Okay, let's try to figure out this quantum mechanics problem about hydrogen atom transitions. The question is about the decay of the |3,0,0⟩ state to |1,0,0⟩ via two dipole transitions. The options are four different routes with different probabilities. Hmm, I need to recall how dipole transitions work in hydrogen atoms.

First, I remember that dipole transitions are governed by the selection rules. The selection rules for hydrogen states |n, l, m⟩ when emitting or absorbing a photon are Δl = ±1 and Δm = 0, ±1. So, the change in the angular momentum quantum number l must be plus or minus 1, and the change in m can be -1, 0, or +1. Also, the parity of the state changes because dipole transitions change the parity. The parity of a state is given by (-1)^l. So, when a dipole transition happens, the parity flips. For example, if the initial state has even parity (l even), the final state after a dipole transition must have odd parity (l odd), and vice versa.

Looking at the initial state |3,0,0⟩. Here, l is 0, so the parity is (-1)^0 = +1. The final state is |1,0,0⟩, which has l=0 again, so parity is +1. Wait, but a dipole transition changes the parity. So if you go from |3,0,0⟩ (parity +1) to an intermediate state via a dipole transition, that intermediate state must have parity -1 (since dipole transitions flip parity). Then, from the intermediate state to |1,0,0⟩, another dipole transition would flip the parity again, ending up with parity +1. But |1,0,0⟩ is parity +1, so this makes sense. So the transitions must go through a state with parity -1.

But wait, the intermediate state after the first dipole transition must have l=1 (since Δl=±1, so from l=0, Δl=+1 gives l=1). So the intermediate state must be either |2,1,1⟩, |2,1,-1⟩, or |2,1,0⟩. Because m can be -1, 0, or +1 when l=1.

Now, the possible first transitions from |3,0,0⟩ would be to |2,1, m⟩ states. Then from there to |1,0,0⟩. But wait, the final state is |1,0,0⟩, which has l=0. So the second transition must be from l=1 to l=0. But according to the selection rules, Δl=±1. So from l=1, Δl=-1 would take us to l=0. That's allowed. So the second transition is from |2,1,m⟩ to |1,0,0⟩.

But wait, the intermediate state after the first transition is |2,1,m⟩. Then the second transition would have Δl=-1, so l=1 to l=0, which is correct. But what about m? For the second transition, Δm can be -1, 0, or +1. The final state has m=0. So the intermediate state's m must be such that m + Δm = 0. Since Δm can be -1, 0, or +1, the intermediate m could be 1 (Δm=-1), 0 (Δm=0), or -1 (Δm=1). But the final state is m=0. So the intermediate state's m must be 1, 0, or -1, but the second transition's m change must result in 0. For example, if the intermediate state is |2,1,1⟩, then the second transition could have Δm=-1 to reach m=0. Similarly, if the intermediate state is |2,1,-1⟩, then Δm=+1 to reach m=0. If the intermediate state is |2,1,0⟩, then Δm=0 to reach m=0. So all three possibilities are allowed.

But wait, the problem states that the decay is via two dipole transitions. So the state |3,0,0⟩ decays to an intermediate state via a dipole transition, then the intermediate state decays to |1,0,0⟩ via another dipole transition. The question is about the transition route and the probability.

But the possible routes given in the options are:

A. |3,0,0⟩ → |2,1,1⟩ → |1,0,0⟩ and 1/4

B. |3,0,0⟩ → |2,1,-1⟩ → |1,0,0⟩ and 1/4

C. |3,0,0⟩ → |2,1,0⟩ → |1,0,0⟩ and 1/3

D. |3,0,0⟩ → |2,1,0⟩ → |1,0,0⟩ and 2/3

Wait, but the options don't include all possible m values. So the question is probably considering that the intermediate state must have m=0, but that's not necessarily the case. Wait, no, the initial state |3,0,0⟩ has m=0. The intermediate state can have m=1, -1, or 0. Then the second transition can take m to 0. So all three possibilities are allowed. But the probabilities depend on the possible transitions.

