Okay, so I need to figure out the momentum a photon must have to cause a diatomic molecule to switch from its fundamental state to the next lowest energy state. The molecule is modeled using the quantum rotation-vibration model, which is also known as the non-rigid rotor model. The given parameters are the masses of the two atoms, X and Y, their bond length R, and the angular frequency w. The choices are A to D, each with a different momentum value. 

First, I remember that in such models, the energy levels of a diatomic molecule are quantized. The fundamental state is the lowest vibrational energy level, but when a photon is absorbed, it can cause a transition to the next vibrational state. Wait, but the question says "switch to the next state with the lowest possible energy." Hmm, the next state would be the first excited vibrational state, which is the first energy level above the fundamental. 

Wait, but in rotational transitions, the photon's energy corresponds to a change in the rotational state. However, the question mentions the molecule is in its fundamental state, which is the lowest vibrational state. So maybe the transition here is between vibrational states, not rotational. Because the fundamental state is the lowest vibrational state, and the next state would be the first excited vibrational state. So the photon's energy would need to match the energy difference between these two vibrational levels. 

Alternatively, maybe the molecule is transitioning between rotational states. But the problem states "the next state with the lowest possible energy," which might imply a vibrational transition. However, the model mentioned is the rotation-vibration model, so both rotational and vibrational states are involved. Since the question is about the momentum of the absorbed photon, perhaps this is related to the rotational transition. Because the momentum would correspond to the photon's energy, which is related to the frequency. 

Wait, but the angular frequency w is given. Let me think. The energy levels in the quantum harmonic oscillator (for the vibrational part) are quantized as E_v = (v + 1/2) h w, where v is the vibrational quantum number. The transition from v=0 to v=1 would require a photon with energy E = h w. However, the momentum of the photon would relate to its energy, since the energy of a photon is E = h f = p c, where p is the momentum. So if the energy required is h w, then p = h w / c. But wait, let me check the units. 

Alternatively, the energy difference between the vibrational levels is h w. So the photon's energy must be h w, so the momentum would be E / c = h w / c. Let me compute that. 

But wait, let's check the numbers. The given w is 4e14 rad/s. Planck's constant h is approximately 6.626e-34 J·s. The speed of light c is 3e8 m/s. So p = (6.626e-34 J·s) * (4e14 rad/s) / (3e8 m/s). Let me compute that. 

First, multiply 6.626e-34 by 4e14: 6.626 *4 = 26.504, and 1e-34 *1e14 = 1e-20. So 26.504e-20 J·s. Then divide by 3e8 m/s: 26.504e-20 / 3e8 = (26.504 /3) *1e-28. 26.504 /3 is approximately 8.835. So 8.835e-28 N·s. Hmm, looking at the options, the closest is C: 1.4e-28, but that's smaller. Wait, that can't be right. Maybe I made a mistake here.

Wait a second, maybe the model here isn't the harmonic oscillator. The question mentions the non-rigid rotor model, which is a rotational model, not vibrational. Wait, maybe the question is about a rotational transition. Let me clarify. The question says "switch to the next state with the lowest possible energy." If the molecule is in the fundamental vibrational state (v=0), the next vibrational state would be v=1. But in the rotational model, the rotational states are quantized with energy E_rot = (J(J+1)ħ²)/(2I), where I is the moment of inertia. The transition between rotational states would require a photon with energy equal to the difference between two rotational levels. However, the question is about the momentum of the photon. 

Wait, but the problem states that the molecule is modeled using the quantum rotation-vibration model, so both vibrational and rotational states are considered. The fundamental state is the lowest vibrational state (v=0), and the next state would be either the next vibrational state (v=1) or a rotational transition within the same vibrational state. The question is ambiguous, but since it's asking about the momentum, which relates to the photon's energy, perhaps it's about the vibrational transition. Alternatively, maybe the rotational transition. 

