Okay, let's try to figure out this quantum mechanics problem. So, the question is about two electrons in p orbitals, and we need to find the probability that both have Lz eigenvalues of -ħ when measured. The coupled state given is |1,1,2,-1>. Hmm, let me start by recalling some concepts.

First, the electrons are in p orbitals, which means their individual orbital angular momentum quantum numbers l1 and l2 are both 1. The total orbital angular momentum is l = 2, and the z-component is m = -1. Since we're ignoring spin angular momentum, we're only dealing with orbital angular momentum here.

The problem asks for the joint probability of measuring L1z and L2z both as -ħ. That means we need to find the probability that both electrons have their z-components of orbital angular momentum equal to -ħ.

I remember that when dealing with coupled angular momenta, the state |l1, l2, l, m> can be expressed as a linear combination of the uncoupled states |l1, m1> |l2, m2>, where m = m1 + m2. So, maybe we need to expand this coupled state into the product basis of individual Lz eigenstates.

The given state is |1,1,2,-1>. Let's think about how to express this state in terms of the product states |m1, m2>. The total m is -1, so m1 + m2 = -1. The possible m1 and m2 values are integers from -1 to 1 since l1 = l2 = 1.

So, possible pairs (m1, m2) that sum to -1 are:
- (-1, 0)
- (0, -1)
- (1, -2) → Wait, m2 can't be -2 because l2=1, so m2 ranges from -1 to 1. So this is invalid.
Similarly, ( -2, 1) is invalid. So the valid pairs are (-1,0) and (0,-1).

Therefore, the state |1,1,2,-1> can be written as a combination of |1, -1>|1, 0> and |1, 0>|1, -1>. Let me check the Clebsch-Gordan coefficients here. The general formula for |l1, l2, l, m> is a sum over m1 and m2 such that m1 + m2 = m of CG coefficients times |l1, m1>|l2, m2>.

For l1 = l2 = 1 and l = 2, the possible m values range from -2 to 2. For m = -1, we need m1 and m2 such that m1 + m2 = -1. The possible pairs are ( -1, 0 ) and (0, -1 ). The Clebsch-Gordan coefficient for each term would be CG(1,1,2,-1 | 1, -1; 1, 0) and CG(1,1,2,-1 | 0, -1; 1, 1). Wait, no, actually, the Clebsch-Gordan coefficients are determined by the possible ways to add the angular momenta. Let me recall the CG coefficients for l1 = l2 = 1 and l = 2.

The state |1,1,2,-1> would be a combination of |1,-1>|1,0> and |1,0>|1,-1>. The coefficients can be found using the CG tables. For adding two angular momenta of 1 each to get 2, the state |m1 + m2 = -1> would be a combination of |m1= -1, m2=0> and |m1=0, m2=-1>.

Looking up the CG coefficients, for l1 = l2 = 1, l = 2, m = -1. The CG coefficients are:

CG(1,1,2,-1 | -1,0) = sqrt( ( (1 + 1 + 2)(1 + 1 - 2)(1 - 1 + 2)( -1 + 1 + 2) ) / ( (2 + 1)(2 + 1)(1 + 1 + 2)(1 + 1 + 2) )) ) Hmm, maybe a better approach is to remember the specific coefficients.

Alternatively, for l1 = l2 = 1 and l = 2, the state |2, -1> can be written as a combination of |1, -1>|1, 0> and |1, 0>|1, -1>. The coefficients are sqrt(1/2) each, but with a specific sign. Let me think: when you combine two spin-1 particles to get spin 2, the m = -1 state is |m1=-1, m2=0> + |m1=0, m2=-1>, each multiplied by sqrt(1/2) or maybe another factor.

Wait, another way: the state |l1, l2, l=2, m=-1> should be orthogonal to the states with higher m. Let's consider the possible combinations.