Wait, but the problem states that the decay is via two dipole transitions. So the first transition is from |3,0,0⟩ to an intermediate state (which must have l=1 and m=±1 or 0), and then from there to |1,0,0⟩. However, the problem states that the transition is via two dipole transitions. So the question is about the possible paths and their probabilities.

But the problem asks for the correct answer to the question: the state |3,0,0⟩ decays to |1,0,0⟩ via two dipole transitions. The transition route and the corresponding probability. So which of the options gives the correct route and the correct probability?

Wait, but the options only have specific m values for the intermediate state. For example, option A is |2,1,1⟩, option B is |2,1,-1⟩, option C is |2,1,0⟩, and option D is |2,1,0⟩. So each option is a different intermediate state. So which intermediate state is correct, and what's the probability?

But according to the selection rules, from |3,0,0⟩, the first transition can be to any of |2,1,1⟩, |2,1,-1⟩, or |2,1,0⟩. Then from each of those, the second transition can be to |1,0,0⟩. But the probability of each path would depend on the possible m transitions.

Wait, but in dipole transitions, the probability is determined by the square of the matrix element of the dipole operator. The dipole transition probability depends on the overlap between the initial and final states. For hydrogen, the dipole matrix elements between |n l m⟩ and |n' l' m'⟩ are non-zero only if certain conditions are met, like the selection rules. But the exact probability depends on the matrix element, which depends on the spherical harmonics.

The dipole operator is proportional to r times the spherical harmonics Y_{1}^{m'}, where m' can be -1, 0, +1. The transition amplitude is proportional to the integral of Y_{n l m}^* (r Y_{1}^{m'}) Y_{n' l' m'} dΩ. Which simplifies using the properties of spherical harmonics. The integral is non-zero only when m = m' + m'' (the m of the dipole operator is m', which is -1, 0, +1). Wait, the dipole operator has components with m'= -1, 0, +1. So the transition from |3,0,0⟩ (m=0) to intermediate state |2,1,m_i⟩, then to |1,0,0⟩ (m=0). The first transition must have m_i such that when the second transition occurs (which can change m by -1, 0, +1), the final m is 0. So m_i must be such that m_i + Δm = 0. So Δm can be -1, 0, +1. Therefore, m_i can be -1, 0, or +1.

But the initial state has m=0. So the first transition can take m from 0 to m_i (Δm = m_i - 0 = m_i). Then the second transition takes m_i to 0. So the second transition must have Δm = -m_i. Since the second transition's Δm can be -1, 0, +1, this requires that -m_i must be in that range. Which it is, because m_i can be -1, 0, +1. So for example, if m_i is +1, then Δm = -1. If m_i is 0, Δm = 0. If m_i is -1, Δm = +1. All are allowed.

But the probability of each transition depends on the matrix elements. For the first transition, the matrix element is proportional to the integral of Y_{3,0,0}^* Y_{2,1,m_i} times Y_{1}^{m'}, where m' is the change in m. Wait, maybe I'm complicating this. Let me think again.

The initial state is |3,0,0⟩. The first transition is to |2,1,m_i⟩. The second transition is from |2,1,m_i⟩ to |1,0,0⟩. The probability of each transition is determined by the square of the matrix element of the dipole operator between the initial and intermediate states, and then between the intermediate and final states.

But the total probability for the path would be the product of the probabilities of each individual transition. However, since the dipole operator is a vector (has components in different m directions), the probabilities will depend on the possible m transitions.

Alternatively, perhaps the selection rules and symmetry can help. For the first transition, from |3,0,0⟩ to |2,1,m_i⟩. The dipole operator can cause a change in m by ±1 or 0. But since the initial m is 0, the possible m_i after the first transition are m_i = 0 ±1, 0. So m_i can be -1, 0, +1. But the parity of the intermediate state must flip. Since the initial state has parity +1 (l=0), the intermediate state must have parity -1 (l=1), which it does. Then the second transition must go to |1,0,0⟩, which has parity +1. So the second transition must flip the parity again, which it does because l goes from 1 to 0 (Δl = -1).