Alternatively, perhaps the question is about the transition from the ground state to an excited state in terms of both vibration and rotation. But the problem states "next state with the lowest possible energy," which would be the next vibrational level. Because the vibrational energy increases with v. So going from v=0 to v=1 would require the lowest possible energy jump. 

But earlier calculation gave p ≈ 8.8e-28 N·s, but the options don't have that. Wait, maybe I'm confusing vibrational and rotational transitions. Let me think again. 

If the transition is vibrational, the photon's energy would be h w, so p = h w / c. Let me compute that again. 

h = 6.626e-34 J·s, w =4e14 rad/s. So h w = 6.626e-34 *4e14 = 2.6504e-19 J. Then p = E / c = 2.6504e-19 / 3e8 = 8.835e-28 kg·m/s. Which is approximately 8.8e-28 N·s. But the options don't have that. The closest is C: 1.4e-28, which is an order of magnitude smaller. So maybe this approach is wrong. 

Alternatively, perhaps the transition is rotational. The energy difference between two rotational states J and J+1 is ΔE = h ( (J+1)(J+2) - J(J+1) ) * ħ²/(2I). Wait, no. The rotational energy levels are E_J = J(J+1)ħ²/(2I). The difference between E_{J+1} and E_J is [ (J+1)(J+2) - J(J+1) ] ħ²/(2I) ) = [ (J² +3J +2) - (J² +J) ] ħ²/(2I) ) = (2J +2) ħ²/(2I) ) = (J+1) ħ²/I. 

So the energy required for a rotational transition from J to J+1 is ΔE = (J+1) ħ²/I. But without knowing the initial J, this seems tricky. However, the question states "the next state with the lowest possible energy," which might imply the transition from J=0 to J=1. 

So if the molecule is in the fundamental vibrational state (v=0), and in a rotational state J=0, then the next state would be either v=1 (vibrational transition) or J=1 (rotational transition). But the question is about the momentum of the photon. 

Alternatively, perhaps the question is considering the vibrational transition. Wait, but the problem mentions the quantum rotation-vibration model, which includes both. However, the question is asking about the momentum of the photon required to switch to the next state with the lowest possible energy. The next state could be either a higher vibrational state or a higher rotational state. But since the vibrational energy is much larger than rotational energy (for typical diatomic molecules), the next lowest energy state after the fundamental vibrational state (v=0) would be the first rotational state in the same vibrational level. Wait, but the question says "the next state with the lowest possible energy," which would be the next possible state with the lowest energy. Wait, but the vibrational states have lower energy than rotational states? Wait, no. The vibrational energy levels are much lower than rotational energy levels. For example, the vibrational energy is on the order of 1e-20 J, while rotational energy is on the order of 1e-34 J. Wait, no, that's not right. Wait, the rotational energy levels are much smaller than vibrational. Wait, the rotational energy levels are quantized as E_J = ħ² J (J+1) / (2 I). The vibrational energy levels are E_v = (v + 1/2) h w. 

So for example, the first vibrational level (v=1) has energy h w, while the first rotational level (J=1) has energy E_J = ħ² (1)(2)/(2 I) = ħ² / I. Let me compute the ratio. Let's take I as the moment of inertia, which is I = μ R², where μ is the reduced mass. 

Given the masses of X (20 amu) and Y (2 amu), the reduced mass μ = (m_X m_Y)/(m_X + m_Y) = (20*2)/(20+2) = 40/22 ≈ 1.818 amu. Wait, but 1 amu is approximately 1.66e-27 kg. So μ ≈ 1.818 *1.66e-27 ≈ 3.02e-27 kg. Then I = μ R². R is 2 angstroms, which is 2e-10 m. So I = 3.02e-27 kg * (2e-10 m)^2 = 3.02e-27 *4e-20 = 1.208e-46 kg·m². 

Then E_J for J=1 is ħ² J (J+1) / (2 I). ħ = h/(2π) ≈ 1.05e-34 J·s. So ħ² = (1.05e-34)^2 ≈ 1.1e-68 J²·s². Then E_J = 1.1e-68 * 1*2 / (2 *1.208e-46) ) = (2.2e-68) / (2.416e-46) ≈ 9.1e-23 J. 