The state |1,1,2,-1> can be written as a linear combination of |1,-1>|1,0> and |1,0>|1,-1>. The coefficients are symmetric because the state is symmetric under exchange of particles, right? Because the total state is symmetric if the orbital part is symmetric. Wait, but wait, electrons are fermions, so the total wavefunction must be antisymmetric. But here, we're only considering orbital angular momentum and ignoring spin, so the total wavefunction is the product of spatial and spin parts. But since spin is ignored here, the orbital part must be symmetric if the spin part is symmetric, but since we're ignoring spin, maybe that's not applicable. Wait, the question says "ignoring there spin angular momentum", so maybe the spatial part is all that's considered. But electrons are fermions, so the total wavefunction (spatial × spin) must be antisymmetric. However, since spin is ignored here, maybe the spatial wavefunction is symmetric. Wait, but the problem doesn't mention spin at all. The question is about the orbital angular momentum components, so perhaps we can treat the spatial part as a state, regardless of the spin, but since we're ignoring spin, the combined state is just the spatial part.

Wait, perhaps the problem is only about the orbital angular momentum, so the state |1,1,2,-1> is the coupled state of the two electrons' orbital angular momenta. The question is about measuring L1z and L2z. So, the question is not about the total spin, but just the orbital parts. So, the electrons are in a state where their total orbital angular momentum is 2ħ (since l=2) and the z-component is -ħ (m=-1). The question is, when we measure each electron's Lz, what's the probability that both are -ħ.

So, the state |1,1,2,-1> is a combination of the product states where m1 + m2 = -1. The possible product states are:

|1,-1>|1,0> and |1,0>|1,-1>

Each with some coefficients. Let's find those coefficients.

The Clebsch-Gordan coefficients for coupling two angular momenta of 1 to get total angular momentum 2 and m = -1 are:

The states are |1,1,2,-1> = sqrt(1/2) (|1,-1>|1,0> + |1,0>|1,-1> )

Wait, let me check. For l1 = l2 = 1, l = 2, m = -1.

The Clebsch-Gordan coefficient for |m1, m2> contributing to |2, -1> is:

For m1 = -1 and m2 = 0, the coefficient is sqrt( ( (1 + 1 + 2)(1 + 1 - 2)(1 - 1 + 2)(-1 +1 + 2) ) / ( (2 + 1)(2 + 1)(1 + 1 + 2)(1 + 1 + 2) )) ) Hmm, that's complicated. Alternatively, from the properties of CG coefficients, when adding two angular momenta of 1, the m = -1 state would be a combination of |m1=-1, m2=0> and |m1=0, m2=-1>, each with coefficients sqrt(1/2). Let me confirm.

Yes, for example, the state |1,1,2,-1> can be written as 1/sqrt(2) [ |1,-1>|1,0> + |1,0>|1,-1> ]

So, the amplitude for each of these product states is 1/sqrt(2). Therefore, the state is an equal superposition of these two product states.

Now, the question is the probability of measuring L1z = -ħ and L2z = -ħ. That would correspond to the product state |1,-1>|1,-1>. But in our coupled state, the product states involved are |1,-1>|1,0> and |1,0>|1,-1>. There's no term where both m1 and m2 are -1. Because m1 + m2 = -1, so if both were -1, their sum would be -2, not -1. Therefore, the coefficient for |1,-1>|1,-1> in the coupled state is zero. Hence, the probability of finding both L1z and L2z to be -ħ is zero.

Wait, but let me make sure. The coupled state is |1,1,2,-1> = (1/sqrt(2)) [ |1,-1>|1,0> + |1,0>|1,-1> ]

So, when we expand this, there's no term where both m1 and m2 are -1. Therefore, when we measure L1z and L2z, the possible outcomes are:

Either L1z = -ħ and L2z = 0, or L1z = 0 and L2z = -ħ. There's no overlap where both are -ħ. Therefore, the probability is zero. So the answer should be B. 0.

But wait, let me think again. Suppose I have a state that is an equal superposition of |1,-1>|1,0> and |1,0>|1,-1>. If I measure L1z and L2z, the possible outcomes are either (-ħ, 0) or (0, -ħ). Since the state is a combination of these two, the probability of each outcome is |1/sqrt(2)|^2 = 1/2 each. But the question is about both being -ħ, which is not possible here. So the probability is zero. Therefore, the correct answer is B. 0.
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