But the probability of each path depends on the possible m transitions. Let's consider the matrix elements. The dipole operator is proportional to r times the spherical harmonics Y_{1}^{m'}, where m' can be -1, 0, +1. The transition amplitude from |n l m⟩ to |n' l' m'⟩ is proportional to the integral of Y_{n l m}^* Y_{1}^{m'} Y_{n' l' m'} dΩ. For the first transition, n=3, l=0, m=0; n'=2, l'=1, m_i. The integral becomes ∫ Y_{3,0,0}^* Y_{1}^{m'} Y_{2,1,m_i} dΩ. Similarly, for the second transition, n'=2, l'=1, m_i; n''=1, l''=0, m''=0. The integral is ∫ Y_{2,1,m_i}^* Y_{1}^{m''} Y_{1,0,0} dΩ.

But this is getting complicated. Maybe there's a simpler way. In hydrogen, the dipole transition probabilities can be calculated using the selection rules and the matrix elements. The key point is that transitions with Δm=±1 and Δm=0 are allowed, and their probabilities depend on the square of the matrix elements.

But perhaps the problem is more about the possible intermediate states and the possible probabilities. The possible intermediate states are |2,1,1⟩, |2,1,-1⟩, and |2,1,0⟩. However, the question is about the transition route. But the options only include these three possibilities. So which intermediate state is correct?

Wait, but the problem states that the decay is via two dipole transitions. So the initial state must decay to an intermediate state via a dipole transition, then the intermediate state decays to the final state via another dipole transition. The possible intermediate states are |2,1,1⟩, |2,1,-1⟩, |2,1,0⟩. The problem is asking which of these is the correct route and what is the probability.

But the possible m values for the intermediate state are 1, -1, or 0. The question is, which of these is possible? But all three are possible, but the problem gives specific options. Wait, the options don't include all three. For example, option A is |2,1,1⟩, option B is |2,1,-1⟩, option C and D are |2,1,0⟩. So the question is, which of these is the correct intermediate state?

But in reality, all three intermediate states are possible, but the probability of each path depends on the matrix elements. However, the problem states "the correct answer to this question", implying that there's only one correct path. So perhaps there's a selection based on symmetry or other factors.

Alternatively, perhaps the total probability is the sum over all possible intermediate states. But the options give specific routes, not sums. So perhaps the problem is considering the possible paths and their probabilities. Wait, but the options are about the route and the probability. So each option gives a specific route (the intermediate state) and the probability.

But the possible routes are three: intermediate states |2,1,1⟩, |2,1,-1⟩, |2,1,0⟩. However, the problem's options only include two of them (A and B) and two others (C and D). Wait, no. Let me check the options again.

The options are:

A. |3,0,0⟩→|2,1,1⟩→|1,0,0⟩ and 1/4

B. |3,0,0⟩→|2,1,-1⟩→|1,0,0⟩ and 1/4

C. |3,0,0⟩→|2,1,0⟩→|1,0,0⟩ and 1/3

D. |3,0,0⟩→|2,1,0⟩→|1,0,0⟩ and 2/3

Wait, options A and B have different intermediate states (m=1 and m=-1), and options C and D have m=0 with different probabilities (1/3 and 2/3). So which of these is correct?

But the problem is about the transition via two dipole transitions. So the initial state is |3,0,0⟩. The first transition must be to |2,1,m_i⟩, then to |1,0,0⟩. The intermediate state must have l=1. The possible m_i are 1, -1, 0. However, the probability of each transition depends on the matrix element.

But perhaps the probabilities are determined by the possible m transitions. For example, when the initial state has m=0, the first transition can take m_i to -1, 0, or +1. Then the second transition must take m_i back to 0. The probability of each path is the product of the probabilities of each transition.

But the dipole operator has components in different m directions. The probability of transitioning from m=0 to m_i is proportional to |<2,1,m_i|Y_1^0|3,0,0>|^2, assuming the dipole operator has a component in the m=0 direction. Wait, no. The dipole operator has components in different m directions. For example, the dipole operator in the x-direction has m=0, but the y-direction has m=±1. Wait, maybe I need to think in terms of spherical harmonics.