On the other hand, the vibrational energy h w is 6.626e-34 *4e14 ≈ 2.65e-19 J. So the vibrational energy is much higher than the rotational energy. Therefore, the next state with the lowest possible energy after the fundamental vibrational state (v=0) would be the first rotational state (J=1) in the same vibrational level. Because the rotational energy is lower than the next vibrational state. 

So if the molecule is in the v=0 state, the next state with the lowest possible energy is J=1, v=0. Therefore, the photon must provide the energy difference between J=0 and J=1 in the same vibrational level. 

So the energy difference ΔE = E_J=1 - E_J=0. 

E_J=0 = ħ² *0*(0+1)/(2 I) = 0. So E_J=0 = 0. 

E_J=1 = ħ² *1*2/(2 I) = ħ² / I. 

So ΔE = ħ² / I. 

Then the momentum p = ΔE / c. 

So let's compute ΔE first. 

We already have I ≈1.208e-46 kg·m². ħ² = (1.05e-34)^2 ≈1.1e-68 J²·s². So ħ² / I ≈1.1e-68 /1.208e-46 ≈9.1e-23 J. 

Then p = E / c =9.1e-23 /3e8 ≈3e-31 kg·m/s. Wait, that's even smaller than the options. The options are all around 1e-27. So this approach must be wrong. 

Wait, maybe the question is about the vibrational transition. Let's go back. The fundamental state is v=0. The next state would be v=1. The energy difference is h w. So p = h w / c. Earlier calculation gave p≈8.8e-28 N·s. But the options don't have that. So perhaps the question is not about a vibrational transition but a rotational transition. But then the energy required would be the rotational energy difference. Wait, but if the molecule is in the fundamental vibrational state (v=0), then the next state with the lowest possible energy would be a rotational state (J=1) in the same vibrational level, which is much lower in energy than the next vibrational state. So that would require a photon with energy 9.1e-23 J, leading to p≈3e-31 kg·m/s, which is not an option. 

Alternatively, perhaps the question is about the transition from the ground state (v=0, J=0) to the first excited state (v=1, J=0). That would be a vibrational transition. The energy difference is h w. So p = h w / c ≈8.8e-28 N·s. But that's not an option. 

Wait, the options are in the range of 1e-27 to 2e-27. Hmm. Maybe I missed a factor. Let's recalculate. 

h = 6.626e-34 J·s. w =4e14 rad/s. So h w =6.626e-34 *4e14 =2.6504e-19 J. Then p = E/c =2.6504e-19 /3e8 =8.835e-28 kg·m/s. But the options are 1.9e-28, 1.1e-27, etc. 

Wait, maybe the question is about the rotational transition within the same vibrational level. But earlier calculation shows that the energy is much smaller. 

Alternatively, perhaps the energy required is the difference between two rotational states in the same vibrational level. For example, from J=0 to J=1. Then ΔE = E_J=1 - E_J=0 = ħ²/(I) * (1*2 - 0*1)/2? Wait, no. The energy levels are E_J = ħ² J (J+1)/(2 I). So E_J=1 = ħ²*2/(2I) = ħ²/I. E_J=0 =0. So ΔE = ħ²/I. 

We computed that as 9.1e-23 J, leading to p≈3e-31. Not matching. 

Alternatively, perhaps the question is about the rotational transition of the molecule, but considering the reduced mass and bond length. Wait, maybe the problem is using the reduced mass in the angular frequency. Let me check. The angular frequency w in the harmonic oscillator model is given by w = sqrt(k/μ), where k is the force constant. But here, the problem gives us w, so perhaps we don't need to compute it. 

Wait, but the problem gives us the angular frequency w. So if the photon's energy is h w, then p = h w / c. The calculation gives 8.8e-28, but the options don't have that. So maybe the question is about the rotational transition, not the vibrational. Let's try that again. 

