2 / (sqrt(6) * sqrt(2)) |² = | -2 / sqrt(12) |² = (4 / 12) = 1/3. Wait, that's the same as before. So P(P=0) is 1/3.

Then, after measuring P=0, the state is (1/√2, 0, -1/√2). Then, measuring Q. The eigenvectors for Q are the standard basis vectors. So the state is (1/√2, 0, -1/√2). The inner product with (0,0,1) is -1/√2. The probability is | -1/√2 |² = 1/2. So the joint probability is 1/3 * 1/2 = 1/6. So the answer should be C. But let me check the options again. The options are A:2/3, B:1/2, C:1/6, D:1/3. So C is 1/6. But wait, maybe I made a mistake in the eigenvectors for Q. Let's check the matrix Q.

Matrix Q is:

First row: 1,0,0

Second row:0,0,0

Third row:0,0,-1

So the eigenvalues are 1, 0, -1. The eigenvectors are:

For Q=1: any vector (a, 0, 0), so (1,0,0) is an eigenvector.

For Q=0: any vector (0, b, 0), so (0,1,0).

For Q=-1: any vector (0,0,c), so (0,0,1).

So when measuring Q, the state (1/√2, 0, -1/√2) is projected onto these eigenvectors. The component in (0,0,1) is -1/√2, so the probability is 1/2. So yes, that's correct.

Therefore, the total probability is 1/3 * 1/2 = 1/6. So the answer is C. But wait, the options are A:2/3, B:1/2, C:1/6, D:1/3. So C is 1/6. But let me check again the steps to make sure I didn't make any miscalculations.

First, the state is (-1, 2, 1). The norm is sqrt(6). The eigenvector for P=0 is (1,0,-1), norm sqrt(2). The inner product is (-1)(1) +2*0 +1*(-1) = -2. So the probability is | -2 |^2 / (6 * 2) )= 4 / 12 = 1/3. Correct.

After that, the state is (1,0,-1) normalized to (1/√2, 0, -1/√2). Then, measuring Q. The eigenvectors for Q are (1,0,0), (0,1,0), (0,0,1). The state is (1/√2, 0, -1/√2). The component in (0,0,1) is -1/√2. The squared magnitude is 1/2. So the probability is 1/2. So the total is 1/3 * 1/2 = 1/6. So answer is C. But wait, the options given are A:2/3, B:1/2, C:1/6, D:1/3. So C is 1/6. But let me check if there's another interpretation.

Wait, the question says "the probability of getting 0 for P and -1 for Q in the respective measurements." Wait, perhaps the user is asking for the probability of getting 0 for P and then -1 for Q, not the joint probability. But in quantum mechanics, that's the same as the joint probability. Because you measure P first, collapse the state, then measure Q. So the joint probability is the product of the two probabilities.

Alternatively, maybe the question is asking for the probability of getting 0 for P and -1 for Q in a single measurement. But that's not possible because you can only measure one observable at a time. So the question must mean the sequence: first measure P and get 0, then measure Q and get -1. So the answer is 1/6, which is option C.

But wait, let me check the initial eigenvectors again. For P=0, the eigenvector is (1,0,-1). For Q=-1, the eigenvector is (0,0,1). But when we measure Q after P=0, the state is (1/√2, 0, -1/√2). The projection onto (0,0,1) is the third component, which is -1/√2. So the probability is | -1/√2 |² = 1/2. So the joint probability is 1/3 * 1/2 = 1/6. So the answer should be C, which is 1/6.

But wait, another way to think: maybe the question is asking for the probability that P=0 and Q=-1 in the same measurement. But that's not possible. Alternatively, maybe there's a misunderstanding in the problem statement. Let me re-examine the problem statement.

The question says: "the probability of getting 0 for P and -1 for Q in the respective measurements?" The term "respectively" might imply that the measurements are done in some order, but the problem states "if someone measures Q just after the measurement of P". So the order is P first, then Q. Therefore, the joint probability is P(P=0) * P(Q=-1 | P=0). Which is 1/3 * 1/2 = 1/6. So the answer is C. But in the options, C is 1/6. So the correct answer should be C.

But wait, maybe I made a mistake in the eigenvector for P=0. Let me double-check. The matrix P is:

[0, 1/√2, 0]

[1/√2, 0, 1/√2]

[0, 1/√2, 0]

For λ=0, the equations are:

Row 1: (0)v1 + (1/√2)v2 + 0*v3 = 0 → v2=0

Row 2: (1/√2)v1 + 0*v2 + (1/√2)v3 = 0 → v1 + v3 = 0

Row 3: 0*v1 + (1/√2)v2 + 0*v3 = 0 → v2=0

So v2=0, and v1 = -v3. So eigenvectors are (v1, 0, -v1). So normalized eigenvector is (1/√2, 0, -1/√2). Correct.

The inner product with the state (-1, 2, 1) is (-1)(1/√2) + 2*0 + 1*(-1/√2) = (-1/√2) + 0 -1/√2 = -2/√2 = -√2. The norm of the state is sqrt(6), so the probability is | -√2 |² / (6 * 2) )= 2 / 12 = 1/6. Wait, wait! Wait a minute. Wait, that's different from what I did before. Wait, no. Wait, the inner product is (-1)(1/√2) + 2*0 +1*(-1/√2) = (-1/√2 -1/√2) = -2/√2 = -√2. The norm of the state is sqrt(6), and the norm of the eigenvector is sqrt(2). So the probability is |<v|ψ>|² / (||v||² ||ψ||²) )= (| -√2 |² ) / (2 * 6) )= (2) / 12 = 1/6. Wait, but earlier I thought the inner product was -2, leading to 1/3. Wait, what's the correct inner product?

Wait, the state is (-1, 2, 1). The eigenvector is (1, 0, -1). The inner product is (-1)(1) + 2*0 + 1*(-1) = -1 +0 -1 = -2. The norm of the state is sqrt(6), the norm of the eigenvector is sqrt(2). So the probability is | -2 |² / (6 * 2) )= 4 / 12 = 1/3. But wait, if the eigenvector is (1/√2, 0, -1/√2), then the inner product is (-1)(1/√2) + 2*0 +1*(-1/√2) = (-1/√2 -1/√2) = -2/√2 = -√2. The norm of the state is sqrt(6), the norm of the eigenvector is sqrt( (1/√2)^2 + (-1/√2)^2 ) = sqrt(1/2 +1/2) = 1. So the probability is | -√2 |² / (6 * 1) )= 2 /6 = 1/3. So that's correct. So the probability of P=0 is 1/3. Then, after that, the state is (1/√2, 0, -1/√2). Then, measuring Q, the probability of getting -1 is 1/2. So total probability is 1/3 * 1/2 = 1/6. So the answer is C. But the options are A:2/3, B:1/2, C:1/6, D:1/3.

But wait, the problem says "the probability of getting 0 for P and -1 for Q in the respective measurements." Maybe the user is asking for the probability of getting 0 for P and then 0 for Q, but that's not what's written. The wording is a bit ambiguous. But according to the problem statement, it's about getting 0 for P and -1 for Q. So the answer should be C. But wait, maybe there's a mistake in the calculation. Let me check again.

Alternatively, perhaps the eigenvector for Q=-1 is (0,0,1). The state after P=0 is (1/√2, 0, -1/√2). The inner product with (0,0,1) is -1/√2. The probability is | -1/√2 |² = 1/2. So the joint probability is 1/3 * 1/2 = 1/6. So yes, the answer is C. Therefore, the correct answer is C. But the options are A:2/3, B:1/2, C:1/6, D:1/3. So the answer is C.
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