The dipole operator is proportional to r times the spherical harmonics Y_{1}^{m'}, where m' can be -1, 0, +1. The matrix element for the first transition from |3,0,0⟩ to |2,1,m_i⟩ is proportional to the integral of Y_{3,0,0}^* Y_{1}^{m'} Y_{2,1,m_i} dΩ. But since Y_{3,0,0} is a real function (spherical harmonic with m=0), and Y_{1}^{m'} has m'=0, ±1, this integral will be non-zero only if m_i = m' (because of the selection rules for angular momentum coupling). Wait, actually, when you multiply Y_{1}^{m'} and Y_{2,1,m_i}, the product can have terms with m = m' + m_i. But since the initial state has m=0, the final m after the first transition is m_i. So the integral will be non-zero only if m' = m_i. Wait, no, because the integral is over all angles. Let me think again.

The matrix element for the transition is:

⟨2,1,m_i | r Y_{1}^{m'} | 3,0,0⟩

= r ∫ Y_{2,1,m_i}^* Y_{1}^{m'} Y_{3,0,0} dΩ

But Y_{3,0,0} is proportional to cosθ, which is Y_{1,0}^0. Wait, no, Y_{l,m} are for each l and m. Y_{1,0}^0 is proportional to cosθ. Wait, Y_{1,0}^0 is actually proportional to cosθ. But Y_{3,0,0} is proportional to cos^3θ. Wait, no, Y_{3,0}^0 is proportional to cos^3θ.

Wait, maybe I need to recall the explicit forms. The spherical harmonics Y_{l,m} are given by:

Y_{l,m}(θ,φ) = (-1)^m \sqrt{(2l+1)/(4π (l-m)!))} P_l^m(cosθ) e^{i m φ}

So Y_{3,0}^0 is proportional to P_3^0(cosθ) = (3/8)(5cos²θ - 1), but wait, no. Wait, P_3^0(x) is (5x³ - 3x)/2. So Y_{3,0}(θ,φ) = N (5cos³θ - 3cosθ), where N is a normalization constant. Similarly, Y_{1,0}(θ,φ) = sqrt(3/(4π)) cosθ.

So the integral becomes:

∫ Y_{2,1,m_i}^* (r) Y_{1}^{m'} (r) Y_{3,0,0} (r) r² sinθ dθ dφ

Wait, no. The dipole operator is r times Y_{1}^{m'}, so the matrix element involves integrating Y_{2,1,m_i}^* Y_{1}^{m'} Y_{3,0,0} dΩ. But Y_{3,0,0} is Y_{3,0}^0, which is different from Y_{1}^{m'}. Let's compute this integral.

But this is getting complicated. Maybe there's a selection rule. The product Y_{1}^{m'} Y_{3,0,0} can be expressed as a sum of spherical harmonics. The integral of Y_{2,1,m_i}^* times that sum will be non-zero only if Y_{2,1,m_i} is present in the expansion. So the integral will be non-zero only if Y_{2,1,m_i} is part of the product Y_{1}^{m'} Y_{3,0,0}.

But Y_{3,0,0} is Y_{3,0}^0, which is a single spherical harmonic. The product Y_{1}^{m'} Y_{3,0}^0 can be expanded into a sum of spherical harmonics with l = 3 ± 1 = 2 or 4. Since we have l=2 in the intermediate state, the product must contain Y_{2, m_i}^* term. So the integral is non-zero only if the product Y_{1}^{m'} Y_{3,0}^0 contains a Y_{2, m_i}^* term. So m_i must be equal to m' because the product Y_{1}^{m'} Y_{3,0}^0 will have terms with m = m' + 0 (since Y_{3,0}^0 has m=0). Therefore, m_i = m'.

But m' can be -1, 0, +1. So m_i must be equal to m'. Therefore, the possible m_i are -1, 0, +1, but with the condition that m_i = m'. Therefore, the possible m_i are -1, 0, +1, but m' must match m_i. Wait, but m' is the m component of the dipole operator. The dipole operator has components in different m directions, so for each transition, m' can be -1, 0, +1. But for the integral to be non-zero, m_i must equal m'. Therefore, the only possible m_i are those where m_i = m'. So the possible m_i are -1, 0, +1, but the dipole operator's m' must match m_i. Therefore, the possible transitions are from m=0 to m_i = m' = -1, 0, +1. Wait, but the initial state is m=0. The first transition changes m from 0 to m_i. Then the second transition changes m from m_i to 0 (since final state is m=0). Therefore, the second transition must have Δm = -m_i. The dipole operator can have m' = -m_i. So m' = -m_i. Therefore, the dipole operator's m component must be -m_i. But the dipole operator's m' can be -1, 0, +1. Therefore, for each m_i, we need to check if -m_i is in the allowed set {-1, 0, +1}.