If the photon's energy is the rotational energy difference between J=0 and J=1, then ΔE = ħ² (1*2)/(2 I) = ħ² / I. 

We have ħ=1.05e-34 J·s. I= μ R². μ is the reduced mass. 

Masses: X is 20 amu, Y is 2 amu. So μ = (20 *2)/(20 +2) =40/22 ≈1.818 amu. 1 amu =1.66e-27 kg. So μ ≈1.818 *1.66e-27 ≈3.02e-27 kg. R=2 angstroms=2e-10 m. 

I=3.02e-27 kg * (2e-10 m)^2 =3.02e-27 *4e-20 =1.208e-46 kg·m². 

Then ħ²/I = (1.05e-34)^2 /1.208e-46 ≈1.1e-68 /1.208e-46 ≈9.1e-23 J. 

p=9.1e-23 /3e8 ≈3e-31 kg·m/s. Which is not in the options. So this approach is wrong. 

Hmm. Alternatively, maybe the question is about the transition between vibrational states, but using the rotational constant. Wait, the rotational constant B is given by ħ²/(2 I c). Then the rotational energy levels are E_J = B J (J+1). But the vibrational energy levels are E_v = h ν (v + 1/2). The transition from v=0 to v=1 requires h ν. But the question is about the momentum of the photon, which would be h ν / c. 

But ν here is given as w/(2π). Because in the harmonic oscillator model, w = 2π ν. So ν = w/(2π). Therefore, h ν = h w/(2π). Then p = h ν / c = h w/(2π c). 

Given w=4e14 rad/s. So h=6.626e-34, 2π=6.283, c=3e8. 

So p=6.626e-34 *4e14 / (6.283 *3e8) 

Let me compute numerator:6.626e-34 *4e14 =2.6504e-19. 

Denominator:6.283*3e8=1.8849e9. 

p=2.6504e-19 /1.8849e9 ≈1.407e-28 kg·m/s. 

Ah, that's option C:1.4e-28. 

So this would make sense. Therefore, the correct answer is C. 

Wait, but earlier I thought the question was about the vibrational transition. But here, using the rotational constant approach gives the correct answer. Let me confirm. 

The question states that the molecule is modeled using the quantum rotation-vibration model. The photon's momentum would correspond to the energy difference between the vibrational states. But the energy difference between vibrational states is h w. However, using the rotational approach, perhaps the photon's energy is related to the rotational constant B. Wait, but B is in terms of energy, which is h ν_rot, where ν_rot is the rotational frequency. 

Alternatively, perhaps the photon's energy corresponds to the rotational energy difference. But in the previous calculation, using the rotational energy difference between J=0 and J=1 gives a momentum that's much smaller. 

Wait, but the calculation using ν = w/(2π) gives p=1.4e-28, which matches option C. And this approach uses the angular frequency w given in the problem. So perhaps the correct approach is to use the relation p = h ν / c, where ν is the vibrational frequency. Since w = 2π ν, ν = w/(2π). Therefore, p = h w/(2π c). 

So with h=6.626e-34, w=4e14, c=3e8, 

p =6.626e-34 *4e14 / (6.283*3e8) 

= (6.626*4)/(6.283*3) *1e-34+14-8 

= (26.504)/(18.849) *1e-28 

≈1.407e-28, which is option C. 

Therefore, the correct answer is C. 

But why is the angular frequency given? Because in the quantum model, the energy levels are quantized, and the transition between vibrational states involves absorbing a photon with energy h ν, where ν is the vibrational frequency. Since w is the angular frequency, ν = w/(2π), so h ν = h w/(2π). The momentum is then h ν /c = h w/(2π c). 

Therefore, the correct answer is C:1.4e-28 N·s. 

I think this is the right approach. The key was recognizing that the photon's energy corresponds to the vibrational transition, calculated using the given angular frequency. The other approach considering rotational transitions didn't match the answer, so this must be the correct path.
</think>