For m_i = +1: Δm = -1. The dipole operator must have m' = -1. Which is allowed.

For m_i = 0: Δm = 0. The dipole operator must have m' = 0. Allowed.

For m_i = -1: Δm = +1. The dipole operator must have m' = +1. Allowed.

Therefore, all three m_i values are possible, as long as the dipole operator's m' is -m_i. Therefore, the matrix elements are non-zero for all three possible m_i. Therefore, all three intermediate states are possible. But the problem's options only give two of them (A and B have m_i=1 and -1, C and D have m_i=0). Therefore, the problem must be considering only the possible m_i and their probabilities.

But the problem states "the correct answer to this question", implying that only one of the options is correct. Therefore, perhaps there's a mistake in the problem, or perhaps I'm missing some consideration.

Alternatively, perhaps the problem is considering that the first transition must have Δm = ±1 or 0, but also the second transition must have Δm = ∓1 or 0. For example, if the first transition has m_i = +1, then the second transition must have Δm = -1. The dipole operator must have m' = -1. Similarly, if the first transition has m_i = -1, the second must have Δm = +1, so m' = +1. If the first transition has m_i = 0, then the second transition can have Δm = 0 or ±1, but since the final state is m=0, Δm must be 0. Therefore, the dipole operator must have m' = 0.

Therefore, the possible paths are:

1. First transition: m_i = +1 (m'= -1), second transition: m_i = +1 → m=0 (m'= -1)

2. First transition: m_i = -1 (m'= +1), second transition: m_i = -1 → m=0 (m'= +1)

3. First transition: m_i = 0 (m'=0), second transition: m_i =0 → m=0 (m'=0)

Therefore, all three paths are possible. However, the problem's options only include two of them (A and B have m_i=1 and -1, C and D have m_i=0). So the problem is asking which of these is the correct route and the probability. But the problem states "the correct answer", implying only one option is correct. Therefore, perhaps the problem is considering that only the m_i=0 path is possible, or that the transition through m_i=1 or -1 is suppressed for some reason.

Alternatively, perhaps the probability is determined by the square of the matrix elements. Let's compute the probabilities for each possible path.

The total probability would be the product of the probabilities for each transition. The probability for each transition is proportional to the square of the matrix element. Let's compute the matrix elements.

The initial state is |3,0,0⟩, which has m=0. The first transition is to |2,1,m_i⟩, which has m=m_i. The dipole operator has components in m'=-1, 0, +1. The transition matrix element is proportional to the integral of Y_{3,0}^* Y_{2,1,m_i} Y_{1}^{m'} dΩ. But since the initial state has m=0 and the dipole operator has m'=m'', the integral will be non-zero only if m_i = m''. Therefore, for each m_i, the dipole operator's m' must be equal to m_i. Wait, no. The dipole operator has components in different m directions, so the integral will be non-zero only if m_i = m''. Therefore, for each m_i, the dipole operator's m' must be equal to m_i. Therefore, the possible transitions are:

- From m=0 to m_i=+1 via m'=+1

- From m=0 to m_i=0 via m'=0

- From m=0 to m_i=-1 via m'=-1

But the second transition must then have m'' = -m_i to go back to m=0. The dipole operator's m' must be -m_i. Therefore, the second transition's m' is -m_i. Therefore, the dipole operator's m' must be m'= -m_i for the second transition. Therefore, the matrix elements for each path are:

Path 1: m_i=+1, m'= -1

Path 2: m_i=-1, m'= +1

Path 3: m_i=0, m'=0

Each of these paths has a matrix element proportional to the integral of Y_{3,0} Y_{2,1,+1} Y_{1}^{-1} dΩ, and similarly for the others.

But calculating these integrals exactly would be time-consuming. However, there's a selection rule for the dipole operator. The dipole operator has angular momentum l=1. The transition from l=0 to l=1 is allowed. Then from l=1 to l=0 is also allowed. The matrix element for the first transition (|3,0,0⟩ → |2,1,m_i⟩) is proportional to the integral of Y_{3,0} Y_{1}^{m'} Y_{2,1,m_i} dΩ. For this integral to be non-zero, the product of Y_{1}^{m'} and Y_{2,1,m_i} must include Y_{3,0}.

But this is getting too complicated. Maybe there's a symmetry consideration. The initial state is spherically symmetric (m=0), so the transition probabilities should depend only on the m_i and the selection rules. But I'm not sure.

Alternatively, perhaps the problem is assuming that the first transition must have m_i=0, because the initial state has m=0, and the dipole operator's m' must be 0. But that's not necessarily true. The dipole operator can have m' = ±1 as well. The initial state is m=0, so the first transition can go to m_i=±1 as long as the dipole operator's m' is -m_i. For example, if m_i=+1, then m'= -1. If m_i=-1, then m'= +1. If m_i=0, then m'=0.

But the problem's options only include m_i=+1, -1, 0. So the possible probabilities are the sum over all possible paths. But the options give specific paths and probabilities. Therefore, perhaps each option is a possible path, and the probability is determined by the square of the matrix element for that path.

Assuming that the matrix elements for the first transition (from |3,0,0⟩ to |2,1,m_i⟩) are the same for all m_i, but that's not true because the spherical harmonics are different. However, perhaps the probabilities are the same for all three m_i, but that's unlikely.

Alternatively, perhaps the transition probabilities depend on the m_i. For example, the transition |3,0,0⟩ → |2,1,0⟩ → |1,0,0⟩ would have a probability that's different from the transitions to m_i=±1. But without exact calculations, it's hard to say.

Alternatively, perhaps the problem is considering that the transition |3,0,0⟩ → |2,1,0⟩ → |1,0,0⟩ is the only allowed path due to some selection rules I'm missing. For example, maybe the parity of the intermediate state must be the same as the final state or something. But parity of |2,1,0⟩ is (-1)^2 = +1, which is the same as the final state |1,0,0⟩. So parity is preserved. Wait, but dipole transitions flip parity. So from |3,0,0⟩ (parity +1) to |2,1,0⟩ (parity -1), then to |1,0,0⟩ (parity +1). So parity is flipped twice, which is allowed. So all three paths are possible.

But the problem's options only include two of the m_i possibilities. So perhaps the problem is considering that the intermediate state must have m_i=0. But why?

Alternatively, perhaps the problem is considering the selection rules for the second transition. For example, the second transition is from |2,1,m_i⟩ to |1,0,0⟩. The dipole operator has m' = 0 (since the final state has m=0). Therefore, the second transition must have m'' = 0. Therefore, the dipole operator's m' must be 0. Therefore, the first transition must have m_i = 0, because the first transition's m' = -m_i (from the second transition's m' = 0). Therefore, m_i = 0. Therefore, the only possible intermediate state is |2,1,0⟩. Therefore, the correct answer is either C or D.

Wait, this seems promising. Let's think again. The second transition must have m' = 0 (since the final state has m=0). Therefore, the second transition's m' is 0. The dipole operator's m' must be 0. Therefore, the first transition's m_i must be m_i = -m' = 0. Because in the second transition, the change in m is Δm = -m_i (since the first transition's m_i is m_i, and the second transition's m' is -m_i to reach m=0). Therefore, m_i = -m' = 0. Therefore, the only possible intermediate state is |2,1,0⟩. Therefore, the correct route is |3,0,0⟩ → |2,1,0⟩ → |1,0,0⟩. Then the probability depends on the matrix elements for these two transitions.

So if that's the case, then the correct answer must be either C or D. Now, the probability is either 1/3 or 2/3. How to determine which one.

The probability is the product of the probabilities of each transition. The first transition is from |3,0,0⟩ to |2,1,0⟩. The second transition is from |2,1,0⟩ to |1,0,0⟩.

The transition probability for each step is proportional to the square of the matrix element. Let's denote the matrix element for the first transition as M1, and for the second as M2. Then the total probability is |M1|^2 * |M2|^2.

But what are these matrix elements?

The first transition matrix element is ⟨2,1,0 | r Y_{1}^0 | 3,0,0⟩. The second transition is ⟨1,0,0 | r Y_{1}^0 | 2,1,0⟩.

But Y_{1}^0 is proportional to cosθ, which is Y_{1,0}^0. The integral for the first transition is ∫ Y_{3,0}^* Y_{1,0} Y_{2,1,0} dΩ. Similarly for the second transition.

But calculating these integrals is time-consuming. However, perhaps there's a selection rule that simplifies this. The initial state |3,0,0⟩ has l=3, m=0. The dipole operator has l=1, m'=0. The intermediate state |2,1,0⟩ has l=1, m=0. The final state |1,0,0⟩ has l=1, m=0.

But wait, the initial state |3,0,0⟩ is l=3, m=0. The dipole operator is l=1, m'=0. The product of the dipole operator and the initial state's wavefunction must be decomposed into spherical harmonics. The integral for the first transition is ∫ Y_{3,0} Y_{1,0} Y_{2,1,0} dΩ. The product Y_{3,0} Y_{1,0} can be expanded into a sum of Y_{l,m} terms. The possible l values are 3±1=4 or 2. Therefore, the product Y_{3,0} Y_{1,0} = a Y_{4,0} + b Y_{2,0}. Therefore, the integral ∫ Y_{3,0} Y_{1,0} Y_{2,1,0} dΩ = ∫ (a Y_{4,0} + b Y_{2,0}) Y_{2,1,0} dΩ. But the integral of Y_{4,0} Y_{2,1,0} is zero (different l and m). The integral of Y_{2,0} Y_{2,1,0} is also zero (different m). Therefore, the integral is zero. Wait, that can't be. Wait, Y_{2,0} and Y_{2,1,0} are orthogonal, so their product integral is zero. Therefore, the matrix element M1 is zero. That can't be right. There must be a mistake here.

Wait, maybe I made a mistake in the expansion. The product Y_{3,0} Y_{1,0} can be expressed as a sum of spherical harmonics. The general formula for the product of two spherical harmonics is:

Y_{l1,m1} Y_{l2,m2} = Σ_{l,m} [ (2l1+1)(2l2+1)/(4π(l1+l2+m)! ... ) ) ] Maybe it's better to recall that the product of Y_{l1,m1} and Y_{l2,m2} can be written as a sum of Y_{l,m} with l ranging from |l1 - l2| to l1 + l2, and m = m1 + m2. Therefore, the product Y_{3,0} Y_{1,0} would be a sum of Y_{l,0} where l ranges from 2 to 4.

Therefore, Y_{3,0} Y_{1,0} = a Y_{4,0} + b Y_{2,0}. Then the integral for the first transition is ∫ Y_{3,0} Y_{1,0} Y_{2,1,0} dΩ = ∫ (a Y_{4,0} + b Y_{2,0}) Y_{2,1,0} dΩ. But Y_{4,0} and Y_{2,1,0} are orthogonal (different l and m), so the first term is zero. The second term is ∫ Y_{2,0} Y_{2,1,0} dΩ. But Y_{2,0} and Y_{2,1,0} have different m (0 vs 1), so their integral is zero. Therefore, the matrix element M1 is zero. Therefore, the first transition from |3,0,0⟩ to |2,1,0⟩ is not allowed. But that contradicts the problem statement, which says the decay is via two dipole transitions. Therefore, there must be an error in this reasoning.

Alternatively, perhaps the initial assumption that the intermediate state must have m_i=0 is incorrect. Let's reconsider. The second transition's m' must be 0 (since the final state has m=0). Therefore, the first transition's m_i must be equal to -m' = 0. Therefore, the intermediate state must have m_i=0. Therefore, the only possible intermediate state is |2,1,0⟩. But according to the previous calculation, the matrix element M1 is zero. Therefore, the transition |3,0,0⟩ → |2,1,0⟩ is not allowed. Therefore, the problem's premise is incorrect, or the options are wrong.

But the problem states that the decay is via two dipole transitions, so the first transition must be allowed. Therefore, perhaps there's a mistake in the earlier reasoning. Let's think again.

The second transition is from |2,1,m_i⟩ to |1,0,0⟩ with m' = 0. Therefore, the first transition's m_i must be equal to -m' = 0. Therefore, the first transition is from |3,0,0⟩ → |2,1,0⟩. But the matrix element for this transition is zero, as we saw earlier. Therefore, this path is not allowed. Therefore, there must be another way. Therefore, the previous conclusion that the intermediate state must have m_i=0 is invalid. Therefore, perhaps the problem is considering that the intermediate state can have any m_i, but the matrix elements are non-zero.

But then, how to determine the probabilities. Since the problem gives specific options, perhaps the correct answer is option C or D, with probabilities 1/3 and 2/3. But which one.

Alternatively, perhaps the transition probabilities are determined by the square of the Clebsch-Gordan coefficients for the respective transitions. For the first transition, from |3,0,0⟩ to |2,1,m_i⟩, the Clebsch-Gordan coefficient would determine the matrix element. Similarly, for the second transition.

But without exact calculations, it's difficult. However, perhaps the transition probabilities are determined by the possible paths. For example, the first transition can go to |2,1,1⟩, |2,1,0⟩, or |2,1,-1⟩, each with certain probabilities. The second transition must then go back to m=0. The total probability would be the sum over all possible m_i of the product of probabilities for each transition.

But the problem's options only consider m_i=1, -1, 0. However, without knowing the exact probabilities, it's hard to choose between options C and D. But perhaps the problem is considering that the transition through m_i=0 is more probable. Alternatively, perhaps the probability is 1/2 for each possible m_i. But the options have 1/4, 1/3, 2/3. Hmm.

Alternatively, perhaps the problem is considering that the intermediate state must have m_i=0, but the calculation shows that the matrix element is zero. Therefore, this path is forbidden. Therefore, the problem's premise is wrong, but since the problem states that the decay is via two dipole transitions, there must be a valid path. Therefore, perhaps the correct answer is option C or D, but which one.

Alternatively, perhaps the problem is considering that the transition probabilities are 1/3 and 2/3 for the two possible paths: one through m_i=0 and another through m_i=1 and m_i=-1. But how?

Alternatively, perhaps the probability is determined by the number of possible intermediate states. For example, if there are three possible intermediate states (m_i=±1,0), and each has an equal probability, then the total probability would be 1/3 for each. But the options have 1/4, 1/3, 2/3. If the transition through m_i=0 has a higher probability, perhaps 2/3, and the others 1/3 each. But this is speculative.

Alternatively, perhaps the problem is considering that the transition from |3,0,0⟩ to |2,1,0⟩ is forbidden (as the matrix element is zero), but the transitions to |2,1,±1⟩ are allowed. Therefore, the only possible paths are through m_i=1 and m_i=-1. But the problem states that the decay is via two dipole transitions, so these two paths are possible. The probability would then be the sum of the probabilities for these two paths. If each has the same probability, the total would be 1/2 each, but the options don't have 1/2. Alternatively, if the path through m_i=1 and m_i=-1 each have probabilities 1/4, and the path through m_i=0 is zero, then the total probability would be 1/2. But the options don't have 1/2. Alternatively, the problem is considering that the transition through m_i=0 is not allowed, so the only possible paths are through m_i=1 and m_i=-1, each with probability 1/2. But again, the options don't have 1/2.

Alternatively, perhaps the transition probabilities are determined by the square of the Clebsch-Gordan coefficients. For the first transition, from |3,0,0⟩ to |2,1,m_i⟩, the Clebsch-Gordan coefficient would be non-zero only for certain m_i. For example, the Clebsch-Gordan coefficient for l=3, m=0 to l=2, m_i would be non-zero only for m_i=0, but earlier calculation suggested that the matrix element is zero. Therefore, this approach is not helpful.

Given the time I've spent and the complexity of the problem, I think the correct answer is option C: |3,0,0⟩→|2,1,0⟩→|1,0,0⟩ with probability 1/3. But I'm not entirely sure. Alternatively, perhaps the correct answer is option D with probability 2/3. But I need to make a decision here.
</think>